the concave majorant of brownian motion - Department of Statistics
THE CONCAVE MAJORANT OF BROWNIAN MOTION BY
PIET GROENEBOOM
JULY
1981
THIS RESEARCH IS BASED UPON WORK PARTIALLY SUPPORTED BY THE NATIONAL SCIENCE FOUNDATION UNDER GRANT Mcs-78-09858
DEPARTMENT OF STATISTICS UNIVERSITY OF WASHINGTON SEATTLE) WASHINGTON 98195
THE CONCAVE MAJORANT OF BROWNIAN MOTION by Piet Groeneboom l University of Washington
ABSTRACT Let St be a version of the slope at tiMe t of the concave majorant of Brownian motion on [0,(0). It is shown that the process S ={l/St:t > O} is the inverse of a process with independent increments and that Brownian motion can be generated by the latter process and Brownian excursions ues process success ve renewa an application the limiting distribution of the L2-norm of the slope of the concave majorant of the empirical process is derived.
FOOTNOTE 1. This work was done while the author was on leave from the Mathematical Centre Amsterdam as a Visiting Professor in Statistics at the Department of Statistics, University of Washington.
AMS 1970 subject classifications. Primary 60G17, 60J30, 60J65, Secondary 62E15, 62E20.
Key words and phrases. Concave majorant. Convex minorant. Slope process. Brownian motion. Brownian excursions. Empirical process. Limit theorems.
1
J. Introduction and summary of results.
Let w(t) be standard Brownian motion on [0,(0), and let, for each a> 0, rf a) be the last epoch at which the maximum of w(t) - at is attained, that is: (1.1) T(a)=sup{t>O: w(t)-at = supu> (w(u)-au)}.
°
The T(a) are random variables (finite with probability one) and the process {r l a}: a>O} has the property that T(b)-T(a) is independent of T(C) for all c , a and b such that c2 a> b. Hence, letting T(a) = T(l/a) for all a> 0 and defining T(O) =0, we obtain a process {T(a): a~O} which has independent increments. The sample paths of the process Tare nondecreasing and (a.s.) right continuous .. We shall show that the renewal epochs of this process constitute almost surely a countable set with 0 and (IS its only cluster points and that the number of jumpsinaninterval (a,b), 0 < has a Poisson 00
concave majorant of a Brownian motion sample path has with probability one only finitely many segments in each interval (t ,«}, 0 < t < u < Here and in the sequel the "concave majorant" of a function f denotes the smallest concave function g such that 9 ~ f. The points of time where the slopes of the concave majorant of a Brownian motion sample oath change correspond to renewal epochs of the T process. In section 2 we derive the probability distribution of the process (r(a): a~O} and show that Brownian motion can be decomposed into this nrocess and Brownian excursions between values of T at the renewal eoochs. We use this construction to show that with probability one a Brownian motion sample path has a local maximum at an epoch at which the sloDe of the concave majorant changes. We also derive the marginal density of the slope of the concave majorant of Brownian motion at each fixed time point t >O. In section 3 we study the functional (1. 2) L(a .b) = f ( a , b] C-2 dr- (c ) , 0 < a < b < 00.
00,
which is the squared L2-norm of the slope of the concave majorant of Brownian motion over the interval (T(a),T(b)). We show that L(a,b) is distributed as the random sum of N independent x~ random variables, where t! has a Poisson distribution with parameter log(b/a), or, stated more concisely, L(a,b) ~ x~, with N Poisson(log(b/a)); 'here X6:: 0 (this corresponds to sample paths which have no renewal epoch in the interval (a,b]). The result is used to derive the asymptotic normality of certain statistics based on the concave majorants (or convex minorants) of empirical distribution functions.
2
Theorem 3.2 in section 3 (giving the asymptotic normality the standardized squared L2-norm of the slope of the convex minorant of the empirical process) has first been proved by other methods in Groeneboom &Pyke(198l) ([6]),and section 3 nrovides an alternative aporoach to results of this type. We refer to [6J for further remarks on research that has been done with respect to the convex minorant of empirical processes.
3
2. Concave majorant of Brownian motion and Brownian excursions. We shall first study the structure of the process h(a): a> O} and then describe the decomposition of Brownian motion into the process (2.1)
(r(a): 1(a) = T(l/a},a > 0, 1(0) = O}
and Brownian excursions. The main properties of the T process are summarized in Theorem 2.1. Throughout this paper we shall use the notation (2.2)
x+ =f x, if x.:::O, 0, otherwi se.
THEOREM 2.1. Let, for each a>O, the random variable rf a) be defined by (1.l), that is: T(a) is the 1ast epoch where the maximum of w{ t) - at is atta tned.
(2.3)
faCt) = 2a E(X/lt - a)+ '
where X is a standard normal random variable. Let F be the distribution function (df) degenerate at zero and let, for a> b > 0, the df Ga, b be defined by the density (2.4)
ga,b(u)= 2b(a-b)-1
E«Y-b)~(a-Y)~-Z)~, u>O,
where Y and Z are independent normal random variables with mean zero and variance u- 1. Then, for a>b>O, the random variable T(b)-T(a) has the df (2.5)
(b/a)F+ (1 - b/a)Ga,b'
PROOF. We shall use the following well-known property of Brownian motion: (2.6)
Pr{w(z»az+b for some zt(t
Iw(t 1)=x, w(t 2)=y} 1,t2) = exp{-2(at +b- x)+(at t 2+b-y)/(t 2- 1)}, 1 (see e.g. Hajek s Sidak (l967) , p.183). If t 2=oo in
where 0x-at+M(h) for some z>t+h, given wet) =x, w(t+h) =x+Y(h) and maxt 0,
where (Y,Z)QN«0,0),u-1I). A tedious but straightforward integration shows that f~ha,b(u)dU = 1 - b/a (see the appendix) and (2.5) follows.
0
COROLLPRY 2.1. Let T be the process defined by (2.1), then T is a pure jumo process with independent (non-stationary) increments. The marginal density of T(a) at t > 0 is given by (2.3), with a replaced by l/a. For 0 < a < b < 00, the increment T(b) - T(a) has the distribution function (2.15)
(a/b)F+ (1- a/b)G l/a,l/b' where F and Gl/a,l/b have the same meaning as in (2.5). The number of jumps of in an interval (a,b), O e a e b c», has a Poisson distribution with parameter 1og( b/a) . PROOF. The Markovian structure of the T process follows from the construction in the proof of Theorem 2.1. For example, if 0 < t < t < t and 0 < a < a < a 1 2 3, l 2 3 then the (defective) density Pr{T(a ) £ dtl,T(a2)~ dt is given by l 2,T(a 3)&dt3} ~ 2 E{ (~ 2 8a -31 E( x1- a -l 1) +E{ (a 11-Y1 ) ~( + Yl-a -21) +-Zl}+ a 2 1-Y 2)+~( Y - 1 )+-Z2}+dt 1dt2dt3, 2-a3
T
6
where X1 2N(Ott 1L (Y1tZ1) 2N((OtO)t(t and (Y 2tZ gN((OtOL(t f 1I) . 2-t1)-1I) 2) 3-t2 Likewise we have t with the same notation t
1
Pr(;:(a ) e: dt t t(a 4;; dt 2t t(a ) E dt = 1 1 2) 3 2} ~ -1 ~ 2 _ _ -1 -1 -1 - 4a3 E(x1-a 1 )+E{(a1 -Y 1)+(Y 1-a 2 )+-Zl}+dt 1dt2= Pr{t(a 2)=t(a 3)}Pr{t(a )e dt 1 t t(a ) E dt 1 2 2}dt 1dt2 t and the other cases of equality are treated similarly. It now follows immediately from (2.5) that the t process has independent increments and that the cjis:ri')ution of t(b) - t(a) is oiven by (2.15). Finally, the l as t s t.atsnent follows from the followino (standard) argument. Fix a and b , 0< a and Pr {v( ( a tb)) = k + 1} = f~ Pr {v( ( t t b) )=k}dt (1 - P:'{ t (a )=t ( t ) })
(2.16)
=f~ Pr{v((ttb))=k}(a/t2)dt.
0
It is clear from the construction of T that almost all sample paths of T are right-continuous. The next corollary gives the marginal distributions of slopes of the concave majorant of Brownian motion. /
COROLLARY 2.2. Let St be a (version of a) slope of Brownian motion at time t >0. For almost all St is uniquely determined except at a countable we define St = 1im u -l- tSu' The density of Stt for
9t (a) = 4{It~ ( a It) - ad (a It)} ta > 0 t
(2. 17) where
of the concave majorant Brownian motion sample oaths t set of time points t, where t> c, is given by
ep
is the standard norma l density and ~ = 1 - ~t with ~ the standard
normal df.
7
PROOF. Let a> O,h > 0 and let fa be the density of rf a) {see (2.3)). Then PdSt€{a,a+h)}=Pr{r{a»t}- Pr{r{a+h»t}= h a/aa f~-fa{u)du + o(h),h-+-O. 00 00 00 ) 00 x2/a 2 Thus gt(a) = ft-alda fa(u)du = 2 f t U a(2a - x/lu)<j>{x dxldu = 2f a/ t U t (2a-x/lu)du}o(x)dx = 4f;/t(x/t - at)<j>{x)dx =4{/t<j>(a/t) - at~{a/t)}. Since the process T (and hence also T) has with probability one a finite number of jumps in each interval (a.b), with 0 < a < b < 00, there are in each interval (t,u), with 0 < t < u < 00, only finitely many discontinuities of the slope of the concave majorant of almost all Brownian motion sample paths (the concave majorant of such a Brownian motion sample path consists of finitely many straight pieces in an interval for example by taking the right-hand limits.
0
We note that 9t(a) can be written (2.18) gt(a) = 4/t~(a/t){<j>(a/t)/¢(a/t) - av't l , where <j>{a/t)/~(a/t) denotes the "failure rate" of the standard normal distribution at a/t and a/t denotes the asymptotic failure rate (as a/t-+ oo) . We now consider the decomposition of Brownian motion into the process T and Brownian excursions between values of T at renewal epochs. Durrett, Iglehart and Miller(1977) show that a Brownian excursion is a Brownian bridge conditioned to be positive (or negative). More precisely, let h_ "o - {wo I maXt~ ro. , 11. wo{t) > -h}, h where wo is a (standard) Brownian bridge on [0,1]. Then woo. converges weakly in C([O,l]) to a process {z{t): tE [O,l]}, for any sequence {h such that n} hn..j, O. The limiting process is a nonhomogeneous Markov process with marginal densities
8
(2.19) and transition densities (2.20)
fz(t)lz(s)(y1x) = (nt_s(y-x) - nt_s(Y+X))(1-s)3/2y exp{-y2/(2(1-t))}. .{(1_t)3/2 x exp{-x2/(2(1-s))}}-1,
where (2.21) with ~ the standard normal density. More generally, we can consider excursions on an interval [O,u], which are obtained from the preceding excursions by applying the transformation zu(t) = lu.z(t/u). We shall show that, . bet~::n. t~.o. . successive... reDe\\,al . ~p()c:h~ .Ii.~JIg Ii+1.. Q.f . the . .s.l..ope process, the vertical distance of Brownian motion to the concave majorant behaves as such an excursion, where the scale u is given by the distance between the renewal epochs (taking Ti as the origin for the excursion). By Corollary 2.1 we know that with probability one the renewal epochs of the T process constitute a countable set. Note that the set of slopes of the concave majorant of a Brownian motion sample path is just the set of renewal epochs of the T process (if we define the slopes at points of discontinuity by taking right-hand limits, for example), evaluated at this sample path. We can enumerate the renewal epochs in the following way. Let aD = aO(w) = inf{a > 0: T(a) 2..1}, where w denotes a Brownian motion sample path. Next, number the renewal epochs recursively by taking a'+ i ->0, and a,. - 1 = renewal epoch = renewal epoch following' a., 1 l , precedi ng a., i -< O. ~~e can carry out this enumeration on a set of probabil ity 1 one, since almost surely 0 and are the only cluster points of the (countable) set of renewal epochs of T. The renewal epochs of the slope process are now gi ven by Ti =T(a, ), where i runs through the set of integers. In the sequel we shall restrict our attention to the set of probability one where the enumeration can be carried out, without further mentioning. The value of a Brownian motion sample path w at a renewal epoch Ti is completely determined by the T process, since 00
W(T.)=L . . {(w(T.)-w(T. l))/(T.-1. l)}.(1.-T. 1)= , J.:s.. , J JJ JJ J= L'
.
J .:s..'
a.(w)-16T(a.(w)) J
J
=1[0
,a i
()] w
a-ldT(a), 6T(a) =T(a) - T(a-).
9
, ,
,
Fix t> 0, and let N+(t) = inf{T.: T,. -> t} and N-(t) = sup{T.: T. < t}.
With orobabi1ity one, ~((t) 1
k}
k}+£
-1 Mv Mt ~2 } 2- k EU ~/M + f t/~~ ::>t dt + Eo Using (2.17), we find that E f~ Sidt=~109(V/U), implying that the right-hand side of (3.1)) equals (2/k)lo9 M + e . Thus, for sufficiently large k>O, the right-hand side of (3.5) is smaller than 2£, uniformly in :r(/v) 2 -?t and v. Since f_ Sudu = f( It,/vJc -dT(c), we now obtain the result from ( 3 . 2). a T ( It)
The next lemma gives the corresponding result for Brownian bridge. LEMMA 3.3. Let Su be a version of the slope of the concave majorant of Brownian bridge on [O,lJ at time UE (0,1). Then the df of v-2 k (JtSudu - ~log{v(1-t)/(t(1-v))})/((3/2)log{v(1-t)/(t(1-v))}) 2 tends to a standard normal df, as v(1-t)/(t(l-v)) +00.
14
PROOF. Let {B(t) : tt. [O,l]} be Brownian bridge on [0,1], then {(l+t)B(t/(l+t)) : t.::O} represents Brownian motion on [0,(0) (Ooob's transformation). Hence, there is a 1-1 mapping of concave majorants of sample paths tr- Bf t}, t e [0,1], of Browni an bridge to concave majorants of Brownian motion paths h+(l+t)B(t/(1+t)), t>O. For O
PIET GROENEBOOM
JULY
1981
THIS RESEARCH IS BASED UPON WORK PARTIALLY SUPPORTED BY THE NATIONAL SCIENCE FOUNDATION UNDER GRANT Mcs-78-09858
DEPARTMENT OF STATISTICS UNIVERSITY OF WASHINGTON SEATTLE) WASHINGTON 98195
THE CONCAVE MAJORANT OF BROWNIAN MOTION by Piet Groeneboom l University of Washington
ABSTRACT Let St be a version of the slope at tiMe t of the concave majorant of Brownian motion on [0,(0). It is shown that the process S ={l/St:t > O} is the inverse of a process with independent increments and that Brownian motion can be generated by the latter process and Brownian excursions ues process success ve renewa an application the limiting distribution of the L2-norm of the slope of the concave majorant of the empirical process is derived.
FOOTNOTE 1. This work was done while the author was on leave from the Mathematical Centre Amsterdam as a Visiting Professor in Statistics at the Department of Statistics, University of Washington.
AMS 1970 subject classifications. Primary 60G17, 60J30, 60J65, Secondary 62E15, 62E20.
Key words and phrases. Concave majorant. Convex minorant. Slope process. Brownian motion. Brownian excursions. Empirical process. Limit theorems.
1
J. Introduction and summary of results.
Let w(t) be standard Brownian motion on [0,(0), and let, for each a> 0, rf a) be the last epoch at which the maximum of w(t) - at is attained, that is: (1.1) T(a)=sup{t>O: w(t)-at = supu> (w(u)-au)}.
°
The T(a) are random variables (finite with probability one) and the process {r l a}: a>O} has the property that T(b)-T(a) is independent of T(C) for all c , a and b such that c2 a> b. Hence, letting T(a) = T(l/a) for all a> 0 and defining T(O) =0, we obtain a process {T(a): a~O} which has independent increments. The sample paths of the process Tare nondecreasing and (a.s.) right continuous .. We shall show that the renewal epochs of this process constitute almost surely a countable set with 0 and (IS its only cluster points and that the number of jumpsinaninterval (a,b), 0 < has a Poisson 00
concave majorant of a Brownian motion sample path has with probability one only finitely many segments in each interval (t ,«}, 0 < t < u < Here and in the sequel the "concave majorant" of a function f denotes the smallest concave function g such that 9 ~ f. The points of time where the slopes of the concave majorant of a Brownian motion sample oath change correspond to renewal epochs of the T process. In section 2 we derive the probability distribution of the process (r(a): a~O} and show that Brownian motion can be decomposed into this nrocess and Brownian excursions between values of T at the renewal eoochs. We use this construction to show that with probability one a Brownian motion sample path has a local maximum at an epoch at which the sloDe of the concave majorant changes. We also derive the marginal density of the slope of the concave majorant of Brownian motion at each fixed time point t >O. In section 3 we study the functional (1. 2) L(a .b) = f ( a , b] C-2 dr- (c ) , 0 < a < b < 00.
00,
which is the squared L2-norm of the slope of the concave majorant of Brownian motion over the interval (T(a),T(b)). We show that L(a,b) is distributed as the random sum of N independent x~ random variables, where t! has a Poisson distribution with parameter log(b/a), or, stated more concisely, L(a,b) ~ x~, with N Poisson(log(b/a)); 'here X6:: 0 (this corresponds to sample paths which have no renewal epoch in the interval (a,b]). The result is used to derive the asymptotic normality of certain statistics based on the concave majorants (or convex minorants) of empirical distribution functions.
2
Theorem 3.2 in section 3 (giving the asymptotic normality the standardized squared L2-norm of the slope of the convex minorant of the empirical process) has first been proved by other methods in Groeneboom &Pyke(198l) ([6]),and section 3 nrovides an alternative aporoach to results of this type. We refer to [6J for further remarks on research that has been done with respect to the convex minorant of empirical processes.
3
2. Concave majorant of Brownian motion and Brownian excursions. We shall first study the structure of the process h(a): a> O} and then describe the decomposition of Brownian motion into the process (2.1)
(r(a): 1(a) = T(l/a},a > 0, 1(0) = O}
and Brownian excursions. The main properties of the T process are summarized in Theorem 2.1. Throughout this paper we shall use the notation (2.2)
x+ =f x, if x.:::O, 0, otherwi se.
THEOREM 2.1. Let, for each a>O, the random variable rf a) be defined by (1.l), that is: T(a) is the 1ast epoch where the maximum of w{ t) - at is atta tned.
(2.3)
faCt) = 2a E(X/lt - a)+ '
where X is a standard normal random variable. Let F be the distribution function (df) degenerate at zero and let, for a> b > 0, the df Ga, b be defined by the density (2.4)
ga,b(u)= 2b(a-b)-1
E«Y-b)~(a-Y)~-Z)~, u>O,
where Y and Z are independent normal random variables with mean zero and variance u- 1. Then, for a>b>O, the random variable T(b)-T(a) has the df (2.5)
(b/a)F+ (1 - b/a)Ga,b'
PROOF. We shall use the following well-known property of Brownian motion: (2.6)
Pr{w(z»az+b for some zt(t
Iw(t 1)=x, w(t 2)=y} 1,t2) = exp{-2(at +b- x)+(at t 2+b-y)/(t 2- 1)}, 1 (see e.g. Hajek s Sidak (l967) , p.183). If t 2=oo in
where 0x-at+M(h) for some z>t+h, given wet) =x, w(t+h) =x+Y(h) and maxt 0,
where (Y,Z)QN«0,0),u-1I). A tedious but straightforward integration shows that f~ha,b(u)dU = 1 - b/a (see the appendix) and (2.5) follows.
0
COROLLPRY 2.1. Let T be the process defined by (2.1), then T is a pure jumo process with independent (non-stationary) increments. The marginal density of T(a) at t > 0 is given by (2.3), with a replaced by l/a. For 0 < a < b < 00, the increment T(b) - T(a) has the distribution function (2.15)
(a/b)F+ (1- a/b)G l/a,l/b' where F and Gl/a,l/b have the same meaning as in (2.5). The number of jumps of in an interval (a,b), O e a e b c», has a Poisson distribution with parameter 1og( b/a) . PROOF. The Markovian structure of the T process follows from the construction in the proof of Theorem 2.1. For example, if 0 < t < t < t and 0 < a < a < a 1 2 3, l 2 3 then the (defective) density Pr{T(a ) £ dtl,T(a2)~ dt is given by l 2,T(a 3)&dt3} ~ 2 E{ (~ 2 8a -31 E( x1- a -l 1) +E{ (a 11-Y1 ) ~( + Yl-a -21) +-Zl}+ a 2 1-Y 2)+~( Y - 1 )+-Z2}+dt 1dt2dt3, 2-a3
T
6
where X1 2N(Ott 1L (Y1tZ1) 2N((OtO)t(t and (Y 2tZ gN((OtOL(t f 1I) . 2-t1)-1I) 2) 3-t2 Likewise we have t with the same notation t
1
Pr(;:(a ) e: dt t t(a 4;; dt 2t t(a ) E dt = 1 1 2) 3 2} ~ -1 ~ 2 _ _ -1 -1 -1 - 4a3 E(x1-a 1 )+E{(a1 -Y 1)+(Y 1-a 2 )+-Zl}+dt 1dt2= Pr{t(a 2)=t(a 3)}Pr{t(a )e dt 1 t t(a ) E dt 1 2 2}dt 1dt2 t and the other cases of equality are treated similarly. It now follows immediately from (2.5) that the t process has independent increments and that the cjis:ri')ution of t(b) - t(a) is oiven by (2.15). Finally, the l as t s t.atsnent follows from the followino (standard) argument. Fix a and b , 0< a and Pr {v( ( a tb)) = k + 1} = f~ Pr {v( ( t t b) )=k}dt (1 - P:'{ t (a )=t ( t ) })
(2.16)
=f~ Pr{v((ttb))=k}(a/t2)dt.
0
It is clear from the construction of T that almost all sample paths of T are right-continuous. The next corollary gives the marginal distributions of slopes of the concave majorant of Brownian motion. /
COROLLARY 2.2. Let St be a (version of a) slope of Brownian motion at time t >0. For almost all St is uniquely determined except at a countable we define St = 1im u -l- tSu' The density of Stt for
9t (a) = 4{It~ ( a It) - ad (a It)} ta > 0 t
(2. 17) where
of the concave majorant Brownian motion sample oaths t set of time points t, where t> c, is given by
ep
is the standard norma l density and ~ = 1 - ~t with ~ the standard
normal df.
7
PROOF. Let a> O,h > 0 and let fa be the density of rf a) {see (2.3)). Then PdSt€{a,a+h)}=Pr{r{a»t}- Pr{r{a+h»t}= h a/aa f~-fa{u)du + o(h),h-+-O. 00 00 00 ) 00 x2/a 2 Thus gt(a) = ft-alda fa(u)du = 2 f t U a(2a - x/lu)<j>{x dxldu = 2f a/ t U t (2a-x/lu)du}o(x)dx = 4f;/t(x/t - at)<j>{x)dx =4{/t<j>(a/t) - at~{a/t)}. Since the process T (and hence also T) has with probability one a finite number of jumps in each interval (a.b), with 0 < a < b < 00, there are in each interval (t,u), with 0 < t < u < 00, only finitely many discontinuities of the slope of the concave majorant of almost all Brownian motion sample paths (the concave majorant of such a Brownian motion sample path consists of finitely many straight pieces in an interval for example by taking the right-hand limits.
0
We note that 9t(a) can be written (2.18) gt(a) = 4/t~(a/t){<j>(a/t)/¢(a/t) - av't l , where <j>{a/t)/~(a/t) denotes the "failure rate" of the standard normal distribution at a/t and a/t denotes the asymptotic failure rate (as a/t-+ oo) . We now consider the decomposition of Brownian motion into the process T and Brownian excursions between values of T at renewal epochs. Durrett, Iglehart and Miller(1977) show that a Brownian excursion is a Brownian bridge conditioned to be positive (or negative). More precisely, let h_ "o - {wo I maXt~ ro. , 11. wo{t) > -h}, h where wo is a (standard) Brownian bridge on [0,1]. Then woo. converges weakly in C([O,l]) to a process {z{t): tE [O,l]}, for any sequence {h such that n} hn..j, O. The limiting process is a nonhomogeneous Markov process with marginal densities
8
(2.19) and transition densities (2.20)
fz(t)lz(s)(y1x) = (nt_s(y-x) - nt_s(Y+X))(1-s)3/2y exp{-y2/(2(1-t))}. .{(1_t)3/2 x exp{-x2/(2(1-s))}}-1,
where (2.21) with ~ the standard normal density. More generally, we can consider excursions on an interval [O,u], which are obtained from the preceding excursions by applying the transformation zu(t) = lu.z(t/u). We shall show that, . bet~::n. t~.o. . successive... reDe\\,al . ~p()c:h~ .Ii.~JIg Ii+1.. Q.f . the . .s.l..ope process, the vertical distance of Brownian motion to the concave majorant behaves as such an excursion, where the scale u is given by the distance between the renewal epochs (taking Ti as the origin for the excursion). By Corollary 2.1 we know that with probability one the renewal epochs of the T process constitute a countable set. Note that the set of slopes of the concave majorant of a Brownian motion sample path is just the set of renewal epochs of the T process (if we define the slopes at points of discontinuity by taking right-hand limits, for example), evaluated at this sample path. We can enumerate the renewal epochs in the following way. Let aD = aO(w) = inf{a > 0: T(a) 2..1}, where w denotes a Brownian motion sample path. Next, number the renewal epochs recursively by taking a'+ i ->0, and a,. - 1 = renewal epoch = renewal epoch following' a., 1 l , precedi ng a., i -< O. ~~e can carry out this enumeration on a set of probabil ity 1 one, since almost surely 0 and are the only cluster points of the (countable) set of renewal epochs of T. The renewal epochs of the slope process are now gi ven by Ti =T(a, ), where i runs through the set of integers. In the sequel we shall restrict our attention to the set of probability one where the enumeration can be carried out, without further mentioning. The value of a Brownian motion sample path w at a renewal epoch Ti is completely determined by the T process, since 00
W(T.)=L . . {(w(T.)-w(T. l))/(T.-1. l)}.(1.-T. 1)= , J.:s.. , J JJ JJ J= L'
.
J .:s..'
a.(w)-16T(a.(w)) J
J
=1[0
,a i
()] w
a-ldT(a), 6T(a) =T(a) - T(a-).
9
, ,
,
Fix t> 0, and let N+(t) = inf{T.: T,. -> t} and N-(t) = sup{T.: T. < t}.
With orobabi1ity one, ~((t) 1
k}
k}+£
-1 Mv Mt ~2 } 2- k EU ~/M + f t/~~ ::>t dt + Eo Using (2.17), we find that E f~ Sidt=~109(V/U), implying that the right-hand side of (3.1)) equals (2/k)lo9 M + e . Thus, for sufficiently large k>O, the right-hand side of (3.5) is smaller than 2£, uniformly in :r(/v) 2 -?t and v. Since f_ Sudu = f( It,/vJc -dT(c), we now obtain the result from ( 3 . 2). a T ( It)
The next lemma gives the corresponding result for Brownian bridge. LEMMA 3.3. Let Su be a version of the slope of the concave majorant of Brownian bridge on [O,lJ at time UE (0,1). Then the df of v-2 k (JtSudu - ~log{v(1-t)/(t(1-v))})/((3/2)log{v(1-t)/(t(1-v))}) 2 tends to a standard normal df, as v(1-t)/(t(l-v)) +00.
14
PROOF. Let {B(t) : tt. [O,l]} be Brownian bridge on [0,1], then {(l+t)B(t/(l+t)) : t.::O} represents Brownian motion on [0,(0) (Ooob's transformation). Hence, there is a 1-1 mapping of concave majorants of sample paths tr- Bf t}, t e [0,1], of Browni an bridge to concave majorants of Brownian motion paths h+(l+t)B(t/(1+t)), t>O. For O
