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Submitted to the Annals of Applied Probability

ON THE CONDITIONAL DISTRIBUTIONS AND THE EFFICIENT SIMULATIONS OF EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS By Jingchen Liu∗ , and Gongjun Xu Columbia University and University of Minnesota In this paper, we consider the extreme behavior of a Gaussian random field f (t) living on a compact set T .RIn particular, we are interested in tail events associated with the integral T ef (t) dt. We construct a (nonGaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field f (given that R f (t) e dt exceeds a large value) in total variation. Based on this approxiT R mation, we show that the tail event of T ef (t) dt is asymptotically equivalent to the tail event of supT γ(t) where γ(t) is a Gaussian process and it is an affine function of f (t) and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of log b to compute R the probability P ( T ef (t) dt > b) with a prescribed relative accuracy.

1. Introduction. Consider a Gaussian random field {f (t) : t ∈ T } living on a d-dimensional domain T ⊂ Rd with zero mean and unit variance, that is, for every finite subset {t1 , ..., tn } ⊂ T , (f (t1 ), ..., f (tn )) is a mean zero multivariate Gaussian random vector. Let µ(t) be a (deterministic) function and σ ∈ (0, ∞) be a scale factor. Define Z I(T ) , eσf (t)+µ(t) dt. (1.1) T

In this paper, we develop a precise asymptotic description of the conditional distribution of f given that I(T ) exceeds a large value b, that is, P (·|I(T ) > b). In particular, we provide a tractable total variation approximation (in the sample path space) for such conditional random fields based on a change of measure technique. In addition to the asymptotic descriptions, we design efficient Monte Carlo estimators that run in polynomial time of log b for computing the tail probabilities Z  v(b) = P (I(T ) > b) = P eσf (t)+µ(t) dt > b (1.2) T

with a prescribed relative accuracy. 1.1. The literature. In the probability literature, the extreme behaviors of Gaussian random fields have been studied extensively. The results range from general bounds to sharp asymptotic approximations. An incomplete list of works includes ∗ This research is supported by Institute of Education Sciences, through Grant R305D100017, NSF CMMI-1069064, and NSF SES-1323977. AMS 2000 subject classifications: Primary 60G15, 65C05 Keywords and phrases: Gaussian process, change of measure, efficient simulation

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[33, 35, 43, 48, 19, 53, 14, 37, 50]. A few lines of investigations on the supremum norm are given as follows. Assuming locally stationary structure, the double-sum method ([46]) provides the exact asymptotic approximation of supT f (t) over a compact set T , which is allowed to grow as the threshold tends to infinity. For almost surely at least twice differentiable fields, [1, 51, 4] derive the analytic form of the expected Euler-Poincar´e Characteristics of the excursion set (χ(Ab )) which serves as a good approximation of the tail probability of the supremum. The tube method ([49]) takes advantage of the Karhune-Lo`eve expansion and Weyl’s formula. A recent related work along this line is given by [45]. The Rice method ([10, 11, 12]) provides an implicit description of supT f (t). Change of measure based rare-event simulations are studied in [2]. The discussions also go beyond the Gaussian fields. For instance, [34] discusses the situations of Gaussian process with random variances. See also [3] for discussions on non-Gaussian cases. The distribution of I(T ) is studied in the literature when f (t) is a Brownian motion ([55, 27]). Recently, [40, 41] derive the asymptotic approximations of P (I(T ) > b) as b → ∞ for three times differentiable and homogeneous Gaussian random fields. Besides the tail probability approximations, rigorous analysis of the conditional distributions of stochastic processes given the occurrence of rare events is also an important topic. In the classic large deviations analysis for light-tailed stochastic systems, the sample path(s) that admits the highest probability (the most likely sample path) under the conditional distribution given the occurrence of a rare event is central to the entire analysis in terms of determining the appropriate exponential change of measure, developing approximations of the tail probabilities, and designing efficient simulation algorithms (see, for instance, standard textbook [28]). For heavy-tailed systems, the conditional distributions and the most likely paths, which typically admit the so-called “one-big-jump” principle, are also intensively studied ([7, 8, 18]). These results not only provide intuitive and qualitative descriptions of the conditional distribution but also shed light on the design of rare-event simulation algorithms ([16, 17, 18]) – the best importance sampling estimator of the rare-event probability uses a change of measure corresponding to the interesting conditional distribution. In addition, the conditional distribution (or the conditional expectations) is also of practical interest. For instance, in risk management, the conditional expected loss given some rare/disastrous event is an important risk measure and stress test. In the literature of Gaussian random fields, the exact Slepian models (conditional field given a local maximum or level crossing of f (t)) are studied intensively for twice differentiable fields. For instance, [36] gives the Slepian model conditioning on an upcrossing of level u at time zero. [38] treats conditioning on a local maximum of height u at time zero. The first rigorous treatment of Slepian models for nonstationary processes is given by [39]. [32] extends the results of [36] for level crossings to the general non-stationary case. This work is followed up by [30]. In the later analysis, we will set an asymptotic equivalence between the conditional distribution given {I(T > b} and that given the high excursion of the supremem of f . The later can be characterized by the Slepain model.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

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1.2. Contributions. In this paper, we pursue along this line for the extreme behaviors of Gaussian processes and begin to describe the conditional distribution of f given the occurrence of the event {I(T ) > b}. In particular, we provide both quantitative and qualitative descriptions of this conditional distribution. Furthermore, from a computational point of view, we construct a Monte Carlo estimator that takes a polynomial computational cost (in log b) to estimate v(b) for a prescribed relative accuracy. Central to the analysis is the construction of a change of measure on the space C(T ) (continuous functions living on T ). The application of the change of measure ideas is common in the study of large deviations analysis for the light-tailed stochastic systems. However, it is not at all standard in the study of Gaussian random fields. The proposed change of measure is not of a classical exponential-tilting form. This measure has several features that are appealing both theoretically and computationally. First, we show that the change of measure denoted by Q approximates the conditional measure P (·|I(T ) > b) in total variation as b → ∞. Second, the measure Q is analytically tractable in the sense that the distribution of f under Q has a closed form representation and the Randon-Nikodym derivative dQ/dP takes the form of a d-dimensional integral. This tractability property has useful consequences. From a methodological point of view, the measure Q provides a very precise description of the mechanism that drives the rare event {I(T ) > b}. This result allows to directly use the intuitive mechanism to provide functional probabilistic descriptions that emphasize the most important elements that are present in the interesting rare events. More technically, the analytical computations associated with the measure Q are easy (compared to the conditional measure) and the expectation E Q [·] is theoretically much more tractable than E[·|I(T ) > b]. Based on this result, we show that the tail event {I(T ) > b} is asymptotically equivalent to the tail event of supT γ(t) where γ(t) is an affine function of f (t) and its derivative field ∂ 2 f (t) and γ(t) implicitly depends on b. Thus, one can further characterize the conditional measure by means of the results on the Slepian model mentioned earlier. Another contribution of this paper lies in the numerical evaluation of v(b). The importance sampling algorithm associated with the proposed change of measure yields an efficient estimator for computing v(b). An important issue concerns the implementation of the Monte Carlo method. The processes considered in this paper are continuous while computers can only represent discrete objects. Inevitably, we will introduce a suitable discretization scheme and use discrete (random) objects to approximate the continuous processes. A naturally raised issue lies in the control of the approximation error relative to the probability v(b). We will perform careful analysis and report the overall computational complexity of the proposed Monte Carlo estimators. A key requirement of the current analysis is the twice differentiability of f . Our change of measure is written explicitly in the form of f , ∂f , and ∂ 2 f . A very interesting future study would be developing parallel results for non-differentiable fields. The technical challenges are two-fold. First, there is lack of asymptotic analysis for the exponential integral of general non-differentiable fields. To the author’s best knowledge, the behavior of I(T ) for non-differentiable processes is investigated

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only when f is a Brownian motion whose techniques cannot be extend to general cases [55, 27]. In addition, there is a lack of descriptive tools (such as derivatives and the Palm model) for non-differentiable processes. This also leads to difficulties in describing the Slepian model for level crossing. To the author’s best knowledge, analytic description of Slepian models for excursion of supT f (t) are available only for twice differentiable fields. Despite of the smoothness limitation, the current analysis has important applications the details of which will be presented in the following section. The rest of this paper is organized as follows. Two applications of this work are given in Section 2. In Section 3, we present the main results including the change of measure, the approximation of P (·|I(T ) > b), and the efficient Monte Carlo estimator of v(b). Proofs of the theorems are given in Sections 4-7. A supplemental material is provided including all the supporting lemmas. 2. Applications. The integral of exponential functions of Gaussian random fields plays an important role in many probability models. We present two such models for which the conditional distribution is of interest and the underlying random fields are differentiable. 2.1. Spatial point process. In spatial point process modeling, let λ(t) be the intensity of a Poisson point process on T , denoted by {N (A) : A ⊂ T }. In order to build in spatial dependence structure and to account for overdispersion, the log-intensity is typically modeled asR a Gaussian random field, that is, log λ(t) = f (t)+µ(t) and then E[N (A)|λ(·)] = A ef (t)+µ(t) dt, where µ(t) is the mean function and f (t) is a zero-mean Gaussian process. For instance, [21] considers the time series setting in which T is a one dimensional interval, µ(t) is modeled as the observed covariate process and f (t) is an autoregressive process; see [24, 20, 56, 22, 23] for more examples in high dimensional domains. For illustration purpose, we consider a very concrete case that the point process N (·) represents the spatial distribution of asthma cases over a geographical domain T . The latent intensity λ(t) (or equivalently f (t)) represents the unobserved (and appropriately transformed) metric of the pollution severity at location t. The mean function can be written as a linear combination of the observed covariates that may affect the pollution level, that is, µ(t) = β > x(t) is treated as a deterministic function. It is well understood that λ(t) is a smooth function of the spatial parameter t at the macro level as the atmosphere mixes well; see, e.g., [47]. One natural question in epidemiology is: upon observing an unusually high number of asthma cases what is their geographical distribution, that is, the conditional distribution of the point process N (·) given that N (T ) > b for some large b. First of all, [41] shows that P (N (T ) > b) ∼ P (I(T ) > b) as b → ∞. Following the same derivations, it is not difficult to establish the following convergence P (·|N (T ) > b) − P (·|I(T ) > b) → 0

in total variation as b → ∞.

The total count N (T ) is a Poisson random variable with mean I(T ). Intuitively speaking, the tail of the integral is similar to a lognormal random variable and

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

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thus is heavy-tailed. Its overshoot over level b is Op (b/ log b).√However, a Poisson random variable with mean I(T ) ∼ b has standard deviation b  b/ log b. Thus, a large number of N (T ) is mainly caused by a large value of I(T ). The symmetric difference of the two sets {N (T ) > b} and {I(T ) > b} vanishes and the probability law of the entire system conditional upon observing that N (T ) > b is asymptotically the same as that given I(T ) > b. Therefore, the conditional distribution of N (·) given N (T ) > b is asymptotically another doubly-stochastic Poisson process whose intensity is λ(t) = eµ(t)+f (t) where f (t) follows the conditional distribution of P (f ∈ · |I(T ) > b). Based on the main results presented momentarily, a qualitative description of the conditional distribution of N (·) is as follows. Given N (T ) > b, the overshoot is of order Op (b/ log b) that is N (T ) = b + Op (b/ log b). The locations of the points are i.i.d. samples approximately following a d-dimensional multivariate Gaussian distribution with mean τ ∈ T and variance Σ/ log b where Σ depends on the spectral moments of f . The distribution of τ is uniform over T if µ(t) is a constant; if µ(t) is not constant, τ has a density l(t) presented in (3.13). 2.2. Financial application. The exponential integral can be considered as a generalization of the sum of dependent lognormal random variables that has been studied intensively from different aspects in the applied probability literature (see [26, 6, 13, 31, 25, 9, 29]). In portfolio risk analysis, consider a portfolio of n assets S1 ,...,Sn . The asset prices are usually modeled as log-normal random variables. That is, let Xi = log Si and (X1 , ...,PXn ) follows a multivariate normal distribun tion. The total portfolio value S = i=1 wi Si is the weighted sum of dependent log-normal random variables. An important question is the behavior of this sum when the portfolio size becomes large and the assets are highly correlated. One may employ a latent space approach used in the literature of social network. More specifically, we construct a Gaussian process {f (t) : t ∈ T } and map each asset i to a latent variable ti ∈ T , that is, log Si = f (ti ). Then, the log-asset prices fall into a subset of the continuous Gaussian process. Furthermore, we construct a (deterministic) function w(t) so i ) = wi . Then, the unit share value of the portfolio is P Pthat w(t 1 1 f (ti ) w S = w(t )e . See [15, 41] for detailed discussions on the random i i i n n field representations of large portfolios. In the asymptotic regime that n → ∞ and the correlations among the asset prices become close to one, the subset {ti } becomes dense in T . Ultimately, we obtain the limit Z n 1X wi Si → w(t)ef (t) h(t)dt n i=1 T where h(t) is the limiting spatial distribution of {ti } inR T . Let µ(t) = log w(t) + log h(t). Then the (limiting) unit share price is I(T ) = T ef (t)+µ(t) dt. The current study provides an asymptotic description of the performance of each asset given the occurrence of the tail event I(T ) > b. This is of great importance in the study of the so-called stress test that evaluates the impact of shocks on and the vulnerability of a system. For instance, consider that another investor holds

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a different portfolio that has a substantial overlap with the current one or it has exactly the same collection of assets but with different weights. Thus, this second portfolio corresponds to a different mean function µ0 (t). Stress test investigates the performance of this second portfolio conditional on that a rare event has occurred to the first, that is, Z Z  0 P ef (t)+µ (t) dt ∈ · ef (t)+µ(t) dt > b . T

T

To characterize the aboveR distribution, we need a precise description of the conditional measure P (f ∈ · | T ef (t)+µ(t) dt > b). 3. Main results. 3.1. Problem setting and notations. Throughout the discussion, we consider a homogeneous Gaussian random field {f (t) : t ∈ T } living on a domain T ⊂ Rd . Let the covariance function be C(t − s) = Cov(f (t), f (s)). We impose the following assumptions: f is stationary with Ef (t) = 0 and Ef 2 (t) = 1. f is almost surely at least two times differentiable with respect to t. T is a d-dimensional compact set of Rd with piecewise smooth boundary. The Hessian matrix of C(t) at the origin is standardized to be −I, where I is the d × d identity matrix. In addition, C(t) has the following expansion when t is close to 0 1 (3.1) C(t) = 1 − t> t + C4 (t) + RC (t), 2 P 1 4 4+δ0 ) for some where C4 (t) = 24 ijkl ∂ijkl C(0)ti tj tk tl and RC (t) = O(|t| δ0 > 0. C5 For each t ∈ Rd , the function C(λt) is a non-increasing function of λ ∈ R+ . C6 The mean function µ(t) falls into either of the two cases: C1 C2 C3 C4

(a) µ(t) ≡ 0; (b) the maximum of µ(t) is unique and is attained in the interior of T and µ(t + ε) − µ(t) = ε> ∂µ(t) + ε> ∆µ(t)ε + O(|ε|2+δ0 ) as ε → 0. We define a set of notations constantly used in the later development and provide some basic calculations. Let Pb∗ be the conditional measure given {I(T ) > b}, that is, Pb∗ (f (·) ∈ A) = P (f (·) ∈ A|I(T ) > b). Let “∂” denote the gradient and “∆” denote the Hessian matrix with respect to t. The notation “∂ 2 ” is used to denote the vector of second derivatives. The difference between ∂ 2 f (t) and ∆f (t) is that ∆f (t) is a d × d symmetric matrix whose diagonal and upper triangle consist of elements of ∂ 2 f (t). Furthermore, let ∂j f (t)

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be the partial derivative with respect to the j-th element of t. Lastly, we define the following vectors = −(∂1 C(t), ..., ∂d C(t)), (3.2)   2 2 µ2 (t) = ∂ii C(t), i = 1, ..., d; ∂ij C(t), i = 1, ..., d − 1, j = i + 1, ..., d ,

µ1 (t)

µ> 02

= µ20 = µ2 (0).


Suppose 0 ∈ T . It is well known that f (0), ∂ 2 f (0), ∂f (0), f (t) is a multivariate Gaussian random vector with mean zero and covariance matrix (c.f. Chapter 5.5 of [4])   1 µ20 0 C(t)   µ02 µ22 0 µ> 2 (t)   >  0 0 I µ1 (t)  C(t) µ2 (t) µ1 (t) 1 where the matrix µ22 is a d(d + 1)/2-dimensional positive definite matrix and contains the 4th order spectral moments arranged in an appropriate order according to the order of elements in ∂ 2 f (0). Let h(x, y, z) be the density function of (f (t), ∂f (t), ∂ 2 f (t)) evaluated at (x, y, z). Then, simple calculations yield that i h −1 1 (x−µ20 µ22 z)2 z +z > µ−1 − 12 y > y+ det(Γ)− 2 −1 22 1−µ20 µ22 µ02 , (3.3) e h(x, y, z) = (d+1)(d+2) 4 (2π) where det(·) is the determinant of a matrix and   1 µ20 Γ= . µ02 µ22 We define u as a function of b such that   d2 d 2π u− 2 eσu = b. σ

(3.4)

Note that the above equation generally has two solutions, one is approximately σ −1 log b and the other is close to zero as b → ∞. We choose u to be the one close to σ −1 log b. For µ(t) and σ appearing in (1.1), we define µσ (t) = µ(t)/σ,

ut = u − µσ (t).

(3.5)

Approximately, ut is the level that f (t) needs to reach so that I(T ) > b. Furthermore, we need the following spatially varying set:  At = f (·) ∈ C(T ) : αt > ut − ηu−1 , (3.6) t where η > 0 is a tuning parameter that will be eventually sent to zero as b → ∞ and αt is a function of f (t) and its derivative fields taking the form of αt

=

f (t) +

|∂f (t)|2 1> f¯t00 Bt + + . 2ut 2σut ut

(3.7)

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In the above equation (3.7), f¯t00 is defined as (with the notations in (3.2)) f¯t00 = ∂ 2 f (t) − ut µ02 .

(3.8)

The term Bt is a deterministic function depending only on C(t), µ(t), and σ, Bt

=

1 X 4 1> ∂ 2 µσ (t) + d × µσ (t) + 2 ∂ C(0) + |∂µσ (t)|2 , 2σ 8σ i iiii

(3.9)

>

where d is the dimension of T , and 1 = (1, ..., 1, 0, ..., 0 ) . Note that αt ≈ f (t). | {z } | {z } d

d(d−1)/2

Thus, on the set At , f (t) ≈ αt > ut − O(u−1 ). Together with the fact that E[∂ 2 f (t)|f (t) = ut ] = ut µ02 , f¯t00 is the standardized second derivative of f given that f (t) = ut . In Section 3.2, we will show that the event {I(T ) > b} is approximately ∪t∈T At . For notational convenience, we write au = O(bu ) if there exists a constant c > 0 independent of everything such that au ≤ cbu for all u > 1, and au = o(bu ) if au /bu → 0 as u → ∞ and the convergence is uniform in other quantities. We write au = Θ(bu ) if au = O(bu ) and bu = O(au ). In addition, we write au ∼ bu if au /bu → 1 as u → ∞. Remark 1. Condition C1 assumes unit variance. R We treat the standard deviation σ as an additional parameter and consider eµ(t)+σf (t) dt. Condition C2 implies that C(t) is at least 4 times differentiable and the first and third derivatives at the origin are all zero. Differentiability is a crucial assumption in this analysis. Condition C3 restricts the results to finite horizon. Condition C4 assumes the Hessian matrix is standardized to be −I, which is to simplify notations. For any Gaussian process g(t) with covariance function Cg (t) and ∆Cg (0) = −Σ and det(Σ) > 0, identity Hessian matrix can be obtained by an affine transformation by letting g(t) = f (Σ1/2 t) and Z Z −1/2 s)+σf (s) eµ(t)+σg(t) dt = det(Σ−1/2 ) eµ(Σ ds. {s:Σ−1/2 s∈T }

T

Condition C5 is imposed for technical reasons so that we are able to localize the integration. For condition C6, we assume that µ(t) either is a constant or attains its global maximum at one place. If µ(t) has multiple (finitely many) maxima, the techniques developed in this paper still apply, but the derivations will be more tedious. Therefore, we stick to the uni-mode case. Remark 2. The setting in (1.2) incorporates the case in which the integral is with respect to other measures with smooth densities. Then, if ν(dt) = κ(t)dt, we will have that Z Z µ(t)+σf (t) e ν(dt) = eµ(t)+log κ(t)+σf (t) dt, A

A

which shows that the density can be absorbed by the mean function.

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3.2. Approximation of the conditional distribution. In this subsection, we propose a change of measure Q on the sample path space C(T ) that approximates Pb∗ in total variation. Let P be the original measure. The measure Q is defined such that P and Q are mutually absolutely continuous. We define the measure Q under two different scenarios: µ(t) is not a constant and µ(t) ≡ 0. Note that the measure Q obviously will depend on b. To simplify the notations, we omit the index b in Q whenever there is no ambiguity. The measure Q takes a mixture form of three measures, which are weighted by (1 − ρ1 − ρ2 ), ρ1 , and ρ2 respectively (a natural constraint is that ρ1 , ρ2 , and 1 − ρ1 − ρ2 ∈ [0, 1]). We define Q through the Radon–Nikodym derivative Z Z Z LR2 (t) dQ = (1−ρ1 −ρ2 ) l(t)·LR(t)dt+ρ1 dt, (3.10) l(t)·LR1 (t)dt+ρ2 dP mes(T ) T T T where ρ1 , ρ2 will be eventually sent to 0 as b goes to infinity at the rate (log log b)−1 , mes(T ) is the Lebesgue measure of T , and
h0,t f (t), ∂f (t), ∂ 2 f (t) LR(t) = , h (f (t), ∂f (t), ∂ 2 f (t))
h1,t f (t), ∂f (t), ∂ 2 f (t) LR1 (t) = , h (f (t), ∂f (t), ∂ 2 f (t)) LR2 (t)

=

2 1 √1 e− 2 (f (t)−ut ) 2π 1 2 √1 e− 2 f (t) 2π

.

(3.11)


The density h f (t), ∂f (t), ∂ 2 f (t) is defined in (3.3), l(t) is a density function on T , h0,t and h1,t are two density functions. Before presenting the specific forms of l(t), h0,t , and h1,t , we would like to provide an intuitive explanation of dQ/dP from a simulation point of view. One can generate f (t) under the measure Q via the following steps: 1. Generate ı ∼Bernoulli(ρ2 ). 2. If ı = 1, then (a) Generate τ uniformly from the index set T , i.e., τ ∼ U nif (T ). (b) Given the realized τ , generate f (τ ) ∼ N (uτ , 1). (c) Given (τ, f (τ )), simulate {f (t) : t 6= τ } from the original conditional distribution under P . 3. If ı = 0 (a) Simulate a random variable τ following the density function l(t). (b) Given the realized τ , simulate f (τ ) = x, ∂f (τ ) = y, ∂ 2 f (τ ) = z from density function hall (x, y, z) =

1 − ρ1 − ρ2 ρ1 h0,τ (x, y, z) + h1,τ (x, y, z). 1 − ρ2 1 − ρ2

(3.12)

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(c) Given τ, f (τ ), ∂f (τ ), ∂ 2 f (τ ) , simulate {f (t) : t 6= τ } from the original conditional distribution under P . Thus, τ is a random index at which we twist the distribution of f and its derivatives. The likelihood ratio at a specific location τ is given by LR(τ ), LR1 (τ ), or LR2 (τ ) depending on the mixture component. The distribution of the rest of the field {f (t) : t 6= τ } given (f (τ ), ∂f (τ ), ∂ 2 f (τ )) is the same as that under P . It is not hard to verify that the above simulation procedure is consistent with the Radon-Nikodym derivative in (3.10). We now provide the specific forms of the functions defining Q. We first consider the situation when µ(t) 6= 0. By condition C6, µ(t) admits its unique maximum at t∗ = arg supt∈T µ(t) in the interior of T . Furthermore, the Hessian matrix ∆µσ (t∗ ) is negative definite. The function l(t) is a density on T such that for t ∈ T l(t) = (1 + o(1)) det(−∆µσ (t∗ ))1/2

 u d/2 t∗

2π

e

u t∗ 2

(t−t∗ )> ∆µσ (t∗ )(t−t∗ )

,

(3.13)

which is approximately a Gaussian density centered around t∗ . As l(t) is defined on a compact set t, the o(1) term goes to zero as b tends to infinity. It is introduced to correct for the integral of l(t) outside the region T that is exponentially small and does not affect the current analysis. The functions h0,t and h1,t are density functions on the vector space where (f (t), ∂f (t), ∂ 2 f (t)) lives on and they are defined as follows (we will explain the following complicated functions momentarily) 2

h0,t (f (t), ∂f (t), ∂ f (t))

2

h1,t (f (t), ∂f (t), ∂ f (t))

  1> f¯00 B −λut f (t)+ 2σut + u t −ut

|∂f (t)|2

t t × e− 2 = IAt × Hλ × ut × e   1/2 2  ¯00 2 −1/2 1 |µ20 µ−1 22 ft | ¯00 − µ22 1 µ f × exp − + , t 22 2 1 − µ20 µ−1 2σ 22 µ02

  1> f¯00 B λ1 ut f (t)+ 2σut + u t −ut

|∂f (t)|2

t t × e− 2 = IAct × Hλ1 × ut × e   1/2 2  ¯00 2 −1/2 µ22 1 1 |µ20 µ−1 00 22 ft | ¯ + µ22 ft − × exp − . 2 1 − µ20 µ−1 2σ 22 µ02 2

1> f¯00

t where I is the indicator function, At = {f (·) : f (t)+ |∂f2u(t)| + 2σutt + B ut > ut −η/ut } t 00 is defined as in (3.6), f¯t is defined as in (3.8), λ < 1 is positive and it will be sent to 1 as b goes to infinity, λ1 is a fixed positive constant (e.g., λ1 = 1), and the normalizing constants are defined as

Hλ = Hλ1 =

e−λη (1 − λ)d/2 λ d

(2π) 2

Z × R

eλ1 η (1 + λ1 )d/2 λ1 d

(2π) 2

d(d+1) 2

e

− 21

Z × R

d(d+1) 2

e



−1 |µ20 µ22 z|2 −1 1−µ20 µ22 µ02

− 21





−1/2

+ µ22

−1 |µ20 µ22 z|2 −1 1−µ20 µ22 µ02



z−

−1/2

+ µ22

1/2 µ22 1 2 2σ

z−



−1 dz

1/2 µ22 1 2 2σ



, (3.14) −1

dz

.

The constants Hλ and Hλ1 ensure that h0,t and h1,t are properly normalized densities.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

11

Understanding the measure Q. The measure Q is designed such that the distribution of f under the measure Q is approximately the conditional distribution of f given I(T ) > b. The two terms corresponding to the probabilities ρ1 and ρ2 are included to ensure the absolute continuity and to control the tail of the likelihood ratio. Thus, ρ1 and ρ2 will be sent to zero eventually. We now provide an explanation of the leading term corresponding to the probability 1 − ρ1 − ρ2 . To understand h0,t , we use the notation αt in (3.7) and rewrite the density function as  
1−λ |∂f (t)|2 h0,t f (t), ∂f (t), ∂ 2 f (t) ∝ IAt exp {−λut (αt − ut )} × exp − 2 2    −1 ¯00 2 −1/2 00 µ1/2 1 |µ20 µ22 ft | 22 1 ¯ , × exp − + µ22 ft − −1 2 1−µ µ µ 2σ 20 22

02

which factorizes into three pieces consisting of αt , ∂f (t), and f¯t00 respectively. We consider the change of variables from (f (t), ∂f (t), ∂ 2 f (t)) to (αt , ∂f (t), f¯t00 ). Then, under the distribution h0,t , the random vectors αt , ∂f (t), and f¯t00 are independent. Note that h0,t is defined on the set At = {αt > ut − ηu−1 t } where η will be send to zero eventually. Then, αt −ut is approximately an exponential random variable with rate λut ; ∂f (t) and f¯t00 are two independent Gaussian random vectors. The density h1,t has a similar interpretation. The only difference is that h1,t is defined on the set {αt − ut < −ηu−1 t } and ut − αt follows approximately an exponential distribution. For the last piece corresponding to ρ2 , the density is simply an exponential tilting of f (t). Under the dominating mixture component, to generate an f (t) from Q, a random index τ is first sampled from T following density l(t), then (f (τ ), ∂f (τ ), ∂ 2 f (τ )) is according to h0,τ . This implies that the large value of the integral R sampled eµ(t)+σf (t) dt is mostly caused by the fact that the field reaches a high level T at τ ; more precisely, ατ reaches a high level of uτ (with an exponential overshoot of rate λuτ ). Therefore, the random index τ localizes the position where the field αt goes very high. The distribution of τ given as in (3.13) is very concentrated around t∗ . This suggests that the maximum of αt (or f (t)) is attained within Op (u−1/2 ) distance from t∗ . We now consider the case where µ(t) ≡ 0. We choose l(t) to be the uniform distribution over set T and have that Z Z Z dQ LR(t) LR1 (t) LR2 (t) = (1 − ρ1 − ρ2 ) dt + ρ1 dt + ρ2 dt, (3.15) dP T mes(T ) T mes(T ) T mes(T ) where mes(·) is the Lebesgue measure. The following theorem states that Q is a good approximation of Pb∗ with appropriate choice of the tuning parameters. Theorem 3. Consider a Gaussian random field {f (t) : t ∈ T } living on a domain T satisfying conditions C1-6. If we choose the parameters defining the change of measure η = ρ1 = ρ2 = 1 − λ = (log log b)−1 , then, we have the following approximation lim sup |Q(A) − Pb∗ (A)| = 0 b→∞ A∈F

12

LIU AND XU

where F is the σ-field where the measures are defined. Remark 4. Theorem 3 is the central result of this paper. We present its detailed proof. The technical developments of other theorems are all based on that of Theorem 3. Therefore, we only layout their key steps and the major differences from that of Theorem 3. Remark 5. The measure Q in the limit of the above theorem obviously depends on the tuning parameters (η, ρ1 , ρ2 , and λ) and the level b. To simplify the notation, we omit the indices of those parameters when there is no ambiguity. Remark 6. The measure corresponding to the last mixture component in (3.10), LR2 (t) dt, has been employed by [41] to develop approximations for v(b). We emT mes(T ) phasize that the measure constructed in this paper is substantially different. In fact, the measure corresponding to LR2 (t) does not appear in the main proof. We included it to control the tail of the likelihood ratio in one lemma. R

To illustrate the application of the measure Q, we provide a further characterization of the conditional distribution Pb∗ by presenting another approximation result which is easier to understand at an intuitive level. Let γu (t) = f (t) +

1> f¯t00 Bt + + µσ (t), βu (T ) = sup γu (t), 2σut ut t∈T

(3.16)

P˜b (f (·) ∈ A) = P (f (·) ∈ A|βu (T ) > u) . The process γu (t) is slightly different from αt . The following theorem states that the measure Q also approximates the distribution P˜b in total variation for b large. Theorem 7. Consider a Gaussian random field {f (t) : t ∈ T } living on a domain T satisfying conditions C1-6. With the same choice of tuning parameters as in Theorem 3, that is η = ρ1 = ρ2 = 1 − λ = (log log b)−1 , Q approximates P˜b in total variation, that is, lim sup |Q(A) − P˜b (A)| = 0.

b→∞ A∈F

3.3. Some implications of the theorems. The results of Theorems 3 and 7 provide both qualitative and quantitative descriptions of Pb∗ . From a qualitative point of view, Theorems 3 and 7 suggest that sup |Pb∗ (A) − P˜b (A)| → 0,

(3.17)

A∈F

as b → ∞. Note that γu (t) itself is a Gaussian process. Thus, the above convergence result connects the tail events of exponential integrals to those of the supremum of another Gaussian random field that is a linear combination of f and its derivative field. We setup this connection mainly because the distribution of Gaussian random fields conditional on level crossing (also known as the Slepian model) is very

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

13

well studied for smooth processes ([30]). For illustration purpose, we cite one result in Chapter 6.2 of [5] when γu (t) is stationary and twice differentiable. Let covariance function of γu (t) be Cγ (t). Conditional on γu (t) achieving a local maximum at location t∗ at level x, we have the following closed form representation of the conditional field γu (t∗ + t) = xCγ (t) − Wx β(t) + g(t) (3.18) where  β(t) =

1 µγ02

µγ20 µγ22

−1

µγ> 2 (t),

µγij ’s are the spectral moments of Cγ (t), Wx is a d(d+1)/2 dimensional random vector whose density can be explicitly written down and g(t) is a mean zero Gaussian process whose covariance function is also in a closed form; see [5] for the specific forms. If we set x > u → ∞, the local maximum is asymptotically the global maximum. Furthermore, thanks to stationarity, the distribution of t∗ is asymptotically uniform over T . The overshoot x − u is asymptotically an exponential random variable. Thus, the conditional field γu (t) can be written down explicitly through the representation (3.18), the overshoot distribution, and the distribution of t∗ . Furthermore, the conditional distribution of f (t) can be implied by (3.16) and conditional normal calculations. From a quantitative point of view, Theorem 3 implies that for any bounded function Ξ : C(T ) → R the conditional expectation E[Ξ(f )|I(T ) > b] can be approximated by E Q [Ξ(f )], more precisely, E[Ξ(f )|I(T ) > b] − E Q [Ξ(f )] → 0

(3.19)

as b → ∞. The expectation E Q [Ξ(f )] is much easier to compute (both analytically and numerically) via the following identity   E Q [Ξ(f )] = E Q E[Ξ(f )| ı, τ, f (τ ), ∂f (τ ), ∂ 2 f (τ )] . (3.20) Note that the inner expectation is under the measure P in that the conditional distribution of f given (f (τ ), ∂f (τ ), ∂ 2 f (τ )) under Q is the same as that under P . Furthermore, conditional on (f (τ ), ∂f (τ ), ∂ 2 f (τ )), the process f (t) is also a Gaussian process and has the expansion 1 f (t) = f (τ ) + ∂f (τ )> (t − τ ) + (t − τ )> ∆f (τ )(t − τ ) + o(|t − τ |2 ). 2 These results provide sufficient tools to evaluate the conditional expectation   E Ξ(f )|ı, τ, f (τ ), ∂f (τ ), ∂ 2 f (τ ) . Once the above expectation has been evaluated, we may proceed to the outer expectation in (3.20). Note that the inner expectation is a function of (ı, τ, f (τ ), ∂f (τ ), ∂ 2 f (τ )), the joint distribution of which is in a closed form. Thus, evaluating the outer expectation is usually an easier task. In fact, the proof of Theorem 3 is an exercise of the above strategy by considering that Ξ(f ) = (dP/dQ)2 .

14

LIU AND XU

Remark 8. According to the detailed proof of Theorem 3, the approximation (3.19) is applicable to all the functions such that supb E[Ξ2 (f )|I(T ) > b] < ∞. To see that, we need to change the statement and the proof of Lemma 13 presented in Section 4. 3.4. Efficient Rare-event Simulation for I(T ). In the preceding subsection we constructed a change of measure that asymptotically approximates the conditional distribution of f given I(T ) > b. In this section, we construct an efficient importance sampling estimator based on this change of measure to compute v(b) as b → ∞. We evaluate the overall computation efficiency using a concept that has its root in the general theory of computation in both continuous and discrete settings [44, 52]. In particular, completely analogous notions in the setting of complexity theory of continuous problems lead to the notion of tractability of a computational problem [54]. Definition 9. A Monte Carlo estimator is said to be a fully polynomial randomized approximation scheme (FPRAS) for estimating v(b) if, for some q1 , q2 , and d > 0, it outputs an averaged estimator that is guaranteed to have at most ε > 0 relative error with confidence at least 1 − δ ∈ (0, 1) in O(ε−q1 δ −q2 | log v(b)|d ) function evaluations. Equivalently, one needs to compute an estimator Zb with complexity O(ε−q1 δ −q2 | log v(b)|d ) such that P (|Zb /v(b) − 1| > ε) < δ. (3.21) In the literature of rare-event simulations, an estimator Lb is said to be strongly efficient in estimating v(b) if ELb = v(b) and supb V arLb /v 2 (b) < ∞. Suppose that (j) a strongly efficient estimator Lb has been obtained. Let {Lb : j = 1, ..., n} be i.i.d. copies of Lb . The averaged estimator n

Zb =

1 X (j) L n j=1 b

p p has a relative mean squared error equal to E(Zb /v(b) − 1)2 = V ar(Lb )n−1/2 v(b)−1 . A simple consequence of Chebyshev’s inequlity yields P (|Zb /v(b) − 1| ≥ ε) ≤

V ar(Lb ) . ε2 nv 2 (b)

Thus, it suffices to simulate n = O(ε−2 δ −1 ) i.i.d. replicates of Lb to achieve the accuracy in (3.21). The so-called importance sampling is based on the identity P (A) = E Q [IA dP/dQ]. The random variable IA dP/dQ is an unbiased estimator of P (A). It is well know that if one chooses Q(·) = P (·|A) then IA dP/dQ has zero variance. The measure Q created in the previous subsection is a good approximation of Pb∗ and thus it naturally leads an estimator for v(b) with small variance.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

15

In addition to the variance control, another issue is that the random fields considered in this paper are continuous objects. Computer can only perform discrete simulations. Thus, we must use a discrete object approximating the continuous field to implement the algorithms. The bias caused by the discretization must be well controlled relative to v(b). In addition, the complexity of generating one such discrete object should also be considered in order to control the overall computational complexity to achieve an FPRAS. We create a regular lattice covering T . Define    id i1 i2 , , ..., : i1 , ..., id ∈ Z . GN,d = N N N For each t = (t1 , · · · , td ) ∈ GN,d , define  TN (t) = (s1 , · · · , sd ) ∈ T : sj ∈ (tj − 1/N, tj ] for j = 1, · · · , d that is the

1 N -cube

intersected with T and cornered at t. Furthermore, let TN = {t ∈ GN,d : TN (t) 6= ∅}.

(3.22)

Since T is compact, TN is a finite set. We enumerate the elements in TN = {t1 , · · · , tM }, where M = O(N d ). We further define X = (X1 , · · · , XM )> , (f (t1 ), · · · , f (tM ))> and use vM (b) = P (IM (T ) > b) as an approximation of v(b) where IM (T ) =

M X

mes(TN (ti )) × eσXi +µ(ti ) .

(3.23)

i=1

We have the following theorem to control the bias. Theorem 10. Consider a Gaussian random field f satisfying conditions in Theorem 3. For any ε0 > 0, there exists κ0 such that for any ε ∈ (0, 1), if N ≥ κ0 ε−1−ε0 (log b)2+ε0 , then for b > 2 |vM (b) − v(b)| < ε. v(b) We estimate vM (b) using a discrete version of the change of measure proposed in the previous section. The specific algorithm is given as follows. 1. Generate a random indicator ı ∼Bernoulli(ρ2 ). If ı = 1, then (a) Generate ι uniformly from {1, ..., M }. (b) Generate Xι ∼ N (utι , 1).

16

LIU AND XU

(c) Given (tι , Xι ), simulate the joint field (f (t), ∂f (t), ∂ 2 f (t)) on the lattice TN \{tι } from the original conditional distribution under P . 2. If ı = 0 (a) If µ(t) is not constant, simulate a random index ι proportional to l(tι ), PM that is, P (ι = i) = l(ti )/κ and κ = i=1 l(ti ); if µ(t) ≡ 0, then ι is simulated uniformly over {1, ..., M }. (b) Given the realized ι, simulate f (tι ) = Xι = x, ∂f (tι ) = y, ∂ 2 f (tι ) = z from density function hall (x, y, z) =

ρ1 1 − ρ1 − ρ2 h0,tι (x, y, z) + h1,tι (x, y, z). 1 − ρ2 1 − ρ2

(c) Given (tι , f (tι ), ∂f (tι ), ∂ 2 f (tι )), simulate the joint field (f (t), ∂f (t), ∂ 2 f (t)) on the lattice TN \{tι } from the original conditional distribution under P. 3. Output ˜b = L

1−ρ1 −ρ2 κ

I{IM (T )>b} PM ρ1 PM i=1 l(ti )LR(ti ) + κ i=1 l(ti )LR1 (ti ) + ρ2 i=1

PM

LR2 (ti ) M

. (3.24)

Let QM be the measure induced by the above simulation scheme. Then, it is not ˜ b = I{I (T )>b} dP/dQM and thus L ˜ b is an unbiased estimator hard to verify that L M of vM (b). The next theorem states the strong efficiency of the above algorithm. Theorem 11. Suppose f is a Gaussian random field satisfying conditions in Theorem 3. If N is chosen as in Theorem 10 and all the other parameters are chosen as in Theorem 3, then there exists some constant κ1 > 0 such that sup b>1

˜2 E QM L b 2 (b) ≤ κ1 . vM

˜ b . According to the results in Theorem Let Zb be the average of n i.i.d. copies of L 10, we have that Zb Zb Zb Zb Zb v(b) − 1 ≤ vM (b) (vM (b)/v(b) − 1) + vM (b) − 1 ≤ ε vM (b) + vM (b) − 1 . The results of Theorem 11 indicate that P (|Zb /vM (b) − 1| ≥ ε) ≤

κ1 . ε2 n

If we choose n = κ1 ε−2 δ −1 , then P (|Zb /v(b) − 1| ≥ 3ε) ≤ δ. Thus, the accuracy level as in (3.21) has been achieved. Note that simulating ˜ b consists of generating a multivariate Gaussian random vector of dimenone L
sion M × (d + 1)(d + 2)/2 = O(N d ) = O (log b)(2+ε0 )d ε−(1+ε0 )d . The complexity of generating such a vector is at
the most O(N 3 ). Thus, the overall complexity is O ε−2−(3+3ε0 )d δ −1 (log b)(6+3ε0 )d . The proposed estimator in (3.24) is a FPRAS.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

17

Remark 12. The proposed algorithm can also be used to compute conditional expectations via the representation E[Ξ(f )|I(T ) > b] = E[Ξ(f ); I(T ) > b]/v(b), where E[Ξ(f ); I(T ) > b] can be estimated by Ξ(f )dP/dQM and v(b) can be estimated by I{I(T )>b} dP/dQM . 4. Proof of Theorem 3 . We use the following simple yet powerful lemma to prove Theorem 3. Lemma 13. Let Q0 and Q1 be probability measures defined on the same σ-field F such thatdQ1 = r−1 dQ0 for a positive random variable r. Suppose that for some ε > 0, E Q1 r2 = E Q0 [r] ≤ 1 + ε. Then, sup E Q1 (X) − E Q0 (X) ≤ ε1/2 . |X|≤1

Proof of Lemma 13. Q E 1 (X) − E Q0 (X) = E Q1 [(1 − r)X] ≤ E Q1 |r − 1| ≤ [E Q1 (r − 1)2 ]1/2 = E Q1 [r2 ] − 1


1/2

≤ ε1/2 .

We also need the following approximations for the tail probability v(b). This proposition is an extension of Theorem 3.4 and Corollary 3.5 in [41]. We layout the key steps of its proof in the supplemental material. Proposition 14. Consider a Gaussian random field {f (t) : t ∈ T } living on a domain T satisfying conditions C1-6. If µ(t) has one unique maximum in T denoted by t∗ , then   (u − µσ (t∗ ))2 v(b) ∼ (2π)d/2 det(−∆µσ (t∗ ))−1/2 G(t∗ ) · ud/2−1 exp − , 2 where u is as defined in (3.4), G(t) is defined as 1

det(Γ)− 2 (2π)

e (d+1)(d+2) 4

1T µ22 1 +Bt 8σ 2

Z R

d(d+1) 2

  1/2 2  2 −1/2 1 |µ20 µ−1 µ22 1 22 z| µ exp − + z − dz. 22 2 1 − µ20 µ−1 2σ 22 µ02

If µ(t) ≡ 0, G(t) is a constant denoted by G. Then, v(b) ∼ mes(T )G · ud−1 e−

u2 2

.

4.1. Case 1: µ(t) is not a constant . To make the proof smooth, we arrange the statement of the rest supporting lemmas in Section 8. We start the proof of Theorem 3 when µ(t) is not a constant. Note that  ∗     dPb 2 dP 2 Q −2 Q E = v(b) E ; I(T ) > b . dQ dQ

18

LIU AND XU

Thanks to Lemma 13, we only need to show that for any ε > 0 there exists b0 such that for all b > b0    i h dP 2 dP 2 Q ; I(T ) > b = E Q Eı,τ ; I(T ) > b ≤ (1 + ε)v(b)2 , EQ dQ dQ Q [ · ] = E Q [ · | ı, τ ] to denote the conditional where we use the notation Eı,τ expectation given ı and τ . τ ∈ T is the random index described as in the simulation scheme admitting a density function l(t) if ı = 0 and mes−1 (T )IT (t) if ı = 1. Note that   2   2  dP dP Q Q Q ; I(T ) > b = Eı,τ ; I(T ) > b f (τ ), ∂f (τ ), ∂ 2 f (τ ) . Eı,τ Eı,τ dQ dQ

For the rest of the proof, we mostly focus on the conditional expectation Q Eı,τ



dP dQ

2

 ; I(T ) > b f (τ ), ∂f (τ ), ∂ 2 f (τ ) .

The rest of the discussion is conditional on ı and τ . To simplify notation, for a given τ , we define f∗ (t) = f (t) − uτ C(t − τ ). On the set {I(T ) > b}, f (τ ) reaches a level uτ , and E[f (t)|f (τ ) = uτ ] = uτ C(t−τ ). Thus, f∗ (t) is the field with the conditional expectation removed. From now on, we work with this shifted field f∗ (t). Correspondingly, we have ∂f∗ (t) = ∂f (t) − uτ ∂C(t − τ ),

∂ 2 f∗ (t) = ∂ 2 f (t) − uτ ∂ 2 C(t − τ ).

We further define notations y = ∂f∗ (τ ), z = ∂ 2 f∗ (τ ), z = ∆f∗ (τ ), ˜ = ∆f∗ (τ ) + µσ (τ )I + ∆µσ (τ ), y˜ = ∂f∗ (τ ) + ∂µσ (τ ), z

w = f∗ (τ ),

wt = f∗ (t),

yt = ∂f∗ (t),

zt = ∂ 2 f∗ (t),

z¯t = ∂ 2 f∗ (t) − ut µ02 . (4.1)

Under the measure Q and a given τ , if ı = 0, (w, y, z) has density function h∗all (w, y, z) =

1 − ρ1 − ρ2 ∗ ρ1 h0,τ (w, y, z) + h∗ (w, y, z); 1 − ρ2 1 − ρ2 1,τ

(4.2)

if ı = 1, then (w, y, z) follows density h∗τ (w, y, z). The forms of the densities can be

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

19

derived from h0,t , h1,t , and h. In particular, their expressions are given as follows     Bτ 1 1> z h∗0,τ (w, y, z) ∝ IAτ × exp −λuτ w + + − |y|2 2σuτ uτ 2   1/2 2  −1 2 −1/2 µ22 1 |µ20 µ22 z| 1 + µ22 z − × exp − 2 1 − µ20 µ−1 2σ 22 µ02     1> z Bτ 1 h∗1,τ (w, y, z) ∝ IAcτ × exp λ1 uτ w + + − |y|2 2σuτ uτ 2   1/2 2  −1 2 −1/2 µ22 1 |µ20 µ22 z| 1 + µ22 z − × exp − 2 1 − µ20 µ−1 2σ 22 µ02 1 n 1h io w − µ20 µ−1 z 2 det(Γ)− 2 22 ∗ > > −1 hτ (w, y, z) = h(w, y, z) = y exp − y + + z µ z , (d+1)(d+2) 22 2 1 − µ20 µ−1 4 (2π) 22 µ02 n o > Bτ 1> z −1 + > −ηu and Aτ = w + y2uτy + 2σu is defined as in (3.6). τ uτ τ In the next step, we will compute dQ/dP in the form of f∗ (t). Basically, we replace f (t) by f∗ (t) + uτ C(t − τ ), ∂f (t) by yt + uτ ∂C(t − τ ), ∂ 2 f (t) by zt + uτ ∂ 2 C(t − τ ), and f¯t00 = ∂ 2 f (t) − ut µ02 by z¯t + uτ ∂ 2 C(t − τ ). For the likelihood ratio terms LR and LR1 in (3.11), note that the |∂f (t)|2 terms in h0,t and h1,t cancel with those in h(f (t), ∂f (t), ∂ 2 f (t)), that is,
 |µ20 µ−1 f¯00 |2 −1/2 1/2 2  µ 1 1> f¯00 B 22 t −λut f (t)+ 2σut + u t −ut − 12 + µ22 f¯t00 − 22 −1 2σ t t 1−µ20 µ22 µ02 Hλ · u t e LR(t) = IAt · .  (f (t)−µ µ−1 ∂ 2 f (t))2  1

det(Γ)− 2

(d+1)(d+2) e

(2π)

20 22 −1 1−µ20 µ22 µ02

− 21

2 +∂ 2 f (t)> µ−1 22 ∂ f (t)

4

We insert the notations in (4.1) and obtain that    1> (¯ zt + µ2 (t − τ )uτ ) Bt LR(t) = IAt · ut Hλ exp −λut wt + uτ C(t − τ ) + + − ut 2σut ut h i 1/2 2 −1 2 |µ µ (¯ z +µ (t−τ )u )| µ 1 τ −1/2 t 20 22 2 − 12 + µ22 (¯ zt +µ2 (t−τ )uτ )− 22 −1 2σ 1−µ20 µ22 µ02 ×e
2 ×h−1 (4.3) x,z wt + uτ C(t − τ ), zt + uτ ∂ C(t − τ ) , where 1

hx,z (x, z) =

det(Γ)− 2 (2π)

(d+1)(d+2) 4

e

− 21



−1 (x−µ20 µ22 z)2 −1 1−µ20 µ22 µ02

+z > µ−1 22 z



,

(4.4)

which is the function h(x, y, z) with the |y|2 term removed. Similarly, we have that    zt + µ2 (t − τ )uτ ) Bt 1> (¯ + − ut LR1 (t) = IAct · ut Hλ1 exp λ1 ut wt + uτ C(t − τ ) + 2σut ut h i −1/2 1/2 2 −1 |µ20 µ22 (¯ zt +µ2 (t−τ )uτ )|2 µ 1 − 12 + µ22 (¯ zt +µ2 (t−τ )uτ )− 22 −1 2σ 1−µ20 µ22 µ02 ×e
2 ×h−1 (4.5) x,z wt + uτ C(t − τ ), zt + uτ ∂ C(t − τ ) .

20

LIU AND XU

With the analytic forms (4.3) and (4.5), we proceed to the likelihood ratio in (3.10) dQ dP where Z K=

(1 − ρ1 − ρ2 )K + ρ1 K1 + ρ2 K2

=

Z l(t)LR(t)dt, K1 =

A∗

Z l(t)LR1 (t)dt, K2 =

(A∗ )c

T

1

(4.6)

2

e− 2 ut +ut wt +ut uτ C(t−τ ) dt. mes(T )

The set A∗ (depending on the sample path f∗ (t)) is defined as   |yt + uτ · ∂C(t − τ )|2 1> (¯ zt + uτ µ2 (t − τ )) Bt η t : wt + C(t − τ )uτ + + + > ut − . 2ut 2σut ut ut We may equivalently define A∗ = {t : f ∈ At }. Note that LR(t) = 0 if f ∈ / At . Thus, the integral K is on the set A∗ and K1 is on the complement of A∗ . Based on the above results, we have that EQ



dP dQ

2

 ; I(T ) > b

i ; I(T ) > b (4.7) 2 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]  i h 1 Q ≤ E Q Eı,τ ; I(T ) > b, A ≥ 0 τ 2 [(1 − ρ1 − ρ2 )K]  i h 1 Q Q +E Eı,τ , 2 ; I(T ) > b, Aτ < 0 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]  h Q ≤ E Q Eı,τ

where Aτ = w +

1

y> y 1> z Bτ + + . 2uτ 2σuτ uτ

(4.8)

Note that the term K2 is not used in the main analysis. In fact, K2 is only used in Lemma 17 for the purpose of localization that will be presented later. The rest of the analysis consists of three main parts.
Part 1 Conditional on ı, τ, f∗ (τ ), ∂f∗ (τ ), ∂ 2 f∗ (τ ) , we study the event Eb = {I(T ) > b} ,

(4.9)

and write the occurrence of this event almost as a deterministic function of f∗ (τ ), ∂f∗ (τ ), and ∂ 2 f∗ (τ ), equivalently, (w,
y, z). Part 2 Conditional on ı, τ, f∗ (τ ), ∂f∗ (τ ), ∂ 2 f∗ (τ ) , we express K and K1 as functions of f∗ (τ ), ∂f∗ (τ ), ∂ 2 f∗ (τ ) with small correction terms. Part 3 We combine the results from the first two parts and obtain an approximation of (4.7). All the subsequent derivations are conditional on ı and τ .

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

21

4.1.1. Preliminary calculations. To proceed, we provide the Taylor expansions for f∗ (t), C(t), and µ(t).
• Expansion of f∗ (t) given f∗ (τ ), ∂f∗ (τ ), ∂ 2 f∗ (τ ) . Let t − τ = ((t − τ )1 , ..., (t −
τ )d ). Conditional on f∗ (τ ), ∂f∗ (τ ), ∂ 2 f∗ (τ ) , we first expand the random function   f∗ (t) = E f∗ (t)|f∗ (τ ), ∂f∗ (τ ), ∂ 2 f∗ (τ ) + g(t − τ ) (4.10) 1 = f∗ (τ ) + ∂f∗ (τ )> (t − τ ) + (t − τ )> ∆f∗ (τ )(t − τ ) 2 +Rf (t − τ ) + g(t − τ ), where
Rf (t − τ ) = O |t|2+δ0 (|f∗ (τ )| + |∂f∗ (τ )| + |∂ 2 f∗ (τ )|)   is the remainder term of the Taylor expansion of E f∗ (t)|f∗ (τ ), ∂f∗ (τ ), ∂ 2 f∗ (τ ) . g(t) is a mean zero Gaussian random field such that Eg 2 (t) = O(|t|4+δ0 ) as t → 0. In addition, the distribution of g(t) is independent of ı, τ, f∗ (τ ), ∂f∗ (τ ), and ∂ 2 f∗ (τ ). • Expansion of C(t): 1 C(t) = 1 − t> t + C4 (t) + RC (t), 2

(4.11)

P 1 4 4+δ0 where C4 (t) = 24 ). ijkl ∂ijkl C(0)ti tj tk tl and RC (t) = O(|t| • Expansion of µ(t): 1 µ(t) = µ(τ ) + ∂µ(τ )> (t − τ ) + (t − τ )> ∆µ(τ )(t − τ ) + Rµ (t − τ ), (4.12) 2 where Rµ (t − τ ) = O(|t − τ |2+δ0 ). We write R(t) = Rf (t) + uτ RC (t) + Rµ (t)/σ to denote all the remainder terms. Choose small constants  and δ such that 0 <   δ  δ0 . By writing xy we mean that x/y is chosen sufficiently small but x/y does not change with b. Let  L = |τ − t∗ | < u−1/2+ , |w| ≤ u1/2+ , |y| < u , |z| < u ,  − −1−δ sup |zt − z| < u , sup |g(t)| < u . (4.13) |t−τ | (t − τ ) + (t − τ )> z(t − τ ) + Rf (t − τ ) + g(t − τ ) 2 t∈T    1 × exp σu − µ(τ ) 1 − (t − τ )> (t − τ ) + C4 (t − τ ) + RC (t − τ ) 2   1 × exp µ(τ ) + ∂µ(τ )> (t − τ ) + (t − τ )> ∆µ(τ )(t − τ ) + Rµ (t − τ ) dt, (4.15) 2 where the first row corresponds to the expansion of wt = f∗ (t), the second and the third correspond to those of C(t) and µ(t) respectively. We write the exponent inside the integral in a quadratic form of (t − τ ) and obtain that n o σ ˜)−1 y˜ I(T ) = exp σu + σw + y˜> (uI − z (4.16) 2 Z n σ
o ˜)−1 y˜)> (uI − z ˜) t − τ − (uI − z ˜)−1 y˜ × exp − (t − τ − (uI − z 2 t∈T × exp{σuτ C4 (t − τ ) + σR (t − τ )} × exp{σg (t − τ )}dt, ˜ are defined as in (4.1). Let a(s) and b(s) be two generic positive where y˜ and z functions. Then, we have the representation of the following integral Z Z a(s)b(s)ds = E[b(S)] a(s)ds T

T

R where S is a random variable taking values in T with density a(s)/ T a(t)dt. Using ˜)1/2 (t − τ ), we write this representation and the change of variable that s = (uI − z

23

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

the big integral in (4.16) as a product of expectations and a normalizing constant, and obtain that I(T )

=

σ ˜)−1/2 exp{σu + σw + y˜> (uI − z ˜)−1 y˜} det(uI − z 2 Z n σ  o −1/2 > −1/2 ˜ ˜ × exp − (s − (uI − z ) s − (uI − z ) y ˜ ) y ˜ ds 1 2 (uI−z)− 2 s+τ ∈T        1  1  1  ˜ ˜ )− 2 S ˜ )− 2 S ˜ )− 2 S +σR (uI−z σg (uI−z σu C (uI−z ×E e . ×E e τ 4

The two expectations in the above display are taken with respect to S and S˜ given ˜)−1/2 s + the process g(t). S is a random variable taking values in the set {s : (uI − z τ ∈ T } with density proportional to e− 2 (s−(uI−z) σ

˜

−1/2

>

y˜)

(s−(uI−z˜ )−1/2 y˜)

(4.17)  ˜)−1/2 s + τ ∈ T and S˜ is a random variable taking values in the set s : (uI − z with density proportional to e

  1  1  ˜ )− 2 s +σR (uI−z ˜ −1/2 y˜)> (s−(uI−z ˜ )−1/2 y˜)+σuτ C4 (uI−z ˜ )− 2 s −σ 2 (s−(uI−z)

Together with the definition of u that b if and only if I(T )

=


2π d/2 σ

u−d/2 eσu = b, we obtain that I(T ) >

˜)−1/2 eσu+σw+ 2 y˜ (uI−z) y˜ det(uI − z Z σ ˜ −1/2 y˜)> (s−(uI−z ˜ )−1/2 y˜) e− 2 (s−(uI−z) × ds 1 σ

>

˜

−1

(uI−z)− 2 s+τ ∈T     1  1  ˜ )− 2 S +σR (uI−z ˜ )− 2 S σuτ C4 (uI−z

×E e  > where

.

2π σ

d/2

−1

· e−u

u−d/2 eσu ,

n h  io 1 ˜)− 2 S˜ ξu = −u log E exp σg (uI − z .

ξu

(4.18)

(4.19)

We take log on both sides and plug in the result of Lemma 20 that handles the big expectation term in (4.18). Then, the inequality (4.18) is equivalent to P 4 ˜ ˜)−1 y˜ log det(I − zu ) ∂ C(0) y˜> (uI − z ξu o(|w| + |y| + |z| + 1) − + i iiii > + w+ .(4.20) 2 2σ 8σ 2 u uσ u1+δ0 /4 On the set L, we further simplify (4.20) using the following facts (see Lemma 21) ∂µσ (τ ) =  ˜ z = log det I − u 

O(u−1/2+ ), 1 1> (z + ∂ 2 µσ (τ )) + d · µσ (τ ) − T r(˜ z) + o(u−1−δ0 /4 ) = − + o(u−1−δ0 /4 ), u u

24

LIU AND XU

where T r is the trace of a matrix. Therefore, on the set L, (4.20) is equivalent to P 4 ∂ C(0) ξu o(|w| + |y| + |z| + 1) y > y 1> (z + ∂ 2 µσ (τ )) + d · µσ (τ ) + + i iiii > + , w+ 2u 2σu 8σ 2 u uσ u1+δ0 /4 and further equivalently (by replacing u with uτ ) P 4 ∂ C(0) y > y 1> (z + ∂ 2 µσ (τ )) + d · µσ (τ ) ξu o(|w| + |y| + |z| + 1) w+ . + + i iiii > + 2 2uτ 2σuτ 8σ uτ uσ u1+δ0 /4 Using the notations defined as in (3.9) and (4.8), I(T ) > b is equivalent to Aτ +

o(|w| + |y| + |z| + 1) ξu > , uσ u1+δ0 /4

where Aτ is defined as in (4.8). Furthermore, with   δ0 and on the set L, o(|y| + |z|)/u−1−δ0 /4 = o(u−1−δ0 /8 ). For the above inequality, we absorb o(wu−1−δ0 /4 ) into Aτ and rewrite it as Aτ > (1 + o(u−1−δ0 /4 ))

hξ

u

σu

i + o(u−1−δ0 /8 ) .

4.3. Part 2. In Part 2, we first consider (1 − ρ1 − ρ2 )K in the first expectation of (4.7) (which is on the set {Aτ ≥ 0}) and then (1 − ρ1 − ρ2 )K + ρ1 K1 in the second expectation of (4.7). Part 2.1: the analysis of K when Aτ ≥ 0. Similar to Part 1, all the derivations are conditional on (ı, τ, w, y, z). We now proceed to the second part of the proof. More precisely, we simplify the term K defined as in (4.6) and write it as a deterministic function of (w, y, z) with a small correction term. Recall that  Z   1> (¯ zt + µ2 (t − τ )uτ ) Bt + − ut K= l(t)ut Hλ exp −λut wt + uτ C(t − τ ) + 2σut ut A∗   1/2 2  zt + µ2 (t − τ )uτ )|2 −1/2 µ22 1 1 |µ20 µ−1 22 (¯ µ (¯ z + µ (t − τ )u ) − × exp − + t 2 τ 22 2 2σ 1 − µ20 µ−1 22 µ02
2 ×h−1 x,z wt + uτ C(t − τ ), zt + uτ ∂ C(t − τ ) dt.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

25

We plug in the forms of hx,z and l(t) that are defined in (4.4) and (3.13) and obtain that K

=

(d+1)(d+2)

d

1

d/2

−2 4 det(Γ) 2 · det(−∆µσ (t∗ ))1/2 ut∗ Hλ (2π)   Z ut∗ · (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) × exp 2 A∗    1> (¯ zt + µ2 (t − τ )uτ ) Bt + − ut ×ut × exp − λut wt + uτ C(t − τ ) + 2σut ut   1/2 2  1 |µ20 µ−1 zt + µ2 (t − τ )uτ )|2 −1/2 µ22 1 22 (¯ × exp − µ + (¯ z + µ (t − τ )u ) − t 2 τ 22 2 2σ 1 − µ20 µ−1 22 µ02
2   1 wt + uτ C(t − τ ) − µ20 µ−1 22 (zt + µ2 (t − τ )uτ ) × exp 2 1 − µ20 µ−1 22 µ02  > −1 + (zt + µ2 (t − τ )uτ ) µ22 (zt + µ2 (t − τ )uτ ) dt.

For some δ 0 such that  < δ 0 < δ, where , δ are the parameters we used to define L, we further restrict the integration region by defining   Z ut∗ (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) (4.21) I2 = exp 2 A∗ ,|t−τ | (¯ zt + µ2 (t − τ )uτ ) Bt ×ut × exp −λut wt + uτ C(t − τ ) + + − ut 2σut ut   1/2 2  −1 2 µ22 1 zt + µ2 (t − τ )uτ )| −1/2 1 |µ20 µ22 (¯ zt + µ2 (t − τ )uτ ) − + µ22 (¯ × exp − 2 2σ 1 − µ20 µ−1 22 µ02
  2 1 wt + uτ C(t − τ ) − µ20 µ−1 22 (zt + µ2 (t − τ )uτ ) × exp 2 1 − µ20 µ−1 22 µ02  + (zt + µ2 (t − τ )uτ )> µ−1 (z + µ (t − τ )u ) dt. t 2 τ 22 Thus, K ≥ (2π)

(d+1)(d+2) −d 4 2

1

d/2

det(Γ) 2 · det(−∆µσ (t∗ ))1/2 ut∗ Hλ · I2 .

For the rest of Part 2.1, we focus on I2 . With some tedious algebra, Lemma 22 writes I2 in a more manageable form, that is, I2 equals   Z ut∗ (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) u2t exp + × ut (4.22) 2 2 A∗ ,|t−τ |
1> µ22 1 o × exp (1 − λ)ut wt + uτ C(t − τ ) − ut + 1 zt − µ02 ut + µ2 (t − τ )uτ − λBt − 2σ 8σ 2 n w + u C(t − τ ) − u
2 − 2 w + u C(t − τ ) − u
µ µ−1 z − µ u + µ (t − τ )u
o t τ t t τ t 20 22 t 02 t 2 τ
× exp dt. −1 2 1 − µ20 µ22 µ02

26

LIU AND XU 0

Lemma 23 implies that {|t − τ | < u−1+δ } ⊂ A∗ . Thus, on the set {Aτ > 0}, we 0 0 have A∗ ∩ {|t − τ | < u−1+δ } = {|t − τ | < u−1+δ } and we can remove A∗ from the 0 integration region of I2 . In addition, on the set L and |t − τ | < u−1+δ , we have that 0

uτ − ut C(t − τ ) = O(u−1+2δ ), µ2 (t − τ ) = µ20 + O(|t − τ |2 ), 0

|uτ µ2 (t − τ ) − ut µ20 | = O(u−1+2δ ), (uτ − ut C(t − τ ))|zt | = o(1). We insert the above estimates to (4.22). Together with the fact that     1 2 ut∗ (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) u2t + = (1 + o(1)) exp ut∗ , exp 2 2 2 we have that  I2

∼ u × exp Z ×

 1 2 1> µ22 1 u − λBt∗ − 2 t∗ 8σ 2  exp (1 − λ)ut [wt + uτ (C(t − τ ) − 1) + (µσ (t) − µσ (τ ))]

|t−τ | z 22 zt + o(1)wt
+ t dt. 2σ 2 1 − µ20 µ−1 22 µ02

Further, we have that 0

−1/2+δ wt2 − 2wt µ20 µ−1 ). 22 zt + o(1)wt = o(1) + u · w · O(u 0

Let ζu = O(u−1/2+δ ) and we simplify I2 to   1> µ22 1 1 2 ut∗ − λBt∗ − I2 ∼ u × exp 2 8σ 2  Z h × exp (1 − λ)(uτ + ζu ) ζu w + wt + uτ (C(t − τ ) − 1) |t−τ | z dt. + (µσ (t) − µσ (τ )) + (1 − λ) 2σ i

In what follows, we insert the expansions in (4.10), (4.11) and (4.12) into the expression of I2 and write the exponent as a quadratic function of t − τ , and obtain that on the set L   1 2 1> µ22 1 I2 ∼ u × exp ut∗ − λBt∗ − 2 8σ 2    1 1> z ˜)−1 y˜ + × exp (1 − λ)(uτ + ζu ) (1 + ζu )w + y˜> (uI − z 2 2σuτ Z > −1 1 ˜ ˜ ˜ −1 × e− 2 (1−λ)(uτ +ζu )(t−τ −(uI−z) y˜) (uI−z)(t−τ −(uI−z) y˜) |t−τ | µ22 1 I2 ∼ (1 − λ)−d/2 u−d+1 exp ut∗ − λBt∗ − 2 8σ 2    1 1> z ˜)−1 y˜ + × exp (1 − λ)(uτ + ζu ) (1 + ζu )w + y˜> (uI − z 2 2σu Z 2 1/2 1/2 −1/2 1 ˜ y˜| × e− 2 |s−(1−λ) (uτ +ζu ) (uI−z) ds s∈Su h i −1/2 ˜ )−1/2 S 0 ) (uτ +ζu )−1/2 (uI−z ×E e(1−λ)(uτ +ζu )g((1−λ) , (4.24) 0

˜)−1/2 s| < u−1+δ } and S 0 is a where Su = {s : |(1 − λ)−1/2 (uτ + ζu )−1/2 (uI − z random variable taking values on the set Su with density proportional to e− 2 |s−(1−λ)

1/2

1

2

˜ )−1/2 y˜| (uτ +ζu )1/2 (uI−z

.

We use κ to denote the last two terms of (4.24), i.e., Z 1/2 1/2 1 ˜ −1/2 y˜|2 κ = e− 2 |s−(1−λ) (uτ +ζu ) (uI−z) ds Su h i −1/2 ˜ )−1/2 S 0 ) (uτ +ζu )−1/2 (uI−z ×E e(1−λ)(uτ +ζu )g((1−λ) .

(4.25)

It is helpful to keep in mind that κ is approximately (2π)d/2 . We insert κ back to ˜)−1 y˜ = |˜ the expression of I2 . Together with the fact that y˜> (uI − z y |2 /u + o(u−1 ), we have   1 2 1> µ22 1 ut∗ − λBt∗ − I2 ∼ κ(1 − λ)−d/2 u−d+1 exp (4.26) 2 8σ 2    |˜ y |2 1> z  × exp (1 − λ)(uτ + ζu ) (1 + ζu )w + + . 2uτ 2σuτ Thus, we have that on the set {Aτ > 0}, K

≥ (2π) =

(d+1)(d+2) −d 4 2

1

d/2

det(Γ) 2 · det(−∆µσ (t∗ ))1/2 ut∗ Hλ · I2

(d+1)(d+2) −d 4 2

1 2

1/2

(4.27) −d/2 −d/2+1

(κ + o(1))(2π) det(Γ) · det(−∆µσ (t∗ )) Hλ · (1 − λ) u    > 2 1 2 1 µ22 1 |˜ y| 1> z  × exp ut∗ − λBt∗ − . + (1 − λ)(uτ + ζu ) (1 + ζu )w + + 2 8σ 2 2uτ 2σuτ

28

LIU AND XU

We further insert the Aτ defined in (4.8) into (4.27) and obtain that K

(d+1)(d+2)

d

1

−2 4 det(Γ) 2 · det(−∆µσ (t∗ ))1/2 Hλ · (1 − λ)−d/2 u−d/2+1 ≥ (κ + o(1))(2π)   >
1 1 µ22 1 2 × exp u2t∗ − Bt∗ − + (1 − λ)u (1 + o(1))A + (1 − λ)ζ · |˜ y | + |z| . τ τ u 2 8σ 2 (4.28)

Part 2.2: the analysis of dP/dQ when Aτ < 0. In this part, we focus mostly on K1 term, whose handling is very similar to that of K. Therefore, we only list out the key steps. For some large constant M , let ˜)−1 y˜| < M u−1 } D = {|t − τ − (uI − z that is the dominating region of the integral. We split the set D = (A∗ ∩D)∪((A∗ )c ∩ D). There are two situations: mes((A∗ )c ∩D) > mes(A∗ ∩D) and mes((A∗ )c ∩D) ≤ mes(A∗ ∩ D). For the first situation, the term K1 is dominating; for the second situation, the term K (more precisely I2 ) is dominating. To simplify K1 , we write it as Z Z (d+1)(d+2) d 1 d/2 − 1/2 4 2 det(Γ) 2 · det(−∆µ (t )) ut∗ Hλ1 × ... + K1 = (2π) σ ∗ (A∗ )c ∩D

, (2π)

(d+1)(d+2) −d 4 2

1

 ...

(A∗ )c ∩D c

d/2

det(Γ) 2 · det(−∆µσ (t∗ ))1/2 ut∗ Hλ1 × [I1,2 + I1,3 ] .

Note that the difference between K1 and K is that the term “−λ” has been replace by “λ1 ”. With exactly the same derivation for (4.22), we obtain that I1,2 equals (by replacing “−λ” in (4.22) by “λ1 ”) 

Z exp (A∗ )c ∩D

ut∗ (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) 1 + u2t 2 2

 × ut

(4.29)

n   (1 + λ1 ) >
1> µ22 1 o × exp (1 + λ1 )ut wt + uτ C(t − τ ) − ut + 1 zt − µ02 ut + µ2 (t − τ )uτ ) + λ1 Bt − 2σ 8σ 2


2 −1 n wt + uτ C(t − τ ) − ut − 2 wt + (uτ C(t − τ ) − ut ) µ20 µ o 22 zt − µ02 ut + µ2 (t − τ )uτ
× exp dt . −1 2 1 − µ20 µ22 µ02

With a very similar derivation as in Part 2.1, in particular, the result in (4.23), we have that   1 2 1> µ22 1 I1,2 ∼ u × exp ut∗ + λ1 Bt∗ − (4.30) 2 8σ 2    1> z 1 ˜)−1 y˜ + × exp (1 + λ1 )(uτ + ζu ) (1 + ζu )w + y˜> (uI − z 2 2σu Z n  1
>
o ˜)−1 y˜ (uI − z ˜) t − τ − (uI − z ˜)−1 y˜ × exp (1 + λ1 )(uτ + ζu ) − t − τ − (uI − z 2 (A∗ )c ∩D n  o × exp (1 + λ1 )(uτ + ζu ) uτ C4 (t − τ ) + R(t − τ ) + g(t − τ ) dt.

29

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

Furthermore, similar to the results in (4.26), we have that I1,2

2

1

d

κ1,2 (1 + λ1 )− 2 u−d+1 e 2 ut∗ +λ1 Bt∗ −

∼

1> µ22 1 8σ 2

  ˜ )−1 y˜+ 1> z (1+λ1 )(uτ +ζu ) (1+ζu )w+ 12 y˜> (uI−z 2σuτ

×e where κ1,2 is defined as Z κ1,2 = t1

e− 2 |s−(1+λ1 )

1/2

1

,

(4.31)

˜ )−1/2 y˜|2 (uτ +ζu )1/2 (uI−z

ds

(s)∈(A∗ )c ∩D

h i −1/2 ˜ )−1/2 S1,2 ) (uτ +ζu )−1/2 (uI−z ×E e(1+λ1 )(uτ +ζu )g((1+λ1 ) , ˜)−1/2 s and the change of variable t1 (s) = τ + (1 + λ1 )−1/2 (uτ + ζu )−1/2 (uI − z ∗ c S1,2 is a random variable taking values in the set {s : t(s) ∈ (A ) ∩ D} with an appropriately chosen density function similarly as in (4.24). In summary, the only difference between I1,2 and I2 lies in that the multiplier −λ is replaced by λ1 . We now proceed to providing a lower bound of (1 − ρ1 − ρ2 )K + ρ1 K1 . Note that max{mes((A∗ )c ∩ D), mes(A∗ ∩ D)} ≥

1 mes(D). 2

Therefore at least one of (A∗ )c ∩ D and A∗ ∩ D is nonempty. If mes((A∗ )c ∩ D) ≥ 1 2 mes(D), we have the bound (1 − ρ1 − ρ2 )K + ρ1 K1 ≥ ρ1 K1 ≥ Θ(1)ρ1 ud/2 I1,2 . Similarly, if mes(A∗ ∩ D) ≥ 21 mes(D), we have that (1 − ρ1 − ρ2 )K + ρ1 K1 ≥ Θ(1)(1 − ρ1 − ρ2 )ud/2 I2 . We further split I2 in Part 2.1 into two parts: Z Z I2 = · · · dt + · · · dt , I2,1 + I2,2 . A∗ ∩D

(4.32)

A∗ ∩D c

Similar to the derivation of I1,2 , we have that I2,1 ∼ κ2,1 (1 − λ)−d/2 u−d+1 e

1> µ22 1 1 2 2 ut∗ −λBt∗ − 8σ 2

  |y| ˜ 2 1> z (1−λ)(uτ +ζu ) (1+ζu )w+ 2uτ + 2σu τ

·e

,

where Z κ2,1

e− 2 |s−(1−λ) 1

=

1/2

2

˜ )−1/2 y˜| (uτ +ζu )1/2 (uI−z

ds

t2 (s)∈A∗ ∩D

h

−1/2

×E e(1−λ)(uτ +ζu )g((1−λ)

˜ )−1/2 S2,1 ) (uτ +ζu )−1/2 (uI−z

i

.

(4.33)

S2,1 is a random variable taking values on the set {s : t2 (s) ∈ A∗ ∩D} with an appropriate density function similarly as in (4.24) and t2 (s) = τ +(1 − λ)−1/2 (uτ + ζu )−1/2 (uI− ˜)−1/2 s. z

30

LIU AND XU

Then combining the above results of I1,2 and I2,1 , we have that for the case in which Aτ < 0 ρ1 K1 + (1 − ρ1 − ρ2 )K ≥ Θ(1)ud/2 [IC1 ρ1 I1,2 + IC2 (1 − ρ1 − ρ2 )I2,1 ]    |y| ˜ 2 1> z (1+λ1 )(uτ +ζu ) (1+ζu )w+ 2uτ + 2σu 1 2 τ ≥ Θ(1)u−d/2+1 e 2 ut∗ × IC1 · ρ1 κ1,2 e + IC2 · (1 − ρ1 − ρ2 )(1 − λ)−d/2 κ2,1 e

  |y| ˜ 2 1> z (1−λ)(uτ +ζu ) (1+ζu )w+ 2uτ + 2σu τ

where C1 = {f (·) : mes((A∗ )c ∩ D) ≥ mes(A∗ ∩ D)} and C2 = C1c . We further insert Aτ defined in (4.8). Note that on the set {Aτ < 0}, (1 + λ1 )Aτ < (1 − λ)Aτ and Bt is bounded away from zero and infinity. Then, (1 − ρ1 − ρ2 )K + ρ1 K1

2 1 2 ≥ Θ(1)u−d/2+1 e 2 ut∗ · e(1+λ1 )(1+ζu )uτ Aτ +ζu ·(|˜y| +|z|) h i × IC1 · ρ1 κ1,2 + IC2 · (1 − ρ1 − ρ2 )(1 − λ)−d/2 κ2,1 .

Part 3. We now put together the results in Part 1 and Part 2 and obtain an approximation for (4.7). Recall that EQ



dP dQ

2

 ; Eb , L

 ; E , L, A ≥ 0 b τ 2 [(1 − ρ1 − ρ2 )K]   1 +E Q ; E , L, A < 0 . (4.35) b τ 2 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]

≤ EQ



1

We consider the two terms on the right-hand-side of the above display one by one. We start with the first term   1 Q E (4.36) 2 ; Eb , L, Aτ ≥ 0 [(1 − ρ1 − ρ2 )K]   1 = EQ ; E , L, A ≥ 0, ı = 0 b τ 2 [(1 − ρ1 − ρ2 )K]   1 Q +E 2 ; Eb , L, Aτ ≥ 0, ı = 1 . [(1 − ρ1 − ρ2 )K] The index τ admits denstiy l(t) when ı = 0 and τ is uniformly distributed over T if ı = 1. Consider the first expectation in (4.36). Note that conditional on τ and ı = 0, on the set L ∩ {Aτ ≥ 0}, (w, y, z) follows density (1 − ρ1 − ρ2 )h∗0,τ (w, y, z)/(1 − ρ2 ) defined as in (4.2). Thus, according to (4.28), we have that the conditional

(4.34)

31

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

expectation  EQ

 1 ı = 0, τ (4.37) ; E , L, A ≥ 0 b τ (1 − ρ1 − ρ2 )2 K 2  −1 2 1 1> µ 1 Hλ det(Γ)− 2 det(−∆µσ (t∗ ))−1/2 d/2 d/2−1 − 12 u2t∗ +Bt∗ + 8σ22 2 ≤ (1 + o(1)) (1 − λ) u e (d+1)(d+2) −d 4 2 (2π) Z
|y|2 1> z 1 − ρ1 − ρ2 ∗ × e−2(1−λ)u (1+o(1))Aτ +o( 2u + 2σu ) · γu (uσAτ ) · h0,τ (w, y, z)dwdydz 1 − ρ2 Aτ >0,L

where  γu (x) = E

 1 −1−δ0 /4 −δ0 /8 ; x > (1 + o(u ı, τ, w, y, z , ))[ξ + o(u )] u (1 − ρ1 − ρ2 )2 κ2

with the expectation taken with respect to the process g(t). We insert the analytic form of h∗0,τ (w, y, z) in (4.2) and obtain that Z
|y|2 1> z 1 − ρ1 − ρ2 ∗ e−2(1−λ)u (1+o(1))Aτ +o( 2u + 2σu ) · γu (uσAτ ) · h0,τ (w, y, z)dwdydz 1 − ρ2 Aτ >0,L Z  (1 − ρ1 − ρ2 )Hλ · uτ = γu (uσAτ ) exp −2(1 − λ + o(1))uAτ + o(|z| + |y|2 ) 1 − ρ2 Aτ >0    1/2 2  2 −1/2 µ22 1 |µ20 µ−1 1−λ > 1 22 z| µ z − + − y y dAτ dydz. × exp − λuτ Aτ − 22 2 1 − µ20 µ−1 2σ 2 22 µ02 (4.38) Thanks to the Borel-TIS inequality (Lemma 16), Lemma 19 and the definition of κ in (4.25), for x > 0, γu (x) is bounded and as b → ∞,   1 −1−δ0 /4 −δ0 /8 E 2 ; x > (1 + o(u ))[ξu + o(u )] → (2π)−d . κ Thus, by the dominated convergence theorem and with Hλ defined as in (3.14), as u → ∞, we have that (4.38) ∼

(2π)−d e−λη λ . (1 − ρ1 − ρ2 )(1 − ρ2 ) 2 − λ

We insert it back to (4.37) and obtain that  1 Q E ; Eb , L, Aτ ≥ 0 (1 − ρ1 − ρ2 )2 K 2 ≤ (1 + o(1))  ×

 ı = 0, τ

(2π)−d e−λη λ (1 − ρ1 − ρ2 )(1 − ρ2 ) 2 − λ

(4.39)

1

Hλ−1 det(Γ)− 2 det(∆µσ (t∗ ))−1/2 (2π)

(d+1)(d+2) −d 4 2

1

2

(1 − λ)d/2 ud/2−1 e− 2 ut∗ +Bt∗ +

1> µ22 1 8σ 2

2 .

32

LIU AND XU

Using the asymptotic approximation of v(b) given by Proposition 14, we obtain that # " 1 + o(1) 1 eλη Q 2 E 2 ; Eb , L, Aτ ≥ 0, ı = 0 ≤ 1 − ρ − ρ λ(2 − λ) v (b). (4.40) 1 2 [(1 − ρ1 − ρ2 )K] We choose ρ1 = ρ2 = η = 1 − λ = 1/ log log b ∼ 1/ log u. Then, the right-hand-side of the above inequality is bounded by (1 + ε)v 2 (b) for b sufficiently large. The handling of the second term of (4.36) is similar except that (w, y, z) follows density h∗τ (w, y, z). Thus, we only mention the key steps. Note that   1 Q ı = 1, τ ; E , L, A ≥ 0 E b τ (1 − ρ1 − ρ2 )2 K 2  −1 2 1 1> µ 1 Hλ det(Γ)− 2 det(−∆µσ (t∗ ))−1/2 d/2 d/2−1 − 12 u2t∗ +Bt∗ + 8σ22 2 = (1 + o(1)) (1 − λ) u e (d+1)(d+2) −d 4 2 (2π)   −1 Z 1 |w−µ20 µ22 z|2 1+o(1) y > y+ +z > µ−1 −2(1−λ)uAτ − 2 det(Γ)− 2 −1 22 z 1−µ20 µ22 µ02 × dAτ dydz γu (uσAτ )e (d+1)(d+2) 4 Aτ ≥0,L (2π) 2

= O(1)(1 − λ)−1 u−1 · ud−2 e−ut∗ .

(4.41)

According to the asymptotic form of v(b) and with ρ2 = 1 − λ = 1/ log log b, we have that   2 1 ; E , L, A ≥ 0, ı = 1 = O(1)ρ2 (1 − λ)−1 u−1 · ud−2 e−ut∗ EQ b τ 2 [(1 − ρ1 − ρ2 )K] = o(1)v 2 (b).

(4.42)

Therefore, combining the results in (4.40) and (4.42), we have the first term in (4.35) is bounded by (1 + 2ε)v 2 (b). The last step is to show that the second term of (4.35) is of a smaller order of v 2 (b). First, we split the expectation   1 ; E , L, A < 0 (4.43) EQ b τ 2 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]   1 Q = E 2 ; Eb , L, Aτ < 0, ı = 1 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]   1 +E Q ; E , L, −η/u < A < 0, ı = 0 b τ τ 2 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]   1 +E Q ; E , L, A ≤ −η/u , ı = 0 . b τ τ 2 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

33

We study these three terms one by one. Let  1 (4.44) γ1,u (x) = E  2 ; IC1 · ρ1 κ1,2 + IC2 · (1 − ρ1 − ρ2 )(1 − λ)−d/2 κ2,1  x > (1 + o(u−1−δ0 /4 ))[ξu + o(u−δ0 /8 )] ı, τ, w, y, z We start with the first expectation in (4.43). Plugging in the lower bound for (1 − ρ1 − ρ2 )K + ρ1 K1 derived in (4.34), we have   1 Q ı = 1, τ E (4.45) ; E , L, A < 0 b τ 2 [(1 − ρ1 − ρ2 )K + ρ1 K1 ]   −1 Z |w−µ20 µ22 z|2 +z > µ−1 z −2(1+λ1 )uAτ − 21 y > y+ −1 22 d−2 −u2t∗ 1−µ20 µ22 µ02 = O(1)u e dAτ dydz. γ1,u (uσAτ )e Aτ −u3/2+ . By Lemma 24, for −u3/2+ < x < 0, there exists a constant δ ∗ > 0 such that   ∗ 1 −1−δ0 /4 −δ0 /8 uδ x E 2 2 ; x > (1 + o(u ))[ξu + o(u )] ı, τ, w, y, z, C1 = O(1)ρ−2 . 1 e ρ1 κ1,2 and  E

 1 −1−δ0 /4 −δ0 /8 ı, τ, w, y, z, C ; x > (1 + o(u ))[ξ + o(u )] 2 u (1 − ρ1 − ρ2 )2 κ22,1

= O(1)(1 − ρ1 − ρ2 )−2 (1 − λ)−d eu

δ∗

x

.

Therefore, the above approximations and the dominated convergence theorem imply that conditional on L, Z ∗ −2d γ1,u (uσAτ )e−2(1+λ1 )uAτ dAτ = O(1) · max{ρ−2 } · u−1−δ . 1 , (1 − λ) Aτ 0 small enough, we consider 0 the case when τ ∈ {t : |t − τ | ≤ u−1/2+δ } ⊂ T and otherwise. For the first situation, τ is “far away” from the boundary of T , which is the important case, the derivation is same as that of the case where µ(t) is not a 0 constant. For the case in which τ is within u−1/2+δ distance from the boundary of T , the contribution of the boundary case is o(v 2 (b)). An intuitive interpretation is that the important region of the integral I(T ) might be cut off by the boundary of T . Therefore, in cases that τ is too close to the boundary, the tail I(T ) is not heavier than that of the interior case. The rigorous analysis is basically repeating the Parts 1, 2, and 3 on a truncated region. Therefore, we omit the details.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

35

5. Proof of Theorem 7. The proof of Theorem 7 is analogous to that of Theorem 3. According to Lemma 18, we focus on the set (for some small 0 > 0) L∗ = L ∩ {

sup

g(t) − 0 u|t|2 < 0}.

(5.1)

|t−τ |>2u−1/2+

A similar three-part procedure is applied here. In Part 1, using the transformation from f to the process f∗ , we have   Bt 1> f¯t00 βu (T ) = sup f (t) + + + µσ (t) 2σut ut t∈T   1> (zt − ut µ02 + uτ µ2 (t − τ )) Bt + + µσ (t) , = sup f∗ (t) + uτ C(t − τ ) + 2σut ut t∈T We insert the expansions in (4.10), (4.11) and (4.12) into the expression of βu (T ) and obtain that βu (T ) equals  1 sup w + y > (t − τ ) + (t − τ )> z(t − τ ) + Rf (t − τ ) + g(t − τ ) 2 t∈T   1 +uτ 1 − (t − τ )> (t − τ ) + C4 (t − τ ) + RC (t − τ ) 2 1 +µσ (τ ) + ∂µσ (τ )> (t − τ ) + (t − τ )> ∆µσ (τ )(t − τ ) + σ −1 Rµ (t − τ ) 2  1> (zt − ut µ02 + uτ µ2 (t − τ )) Bt + + 2σut ut 
1 > 1 ˜)−1 y˜ − (t − τ − (uI − z ˜)−1 y˜)> (uI − z ˜) t − τ − (uI − z ˜)−1 y˜ = sup u + w + y˜ (uI − z 2 2 t∈T  1> (zt − ut µ02 + uτ µ2 (t − τ )) Bt +uτ C4 (t − τ ) + R (t − τ ) + g(t − τ ) + + . 2σut ut Note that the above display is approximately a quadratic function of t − τ and is ˜)−1 y˜. In addition, on the set L∗ , we maximized approximately at t − τ = (uI − z −1/2+ have that |τ − t∗ | < 2u and thus y˜ = y + O(u−1/2+ ). Therefore, on the set L∗ , we have the following approximation of βu (T ) Aτ +

inf

|t−τ | u ≤ (1 + ε)v 2 (b). EQ dQ Additionally, Lemma 18 provides an approximation that P (βu (T ) > u) ∼ v(b). Thus, we use Lemma 13 (presented at the beginning of Section 4) and complete the proof.

g(t).

36

LIU AND XU

6. Proof of Theorem 10. For the bias control, we need the following result ([42]). Proposition 15. Suppose that conditions RC1-6 are satisfied. Let F 0 (x) be the probability density function of log I(T ) = log T eσf (t)+µ(t) dt. Then the following approximation holds as x → ∞ F 0 (x) ∼ σ −2 x · v(ex ). Thus, for any small ε, P (b < I(T ) < b(1 + ε/ log b) | I(T ) > b) = Θ(ε).

(6.1)

Similar to the log-normal distribution, the overshoot of I(T ) is Θ(b/ log b). Note that |vM (b) − v(b)| ≤ P (I(T ) > b, IM (T ) < b) + P (I(T ) < b, IM (T ) > b). Let Lε = {sup |∂f (t)| ≤ 2(1 − u−2 log ε)u}. t∈T

Note that ∂f (t) is a d-dimensional Gaussian process. Using Borel-TIS lemma, we obtain that P (Lcε ) = o(1)ε · v(b). Therefore, it is sufficient to control P (I(T ) > b, IM (T ) < b, Lε ) and P (I(T ) < b, IM (T ) > b, Lε ). By the definition of IM in (3.23), there exists a constant c1 > 0 such that ∆ = |I(T ) − IM (T )|

≤

M Z X i=1

≤

TN (ti )

eσf (t)+µ(t) dt − mes(TN (ti )) · eσf (ti )+µ(ti )

c1 min{IM (T ), I(T )} · sup |∂f (t)|/N. t∈T

Then we have, on the set Lε , ∆ ≤ 2c1 min{IM (T ), I(T )}(1 − u−2 log ε)u/N , which implies that b(1 + 2(1 − u−2 log ε)u/N )) u(1 − u−2 log ε) log b v(b). = O(1) N

P (I(T ) > b, IM (T ) < b, Lε ) ≤ P (b < I(T )
b, Lε ) is completely analogous.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

37

7. Proof of Theorem 11. The proof of Theorem 11 is similar to that of Theorem 3. Therefore, we only layout the key steps. The only difference is that we replace the integral by a finite sum over TN . Recall that the proof of Theorem 3 consists of three parts: first, we write the event {IM (T ) > b} as a function of (w, y, z) (with an ignorable correction term); second, we write the likelihood ratio as a function of (w, y, z) (with an ignorable correction term); third, we integrate the likelihood ratio with respect to (ı, τ, w, y, z). For the current proof, we also have similar 3 parts. Part 1. For the first step in the proof of Theorem 3, we write I(T ) > b if and only > u−1 σ −1 ξu . With the current discretization size, as proved if Aτ + o(|w|+|y|+|z|+1) u1+δ0 /4 in Theorem 10, log I(T ) − log IM (T ) = o(u−1−ε0 /2 ). +o(u−1−ε0 /2 ) > Thus, we reach the same result that IM (T ) > b if Aτ + o(|w|+|y|+|z|) u1+δ0 /4 −1 −1 u σ ξu . Part 2. Consider the likelihood ratio  Z  ρ2 dQ = (1 − ρ1 − ρ2 )l(t)LR(t) + ρ1 l(t)LR1 (t) + LR2 (t) dt. dP mes(T ) T Under the discretization setup, we have M M M X ρ1 X 1 dQM 1 − ρ1 − ρ2 X l(ti )LR(ti ) + l(ti )LR1 (ti ) + ρ2 = LR2 (ti ), dP κ κ i=1 M i=1 i=1

which is a discrete approximation of dQ/dP . In the proof of Theorem 3, after taking all the terms not consisting of t out of the integral (such as that in (4.23)), the discrete sum is essentially approximating the following integral Z > (1−λ)(uτ +ζu ) (t−τ −(uI−z˜ )−1 y˜) (uI−z˜ )(t−τ −(uI−z˜ )−1 y˜) dt. 2 e− |t−τ | u2 , the discretized likelihood ratio in dQM /dP approximate dQ/dP up to a constant in the sense that dQM dQ = Θ(1) . (7.1) dP dP Part 3. With the results of Parts 1 and 2, the analysis of Part 3 is completely analogous to Part 3 in the proof of Theorem 3. Thus, we conclude that ˜ 2b ) ≤ κ1 v(b)2 , E QM (L where the constant κ1 depends on the Θ(1) in (7.1).

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LIU AND XU

8. Appendix: The Lemmas. In this section, we state all the lemmas used in the previous sections. To facilitate reading, we move several lengthy proofs (Lemmas 17, 18, 20, 22, 23, and 24) to the supplemental materials, as those proofs are not particularly related to the proof of the theorems and mostly involve tedious elementary algebra. The first lemma is known as the Borel-TIS lemma, which was proved independently by [19, 53]. Lemma 16 (Borel-TIS). Let f (t), t ∈ U, U is a parameter set, be a mean zero Gaussian random field. f is almost surely bounded on U. Then, E(supU f (t)) < ∞, and   2   − b2 P max f (t) − E max f (t) ≥ b ≤ e 2σU , t∈U

where

σU2

t∈U

= maxt∈U V ar[f (t)].

Lemma 17.

Conditional on the set L as defined in (4.13), we have that  2  dP Q c E ; I(T ) > b, L = o(1)v 2 (b). dQ

On the set L∗ as defined in (5.1), we have that for k = 1 and 2  k  dP k EQ ; βu (T ) > u, Lc∗ = o(1)P (βu (T ) > u) . dQ

Lemma 18.

In addition, we have the approximation P (βu (T ) > u) ∼ v(b). Lemma 19. Let ξu be as defined in (4.19), then there exist small constants δ ∗ , λ0 , λ00 > 0 such that for all x > 0 and sufficiently large u 0

P (|ξu | > x) ≤ e−λ u

δ∗

x2

00

+ e−λ

u2

.

Proof of Lemma 19. For δ < δ0 /10, we split the expectation into two parts ˜ ≤ uδ } and {|S| ˜ > uδ , τ + (uI − z)−1/2 S˜ ∈ T }. Note that |S| ≤ κuδ and g(t) is a {|S| mean zero Gaussian random field with V ar(g(t)) = O(|t|4+δ0 ). A direct application of the Borel-TIS inequality (Lemma 16) yields the result of this lemma.  Lemma 20. Let S be a random variable taking values in s : (uI − z)−1/2 s + τ ∈ T with density proportional to e− 2 (s−(uI−z) σ

−1/2

y˜)

>

(s−(uI−z)−1/2 y˜) .

If |y| ≤ u1/2+ and |z| ≤ u1/2+ and   δ0 , then     1  1  σ(u−µσ (τ ))C4 (uI−z)− 2 S +σR (uI−z)− 2 S log Ee =

1 X 4 o (|w| + |y| + |z| + 1) ∂ C(0) + , 8σu i iiii u1+δ0 /4

where the expectation is taken with respect to S.

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

39

Lemma 21. 1 log(det(I − u−1 z)) = −u−1 T r(z) + u−2 I2 (z) + o(u−2 ), 2 Pd where T r is the trace of a matrix, I2 (z) = i=1 λ2i , and λi ’s are the eigenvalues of z. Proof of Lemma 21. The result is immediate by noting that det(I − u−1 z) = Qd Pd i=1 (1 − λi /u), and T r(z) = i=1 λi . On the set L, I2 defined as in (4.21) can be written as   ut∗ (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) u2t exp + × ut 2 2 A∗ ,|t−τ | µ22 1 > ×e(1−λ)ut wt +uτ C(t−τ )−ut + 2σ 1 zt −µ02 ut +µ2 (t−τ )uτ −λBt − 8σ2
2

−1 wt +uτ C(t−τ )−ut −2 wt +uτ C(t−τ )−ut µ20 µ22 zt −µ02 ut +µ2 (t−τ )uτ
−1 2 1−µ20 µ22 µ02 ×e dt.

Lemma 22. Z

Lemma 23.

For η = 1/ log log b, on the set L, if Aτ ≥ 0, then 0

{|t − τ | ≤ u−1+δ } ⊆ A∗ . Lemma 24. x < 0, E



E



1

;x ρ21 κ21,2

On the set L, there exists some δ ∗ > 0 such that for all −u3/2+ < ∗  uδ x , > (1 + o(u−1−δ0 /4 ))[ξu + o(u−δ0 /8 )] ı, τ, w, y, z, C1 = O(1)ρ−2 1 e

 1 ; x > (1 + o(u−1−δ0 /4 ))[ξu + o(u−δ0 /8 )] ı, τ, w, y, z, C2 2 2 (1 − ρ1 − ρ2 ) κ2,1

= O(1)(1 − ρ1 − ρ2 )−2 (1 − λ)−d eu

δ∗

x

,

where C1 = {mes(Ac ∩ D) ≥ mes(A ∩ D)} and C2 = {mes(Ac ∩ D) < mes(A ∩ D)}. References. [1] R.J. Adler. The Geometry of Random Fields. Wiley, Chichester, U.K.; New York, U.S.A., 1981. [2] R.J. Adler, J.H. Blanchet, and J. Liu. Efficient monte carlo for large excursions of gaussian random fields. Annals of Applied Probability, 22(3):1167–1214, 2012. [3] R.J. Adler, G. Samorodnitsky, and J.E. Taylor. High level excursion set geometry for nongaussian infinitely divisible random fields. preprint, 2009. [4] R.J. Adler and J.E. Taylor. Random fields and geometry. Springer, 2007. [5] R.J. Adler, J.E. Taylor, and K.J. Worsley. Applications of Random Fields and Geometry: Foundations and Case Studies. preprint, 2009. [6] S.M. Ahsan. Portfolio selection in a lognormal securities market. Zeitschrift Fur Nationalokonomie-Journal of Economics, 38(1-2):105–118, 1978. [7] S. Asmussen. Ruin Probabilities. World Scientific, London, UK, 2000.

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of Statistics Series A, 32(Dec):369–378, 1970. [36] M.R. Leadbetter, G. Lindgren, and H. Rootz´ en. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, N.Y., 1983. [37] M. Ledoux and M. Talagrand. Probability in banach spaces : isoperimetry and processes. 1991. [38] G. Lindgren. Some properties of a normal process near a local maximum. Ann. Math. Statist., 41:1870–1883, 1970. [39] G. Lindgren. Prediction of level crossings for normal processes containing deterministic components. Adv. Appl. Prob., 11:93–117, 1979. [40] J. Liu. Tail approximations of integrals of gaussian random fields. Annals of Probability, 40:1069–1104, 2012. [41] J. Liu and G. Xu. Some asymptotic results of gaussian random fields with varying mean functions and the associated processes. Annals of Statistics, 40:262–293, 2012. [42] J. Liu and G. Xu. On the density functions of integrals of gaussian random fields. Advances in Applied Probability, accepted, 2013. [43] M. B. Marcus and L. A. Shepp. Continuity of gaussian processes. Transactions of the American Mathematical Society, 151(2), 1970. [44] M. Mitzenmacher and E. Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005. [45] Y. Nardi, D. O. Siegmund, and B. Yakir. The distribution of maxima of approximately gaussian random fields. Annals of Statistics, 36(3):1375–1403, 2008. [46] V. I. Piterbarg. Asymptotic methods in the theory of Gaussian processes and fields. American Mathematical Society, Providence, R.I., 1996. [47] Aberg S. and Guttorp P. Distribution of the maximum in air pollution fields. Environmetrics, 19:183–208, 2008. [48] V.N. Sudakov and B.S. Tsirelson. Extremal properties of half spaces for spherically invariant measures. Zap. Nauchn. Sem. LOMI, 45:75–82, 1974. [49] J. Y. Sun. Tail probabilities of the maxima of gaussian random-fields. Annals of Probability, 21(1):34–71, 1993. [50] M. Talagrand. Majorizing measures: The generic chaining. Annals of Probability, 24(3):1049– 1103, 1996. [51] J. Taylor, A. Takemura, and R. J. Adler. Validity of the expected euler characteristic heuristic. Annals of Probability, 33(4):1362–1396, 2005. [52] J. Traub, G. Wasilokowski, and H. Wozniakowski. Information-Based Complexity. Academic Press, New York, NY, 1988. [53] B.S. Tsirelson, I.A. Ibragimov, and V.N. Sudakov. Norms of Gaussian sample functions. Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), 550:20–41, 1976. [54] H. Wozniakowski. Computational complexity of continuous problems, 1996. Technical Report. [55] M. Yor. On some exponential functionals of brownian-motion. Advances in Applied Probability, 24(3):509–531, 1992. [56] S. L. Zeger. A regression-model for time-series of counts. Biometrika, 75(4):621–629, 1988.

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Supplementary materials: Proof of the Lemmas

Proof of Proposition 14. The proof of this proposition basically replicates that of Theorem 3.4 in [41] under slightly bigger remainder terms of the Taylor expansions. Therefore, we only layout the key steps. We consider the following change of measure Z 1 dQ2 , LR2 (t)dt, dP T mes(T ) where LR2 is defined in (3.11). Define another set  L1 = |f (τ ) − u| < u1/2+ , |y| < u1/2+ , |z| < u1/2+ ,

sup

 |g(t)| < u−1−δ . (8.1)

|t−τ | b, Lc1 ) = E Q2 ; I(T ) > b, Lc1 = o(1)v(b). dQ2 In what follows, we approximate E Q2 [dP /dQ2 ; I(T ) > b, L1 ]. Note that for τ distributed on T ,   dP P (I(T ) > b) = E Q2 ; I(T ) > b, L1 dQ2  Z −1 Z 1 2 ; I(T ) > b, L1 = E Q2 e− 2 ut +[wt +uτ C(t−τ )]ut dt T

T

uniformly

 τ dτ. 0

For a given δ 0 > 0 small enough, we consider two cases for τ : first, {t : |t−τ | ≤ u−1/2+δ } ⊂ T and otherwise. For the first situation, the derivation for the above expectation takes a similar three-step procedure as that in the proof of Theorem 3. We only state the key steps here. Part 1. On the set L1 and for a given τ , following Part 1 in the proof of Theorem 3, in particular results in (4.20), we have I(T ) > b if and only if A0 > u−1 σ −1 ξu , where A0 = w +

1 1 X 4 1 > ˜)−1 y˜ − ˜) + 2 y˜ (uI − z log det(I − u−1 z ∂iiii C(0) + u−1−δ0 /4 o(|w| + |y| + |z|). 2 2σ 8σ u i

Part 2. Let λu = u−1/2+δ ,  < δ < δ 0 . We first split the integral into two parts, that is, Z Z 2 1 2 1 2 e− 2 ut +[wt +uτ C(t−τ )]ut dt = e−u /2+ut [wt +uτ C(t−τ )+µσ (t)]+ 2 µσ (t) dt T ZT Z = ... + ... |t−τ |λu

I2 + I3 .

For I2 term, note that |t − τ | ≤ λu = u−1/2+δ . We insert the Taylor expansion of µσ (t) Z 2 2 I2 = (1 + o(1))eu /2−uµσ (τ )+µσ (τ )/2 eut [wt +uτ C(t−τ )−u+µσ (τ )] dt. |t−τ | b, L1 ]. This part is completely analogous to Part 3 of the proof of Theorem 3.   dP E Q2 ; σuA0 > ξu , L1 τ dQ2   1 Q2 0 = mes(T )E ; σuA > ξu , L1 τ I2 + I3   Z 1 = mes(T ) E ; σuA0 > ξu τ, w, y, z h(w, y, z)dwdydz, I2 + I3 L

.

44

LIU AND XU


where h is the density function of f (τ ), ∂f (τ ), ∂ 2 f (τ ) under the measure P . Plugging in (8.2) and h(w, y, z), a similar derivation as in Part 3 in the proof of Theorem 3 yields that mes(T )−1 E Q2 [dP /dQ2 ; I(T ) > b, L1 |τ ] equals 1 T r(µ (τ )I+∆µ (τ ))+|∂µ (τ )|2 + (u−µσ (τ ))2 + 2σ −1 σ σ σ 2

(1 + o(1))ud−1 e Z |Γ|−1/2 (2π)

(d+1)(d+2) 4

1> µ22 1 + 12 8σ 2 8σ

P

i

4 C(0) ∂iiii

2    −1/2 |µ20 µ−1 ˜|2 1 1/2 1 22 z µ exp − + z ˜ − µ d˜ z. 22 22 −1 2 1 − µ20 µ22 µ02 2σ z ˜∈Rd(d+1)/2

Note that τ is uniformly distributed over T and thus we conclude the situation when τ is 0 at least u−1/2+δ away from the boundary. 0 For the case in which τ is within u−1/2+δ distance from the boundary of T , we need to follow a similar derivation as the proof of Theorem 3.4 in [41]. We do not provide the R details. An intuitive interpretation is that the important region of the integral ef (t) dt might be cut off by R the boundary of T . Therefore, in cases that τ is too close to the boundary, the tail ef (t) dt is not heavier than that of the interior case. In fact, for the purpose of proving our main theorem, we only need to establish a lower bound of v(b), thus, the boundary case can be in fact omitted. Thereby, we conclude the proof. Proof of Lemma 17. Define another set  L1 = |f (τ ) − u| < u1/2+ , |y| < u1/2+ , |z| < u1/2+ ,

sup

|g(t)| < u

−1−δ

 . (8.3)

|t−τ | b, Lc1 = o(1)v(b)2 . T

In this proof, we mainly use the last component of the mixture LR R 2 (t) that has not been used in the main proof of the theorem. Note that dQ/dP ≥ ρ2 T LR2 (t)/mes(T )dt and therefore  2 Z  dP (8.4) ; eµ(t)+σf (t) dt > b, Lc1 EQ dQ T  Z −2  1 ≤ EQ ρ2 LR2 (t)dt ; I(T ) > b, Lc1 mes(T ) T   −2 Z I(T ) 1 Q 1 b −1 c u2 +ut f (t) 2 t = E e dt ; > , L 1 . ρ22 mes(T ) T mes(T ) mes(T ) On the set

1 mes(T )

Z

eσ(µσ (t)+f (t)) dt = T

I(T ) b > mes(T ) mes(T )

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

45

we have for large b, Z 1 2 1 e− 2 ut +ut f (t) dt (8.5) mes(T ) T Z 1 2 1 2 1 e(u−µσ (t))·(f (t)+µσ (t))+ 2 µσ (t) dt = e− 2 u × mes(T ) T Z 2 1 2 1 1 ≥ e− 2 u + 2 mint∈T µσ (t) × × e(u−maxt∈T µσ (t))(f (t)+µσ (t)) dt mes(T ) T ∩{f (t)+µσ (t)≥0}  Z + e(u−mint∈T µσ (t))(f (t)+µσ (t)) dt T ∩{f (t)+µσ (t) 0 and thus  2  dP −u2 − ω u1+2 2 ; I(T ) > b, Lc1 = o(1)ρ−2 = o(1)v 2 (b). EQ 2 e dQ

for some

We now proceed to bound for the rest of the expectation, that is,  2  dP EQ ; I(T ) > b, Lc ∩ L1 . dQ This requires some fine analysis. Similar to the proof on the set L1 , we still focus on the last component of the likelihood ratio corresponding to LR2 . Let L2 = Lc ∩ L1 . Since L2 ⊂ L1 , the rest of the derivations are on the set L1 . Note that  2   Z −2  dP Q −1 u2 +[wt +uτ C(t−τ )]ut 2 t ; I(T ) > b, L2 ≤ mes2 (T )ρ−2 EQ E e dt ; I(T ) > b, L 2 . 2 dQ T The derivation for the above expectation takes a similar three-step procedure as that in the proof of Theorem 3. We only state the key steps here. Part 1. On the set L1 and for a given τ , following Part 1 in the proof of Theorem 3, in particular results in (4.20), we have I(T ) > b if and only if A0 > u−1 σ −1 ξu ,

46

LIU AND XU

1 ˜)−1 y˜ − 2σ ˜) + 8σ12 u where A0 = w + 12 y˜> (uI − z log det(I − u−1 z |y| + |z|). Part 2. Let λu = u−1/2+δ and we have that Z 1 2 e− 2 ut +[wt +uτ C(t−τ )]ut dt T Z 2 1 2 eut [wt +uτ C(t−τ )+µσ (t)]+ 2 µσ (t) dt ≥ e−u /2

P

i

4 ∂iiii C(0) + u−1 o(|w| +

|t−τ | b, L2 = o(1)v 2 (b). T

Thereby, we conclude the proof. Proof of Lemma 18. The proof of Lemma 18 is analogous to that of Lemma 17. We only show the key steps here. Consider the set   L3 = sup |f (t)| < M u, sup |∂f (t)| < M 2 u, sup |∂ 2 f (t)| < M 2 u , t∈T

t∈T

t∈T

where M is some big constant. By the Borel-TIS lemma, we have   dP ; βu (T ) > u, Lc3 = P (βu (T ) > u, Lc3 ) ≤ P (Lc3 ) = o(1)v(b). EQ dQ For the second moment, we have that  2  dP EQ ; βu (T ) > u, Lc3 dQ



= ≤ ≤

 dP ; βu (T ) > u, Lc3 dQ −1   Z 1 2 ; Lc3 E e− 2 ut +ut f (t) dt T h 2 i u +u sup |f (t)| E e ; Lc3 .

We can always choose M large enough such that the term is of order o(v 2 (b)). On the set L3 , let tsup be the maximum of f (t) that is f (tsup ) = supt∈T f (t). Then, there exists constant c1 such that Z Z 1 2 LR2 (t)dt = e− 2 ut +ut f (t) dt T T Z u2 +c1 u 1 2 ≥ e− 2 ut +ut f (t) dt ≥ e− 2 +uf (tsup ) . (8.7) |t−tsup | u, Lc1 ∩ L3 dQ  Z −2  1 EQ ρ2 LR2 (t)dt ; βu (T ) > u, Lc1 ∩ L3 T mes(T )  Z −2  2 mes (T ) Q −1 u2 +ut f (t) c 2 t e E dt ; β (T ) > u, L ∩ L u 3 . 1 ρ22 T EQ

≤ =



(8.8)

Then, on the set L3 , there exists a constant c2 such that −2   Z mes2 (T ) Q −1 u2 +ut f (t) c 2 t e dt ; f (t ) > u − c , L ∩ L (8.8) ≤ E sup 2 3 1 ρ22 T   −2 u2 +c1 u mes2 (T ) Q c − +uf (tsup ) 2 ≤ ; f (tsup ) > u − c2 , L1 ∩ L3 , E e ρ22 where in the last step we plugged in the bound in (8.7). Therefore there exists a constant c3 such that 2 (8.8) ≤ ρ2 −2 e−u +2c3 u log u Q (Lc1 ∩ L3 ) . Then the Borel-TIS Lemma implies that on the set Lc1 , there exists a positive constant $ such that  2  2 1+$ dP EQ ; βu (T ) > u, Lc1 = o(1)ρ2 −2 e−u −u = o(1)v 2 (b). dQ With exactly the same argument, we have that   dP ; βu (T ) > u, Lc1 = o(1)v(b). EQ dQ Now on L1 , with a similar three-step procedure explored in the proof of Lemma 17, we can obtain that    2  dP dP ; βu (T ) > u, Lc∗ = o(1)v 2 (b), E Q ; βu (T ) > u, Lc∗ = o(1)v(b). EQ dQ dQ With a similar derivation as in Theorem 3, we obtain that P (βu (T ) > u) ∼ v(b). The detailed derivation of the above asymptotic approximation is omitted. Proof of Lemma 20. For δ > , we first split the expectation into two parts: h n    oi 1 1 E exp σ(u − µσ (τ ))C4 (uI − z)− 2 S + σR (uI − z)− 2 S h i h i = E ...; |S| ≤ uδ + E ...; |S| > uδ =

J1 + J2 .

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

49

1

Let t = (uI − z)− 2 s. Given that C(t) is a monotone non-increasing function, we have that for all |s| = O(uδ ) >   σ s − (uI − z)−1/2 y˜ s − (uI − z)−1/2 y˜ 2     1 1 +σ(u − µσ (τ ))C4 (uI − z)− 2 s + σR (uI − z)− 2 s

> σ − t − (uI − z)−1/2 y˜ (uI − z) t − (uI − z)−1/2 y˜ + σ(u − µσ (τ ))C4 (t) + σR(t) 2 −λu|t|2 = −λs2 . −

= ≤

Therefore, for λ0 small 0 2δ

J2 = O e−λ

u


.

We now consider the leading term J1 . On the set that |S| ≤ uδ and |w| + |y| + |z| = O(u1/2+ ), we have that for sufficiently small  and δ,

=



1 1 σ(u − µσ (τ ))C4 (uI − z)− 2 S + σR (uI − z)− 2 S

O u−1+4δ + O u(−1/2+δ)(2+δ0 ) u1/2+ = o(u−1/2−δ0 /8 ).

By using Taylor’s expansion twice, we can essentially move the expectation into the exponent and obtain that log J1 equals h     i   1 1 E σ(u − µσ (τ ))C4 (uI − z)− 2 S + σR (uI − z)− 2 S ; |S| ≤ uδ + o u−1−δ0 /4 . Note that (uI − z)−1/2 y˜ = u−1/2 y˜ + O(u−3/2 |˜ y ||z|) and y = y˜ + O(1). Let Z = (Z1 , ..., Zd ) be a multivariate Gaussian random vector with mean zero and covariance function σ −1 I. Then, S is equal in distribution to Z + u−1/2 y˜ + O(u−3/2 |˜ y ||z|). Therefore, for sufficiently small  and δ, we obtain that

=

log J1 h i σ + O(u−1+ ) X 4 ∂ijkl C(0)E (u−1/2 y˜i + Zi )(u−1/2 y˜j + Zj )(u−1/2 y˜k + Zk )(u−1/2 y˜l + Zl ) 24u ijkl

−1−δ0 /4

=

+u o(|w| + |y| + |z| + 1) h i σ −1 X 4 u ∂ijkl C(0)E (u−1/2 y˜i + Zi )(u−1/2 y˜j + Zj )(u−1/2 y˜k + Zk )(u−1/2 y˜l + Zl ) 24 ijkl

+u

−1−δ0 /4

o(|w| + |y| + |z| + 1),

where the expectations are taken with respect to Z. Then

= =

log J1 1 −1 X 4 1 X 4 u 6∂iikl C(0)u−1 y˜k y˜l + ∂iiii C(0) + u−1−δ0 /4 o(|w| + |y| + |z| + 1) 24 8σu iiii iikl 1 X 4 ∂iiii C(0) + u−1−δ0 /4 o(|w| + |y| + |z| + 1). 8σu iiii

50

LIU AND XU

Proof of Lemma 22. This proof only consists of elementary algebra. We expand the exponent in the second row and the second term in the third row of (4.21). Furthermore, we move the first term in the third row to the last row and move the last term in the fourth row up to the third row. Then, we obtain that I2 equals   Z ut∗ (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) exp × ut 0 2 A∗ ,|t−τ | (zt − µ02 ut + µ2 (t − τ )uτ ) 1> µ22 1 − λBt − × exp −λut wt + λut (ut − uτ C(t − τ )) + (1 − λ) 2σ 8σ 2  > −1 −(zt − µ02 ut + µ2 (t − τ )uτ ) µ22 (zt − µ02 ut + µ2 (t − τ )uτ ) × exp 2  (zt + µ2 (t − τ )uτ )> µ−1 (z + µ t 2 (t − τ )uτ ) 22 + 2 ( )
2 2 wt + uτ C(t − τ ) − µ20 µ−1 |µ20 µ−1 22 (zt + µ2 (t − τ )uτ ) 22 (zt − µ02 ut + µ2 (t − τ )uτ )| × exp − dt. 2(1 − µ20 µ−1 2(1 − µ20 µ−1 22 µ02 ) 22 µ02 ) We now work on the exponents in the third and the last row. In particular, we have that −(zt − µ02 ut + µ2 (t − τ )uτ )> µ−1 (zt + µ2 (t − τ )uτ )> µ−1 22 (zt − µ02 ut + µ2 (t − τ )uτ ) 22 (zt + µ2 (t − τ )uτ ) + 2 2
2 2 wt + uτ C(t − τ ) − µ20 µ−1 |µ20 µ−1 22 (zt + µ2 (t − τ )uτ ) 22 (zt − µ02 ut + µ2 (t − τ )uτ )| − + −1 −1 2(1 − µ20 µ22 µ02 ) 2(1 − µ20 µ22 µ02 ) =

2 µ20 µ−1 22 µ02 ut + ut µ20 µ−1 22 (zt − µ02 ut + µ2 (t − τ )uτ ) 2
2 wt + uτ C(t − τ ) − µ20 µ−1 22 µ02 ut + 2(1 − µ20 µ−1 22 µ02 )
−1 2 wt + uτ C(t − τ ) − µ20 µ−1 22 µ02 ut µ20 µ22 (zt − µ02 ut + µ2 (t − τ )uτ ) − . −1 2(1 − µ20 µ22 µ02 )

We further expand the last two rows and obtain that

+

1 2 ut + ut wt + ut (uτ C(t − τ ) − ut ) 2 (wt + uτ C(t − τ ) − ut )2 − 2 (wt + uτ C(t − τ ) − ut ) µ20 µ−1 22 (zt − µ02 ut + µ2 (t − τ )uτ ) . 2(1 − µ20 µ−1 µ ) 02 22

We put it back to the expression of I2 and obtain that I2 equals   Z ut∗ (t − t∗ )> ∆µσ (t∗ )(t − t∗ ) u2 exp + t × ut 0 2 2 A∗ ,|t−τ | 1 (zt − µ02 ut + µ2 (t − τ )uτ ) 1> µ22 1 − λBt − × exp (1 − λ)ut [wt + uτ C(t − τ ) − ut ] + (1 − λ) 2σ 8σ 2   2 −1 (wt + uτ C(t − τ ) − ut ) − 2 (wt + uτ C(t − τ ) − ut ) µ20 µ22 (zt − µ02 ut + µ2 (t − τ )uτ ) × exp dt. 2(1 − µ20 µ−1 22 µ02 )

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

51 0

Proof of Lemma 23. We now consider a particular t ∈ A∗ ∩ {t : |t − τ | < u−1+δ }. 0 On the localization set L and |t − τ | < u−1+δ , we have that |zt − z| = O(u− ),

|zt | = O(u ),

|yt − y| = O(u−1+δ

0

+

).

(8.9)

0

In what follows, we show that, for all t ∈ {t : |t − τ | < u−1+δ }, Aτ > 0 implies t ∈ A∗ . 0 On the set {t : |t − τ | < u−1+δ } and the set L, by definition, t ∈ A∗ if wt +uτ C(t−τ )−ut +

|yt + uτ ∂C(t − τ )|2 1> (zt − µ02 ut + µ2 (t − τ )uτ ) Bt + + > −ηu−1 t . 2ut 2σut ut

We insert the expansions of f∗ (t), C(t − τ ) and ∂C(t − τ ) into the above inequality and obtain that (t − τ )> z(t − τ ) uτ |t − τ |2 + Rf (t − τ ) + g(t − τ ) − + uτ C4 (t − τ ) 2 2 2 2 2 |yt | − 2yt uτ (t − τ ) + uτ |t − τ | + O(|t − τ |3 uτ yt ) +uτ RC (t − τ ) − µσ (τ ) + µσ (t) + 2ut 1> (zt − µ02 ut + µ2 (t − τ )uτ ) Bt + + > −ηu−1 t . 2σut ut w + y > (t − τ ) +

We further insert Aτ in and obtain that >  2   |yt |2 |y|2 O(|t − τ |3 uτ yt ) uτ uτ yt uτ (t − τ ) + |t − τ |2 + Aτ + y − − − + ut 2ut 2 2ut 2uτ 2ut 1> (zt − µ02 ut + µ2 (t)uτ ) Bt Bτ 1> z − + − − µσ (τ ) + µσ (t) ut uτ 2σut 2σuτ 1 + (t − τ )> z(t − τ ) + Rf (t − τ ) + g(t − τ ) + uτ C4 (t − τ ) + uτ RC (t − τ ) 2 > −ηu−1 t . +

0

Thus, on the set L ∩ {t : |t − τ | < u−1+δ }, (8.9) implies that t ∈ A∗ if Aτ + ηu−1 > −g(t − τ ) + O(u−1− ). 0

In addition, sup|t−τ | −ηu−1 + O(u−1− ). t Note that η ∼ 1/ log u. Thus, we have 0

Aτ > 0 ⇒ {t : |t − τ | < u−1+δ } ⊂ A∗ .

Proof of Lemma 24. Note that δ ∗ can be chosen arbitrarily small so that −2u−δ0 /8 < δ∗ x < 0 implies eu x ≥ 1/2. Thus, on the set −2u−δ0 /8 < x < 0, we have

≤

E



E



1 ρ21 κ21,2 1 ρ21 κ21,2

 ; x > (1 + o(u−1−δ0 /4 ))[ξu + o(u−δ0 /8 )] ı, τ, w, y, z, C1 ∗  uδ x ı, τ, w, y, z, C1 = O(1)ρ−2 , 1 e

52

LIU AND XU

and  1 ; x > (1 + o(u−1−δ0 /4 ))[ξu + o(u−δ0 /8 )] ı, τ, w, y, z, C2 (1 − ρ1 − ρ2 )2 κ22,1   δ∗ 1 ı, τ, w, y, z, C2 = O(1)(1 − ρ1 − ρ2 )−2 (1 − λ)−d eu x . E 2 2 (1 − ρ1 − ρ2 ) κ2,1

E ≤



We now proceed to the case x < −2u−δ0 /8 . Under this setting, we have that x > (1 + o(u−1−δ0 /4 ))[ξu + o(u−δ0 /8 )] implies x/2 > ξu . Thus, it is sufficient to provide bounds for E

1



ρ21 κ21,2

 ; x > ξu ı, τ, w, y, z, C1 ,

and

E



 1 ; x > ξu ı, τ, w, y, z, C2 . 2 2 (1 − ρ1 − ρ2 ) κ2,1

To simplify the notation, let τ = 0. For other values of τ , the derivation is completely analogous. We use the notation Ew,y,z,Ci [·] to denote E [·|w, y, z, Ci ] and Pw,y,z to denote the conditional probability given w, y, z. Note that h i −1/2 S) e−ξu /u = E eg((uI−z) , where S is a random variable with density proportional to (4.17). The statement of the lemma considers x < 0. In the proof, we slip the sign and consider 0 < x < u3/2+ . Thus, we have that {−x > ξu } = {A1 + A2 > ex/u }, where A1

=

A2

=

h i −1/2 S) E eg((uI−z) ; |S| ≤ uδ , h i −1/2 S) E eg((uI−z) ; |S| > uδ , (uI − z)−1/2 S ∈ T .

Furthermore, we have that log A1 ≤

sup

g(s),

|s|≤u−1/2+δ

and by the Borell-TIS inequality (Lemma 16), there exists λ0 , δ ∗ > 0 such that ! P

u·

sup

g(s) > x

0 δ∗

≤ e−λ

u

x2

.

|s|≤u−1/2+δ

In addition, on the set L, the process g is localized and   O u·sup|s|≤u−1+δ |g(s)|

κ−2 1,2 = e

−δ

= eO(u

)

.

(8.10)

00

Therefore, for c > 0 there exists λ > 0 such that h i 2δ x/u Ew,y,z,C1 κ−2 , 0 < A2 < e−cu 1,2 ; A1 + A2 > e "   ≤

O u·sup|s|≤u−1+δ |g(s)|

Ew,y,z,C1 e

; u·

sup |s|≤u−1/2+δ

≤

00 δ ∗

e−λ

u

x2

.



g(s) + O u · e

−cu2δ



# >x

53

EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS

Similarly, since   O u·sup|s|≤u−1+δ |g(s)|

−d κ−2 e 2,1 = Θ(1)(1 − λ)

−δ

= Θ(1)(1 − λ)−d eO(u

)

,

(8.11)

we have h i 2δ 00 δ ∗ 2 x/u , 0 < A2 < e−cu = O(1)(1 − λ)−d e−λ u x . Ew,y,z,C2 κ−2 2,1 ; A1 + A2 > e For the remainder terms, consider C1 first i h 2δ x/u Ew,y,z,C1 κ−2 , A2 ≥ e−cu 1,2 ; A1 + A2 > e h i −cu2δ ≤ Ew,y,z,C1 κ−2 1,2 ; A2 ≥ e " # √ 1 2 2δ ; sup ≤ Ew,y,z,C1 κ−2 g(s/ s > −cu . u) − 1,2 √ 2 uδ ≤s≤ u

(8.12)

Since for all 0 < x < u3/2+ there exists λ0 sufficiently small such that   2δ Pw,y,z A2 > e−cu

≤

Pw,y,z

≤

e−λ

0 2

u

√ 1 sup√ g(s/ u) − s2 > −cu2δ 2 uδ ≤s≤ u 0 1/2−

≤ e−λ

u

x

!

,

therefore (8.12)

≤

= =

# √ 1 2 2δ sup√ g(s/ u) − s > −cu e Ew,y,z,C1 2 uδ ≤s≤ u "  #  u 2 O u·sup|s|≤u−1+δ |g(s)| −λ0 u2 2δ e Ew,y,z,C1 e sup g(s) − s > −cu 2 s∈T,|s|>u−1/2+δ "

−λ0 u2

0 2

e−λ

u



−2 κ1,2

−δ

eO(u

)

δ∗

≤ e−u

x

.

Similarly, for C2 is true, by (8.11), the following bound holds. h i 2δ x/u , A2 ≥ e−cu Ew,y,z,C2 κ−2 2,1 ; A1 + A2 > e " # u 2 −λ0 u2 −2 2δ ≤ e Ew,y,z,C2 κ2,1 sup g(s) − s > −cu 2 s∈T,|s|>u−1/2+δ =

0 2

O(1)(1 − λ)−d e−λ

u

δ∗

= O(1)(1 − λ)−d e−u

x

.

Combining the above results, we have the conclusion.

Jingchen Liu Department of Statistics, Columbia University, 1255 Amsterdam, New York, NY E-mail: [email protected]

Gongjun Xu School of Statistics, University of Minnesota, Twin Cities 367 Ford Hall, 224 Church St SE, Minneapolis, MN 55455. E-mail: [email protected]