general central limit theorems via negligibility - Department of Statistics
GENERAL CENTRAL LIMIT THEOREMS VIA NEGLIGIBILITY
by Galen
TECHNICAL
R.
Shorack
REPORT
November
Department BOx
University Seattle,
of
No.
338
1998
Statistics
354322
of
Washington
Washington
98195
USA
General Central Limit Theorems Via Negligibility Galen R. Shorack University of Washington
Abstract The central limit theorem (CLT) for the sample mean of iid rv's is known to be equivalent to the asymptotic normality condition (ANC) of Levy. And Levy's ANC is well known to be equivalent to an alternative ANC of Feller. Both are equivalent to a negligibility requirement, considered by O'Brien. More recently, additional equivalences have been developed in terms of the quantile function, by Csorgo , Haeusler and Mason. [Other useful and informative equivalences have been developed. Many depend only on comparisons of areas. Others can be developed in the simpler context of the weak law of large numbers, because the asymptotic normality above is equivalent to appropriately phrased consistency of the sample second moment. Roughly, one can learn all about asymptotic normality by studying equivalences in either of the two simpler settings. Both of these tacks were followed elsewhere.] Here, these two earlier threads are tied together with the CLT problem itself.
o.
Introduction
The central limit theorem (CLT) is known to hold for iid rv's if and only if one has Feller's asymptotic normality condition (ANC) that the partial second moment U(x) == fro,xl y 2 dFlxl(y) is slowly varying at infinity (see condition (2.19) below). Levy's(1937) earlier form of the ANC is the requirement that P(IXI > x)/U(x) -+ 0 as x -+ 00 (see (2.21) below). A refined source for both is Feller(1966, p. 303). Versions of the ANC expressed in terms of the quantile function (qf) are found in S. Csorgo, Haeusler and Mason(1988, 1989), where conditions (2.6), (2.13) and (2.14) figured heavily; they approached CLT's via weighted constructions of empirical processes. Equivalences (2.31)(a) and/or (b) can be found in O'Brien(1980), S. Csorgo and Mason(1989) and Gine and Gotze(1995). Feller (1966, p.233) established a condition equivalent to the consistency of the sam pie mean of non negative observations. When this condition is rephrased in terms of the consistency of the sample mean X~ of the squared values (with all Xf ;::: 0, always) it is seen to be the Levy ANC of (2.21). Maller(1979) also obtained the result of Feller's problem, and discussed the X 2 version of it in Maller(1980). See also Raikov(1938). Thus the ANC can be studied in the context of this simpler WLLN problem, and this is therefore a preferable context in which to study and develop equivalences. Shorack(1998b) was written in a way that emphasizes this. This problem of establishing useful equivalences could be made harder, just by examining it in the context of the CLT and/or in the context of the general theory of slowly varying functions. But in fact, many equivalences follow in an elementary fashion from simple pictures and application of Cauchy-Schwarz. This was done systematically in Shorack(1998a), where many equivalences are enunciated. The net result of the various authors is that many conditions equivalent to the ANC are now known, some are in the df domain, some are in the qf domain, some are probabilistic in nature, and some are purely geometric in nature. Then, using a few equivalences, they can all be equated to the CLT. This is part of what will be done herein. But this paper really begins with consideration of row independent triangular arrays. Useful theorems are developed in a qf format. Via Winsorization, the Berry-Esseen theorem is made to play an unuually large role. In Section 2 the earlier work on equivalences is used to quickly deal with necessary and sufficient conditions in the iid case. The key step is to apply the usual proof of the converse half of the Lindeberg-Feller theorem in qf notation. Theorem 0.1 below expresses one very nice resulting ANC equivalence. In Section 3 we show how the standard weak and strong bootstrap results follow. In Section 4 we obtain a universal uniform studentized CLT, as well as its bootstrap version; the key step in the formulation is to Winsorize a slowly growing fraction of observations. (Trimming such slowly growing fractions is a key part of S. Csorgo , Haeusler and .)
1.
Winsorization and Truncation
Weak Negligibility. Let X n1, ... , X nn be independent with df's Fn1, ... , . Let B > 0 be , X()n] to be the smallest dosed and symmetric interval to given. Define X()n by requiring which t; == I:~ Fnkl n assigns probability at least 1 - Oln. Let Pn(x) == I:~ P(IXnkl > x)/n denote the average tail probability, and then let K n denote the qf of the df 1 - r, (.). Note the quantile relationship X()n = K n(1- Oln). For any E > 0, let P~k == P(IXnkl > E). Now the maximum J'v[n == [max1S:k~n IXnkl] satisfies 1 - exp( - I:~ P~k) ::; P(Mn > E)
=1-
n~ (1 - P~k) ::; I:~ P~k'
[The equality is simple, and the bounds follow from it using first 1 - x ::; exp( -x) and then
ni A k == [UI Ak]C; see Loeve(1955, p. 316).] This gives the standard result that (1.1)
M; -+p 0
if and only if
nPn(E) = I:~ P~k -+ 0 for all
E
> O.
Suppose 0 < E ::; 1 and 0 > 0 are fixed, and that we are considering all n exceeding some Since K n(1- ()In) = inf{x : 1- Pn(x) ~ 1- Bin} = inf{x: Fn(x) ::; ()In}, we have
n€,()'
if and only if Thus (1.2) gives (1.3)
u; -+p 0
if and only if
.T()n
= K n (1 - B) -+ 0
for all () > 0
°
o
Weak negligibility in the CLT context. Now let ./Ynk denote Xnk Winsorized outside [-X()on, X()on] (often, 00 == 1); and let iink and a;k denote the resulting means and variances, and then set tln == I:~ iinkln and a; == I:~ a~kln. Applying the previous paragraph to the rv's IXnkllJn an, whose () In-th quantile is X()n/ vn an, gives the equivalence (
1.4 )
a.•
()
= max
L1" lVl n
l~k~n
IXnkl r:::; _ -+p 0 V
n a.,
Of an d on lv if y 1·
1
(b).o V
X()n_ -'-{ r:::;. n (Tn
0 f or au> II {} 0 .
When (1.4)(a)(b) holds then [maxklXnk - tlnkl]/vnan -+p 0 also (as all iink E [-X()on,X()on), and so the triangle inequality establishes this). If M n --7p 0, then the terms are called weakly negligible. If max, P(IXnkl]/vn an > c) -+ 0 for all E > 0, then the terms are called uniformly asymptotically neglible (abbreviated as uan). Let == vn[./Yn - iin]/an. . For 0 < o ::; 00 let X~k denote x.; Winsorized outside XBn]; and let ii~k and a~k denote the resulting means and stan~ard deviations,_ and define both == I:~ ii~kln == I:~(a~k)2In. X~k/n, let Z()n == third rplit.l"::l.l moments 13
z;
o
Proof.
Set
eo = 1.
Consider (i). (Let a
= b c mean that la bl S; c.] Observe that P(L~ x., =1= L~ .'Y~k)) .
O.
Let n(a n 1\ a~) ---t
0 00.
Then uniformly in F n :
9/vn(an I\a~) ---t O.
---t O.
11 ---t O.
Also, every non degenerate df F is eventually in all further F n if we also require (an V a~) ---t O. If all Fn are Bernoullif lfl "!"}, then n must be huge before o-n(a n)
Example 4.1
> O.
0
Proof. That IWzn - 11 ::; 9 in/ Vii o-~ is immediate from the Berry-Esseen theorem. Maximizing lI
by Galen
TECHNICAL
R.
Shorack
REPORT
November
Department BOx
University Seattle,
of
No.
338
1998
Statistics
354322
of
Washington
Washington
98195
USA
General Central Limit Theorems Via Negligibility Galen R. Shorack University of Washington
Abstract The central limit theorem (CLT) for the sample mean of iid rv's is known to be equivalent to the asymptotic normality condition (ANC) of Levy. And Levy's ANC is well known to be equivalent to an alternative ANC of Feller. Both are equivalent to a negligibility requirement, considered by O'Brien. More recently, additional equivalences have been developed in terms of the quantile function, by Csorgo , Haeusler and Mason. [Other useful and informative equivalences have been developed. Many depend only on comparisons of areas. Others can be developed in the simpler context of the weak law of large numbers, because the asymptotic normality above is equivalent to appropriately phrased consistency of the sample second moment. Roughly, one can learn all about asymptotic normality by studying equivalences in either of the two simpler settings. Both of these tacks were followed elsewhere.] Here, these two earlier threads are tied together with the CLT problem itself.
o.
Introduction
The central limit theorem (CLT) is known to hold for iid rv's if and only if one has Feller's asymptotic normality condition (ANC) that the partial second moment U(x) == fro,xl y 2 dFlxl(y) is slowly varying at infinity (see condition (2.19) below). Levy's(1937) earlier form of the ANC is the requirement that P(IXI > x)/U(x) -+ 0 as x -+ 00 (see (2.21) below). A refined source for both is Feller(1966, p. 303). Versions of the ANC expressed in terms of the quantile function (qf) are found in S. Csorgo, Haeusler and Mason(1988, 1989), where conditions (2.6), (2.13) and (2.14) figured heavily; they approached CLT's via weighted constructions of empirical processes. Equivalences (2.31)(a) and/or (b) can be found in O'Brien(1980), S. Csorgo and Mason(1989) and Gine and Gotze(1995). Feller (1966, p.233) established a condition equivalent to the consistency of the sam pie mean of non negative observations. When this condition is rephrased in terms of the consistency of the sample mean X~ of the squared values (with all Xf ;::: 0, always) it is seen to be the Levy ANC of (2.21). Maller(1979) also obtained the result of Feller's problem, and discussed the X 2 version of it in Maller(1980). See also Raikov(1938). Thus the ANC can be studied in the context of this simpler WLLN problem, and this is therefore a preferable context in which to study and develop equivalences. Shorack(1998b) was written in a way that emphasizes this. This problem of establishing useful equivalences could be made harder, just by examining it in the context of the CLT and/or in the context of the general theory of slowly varying functions. But in fact, many equivalences follow in an elementary fashion from simple pictures and application of Cauchy-Schwarz. This was done systematically in Shorack(1998a), where many equivalences are enunciated. The net result of the various authors is that many conditions equivalent to the ANC are now known, some are in the df domain, some are in the qf domain, some are probabilistic in nature, and some are purely geometric in nature. Then, using a few equivalences, they can all be equated to the CLT. This is part of what will be done herein. But this paper really begins with consideration of row independent triangular arrays. Useful theorems are developed in a qf format. Via Winsorization, the Berry-Esseen theorem is made to play an unuually large role. In Section 2 the earlier work on equivalences is used to quickly deal with necessary and sufficient conditions in the iid case. The key step is to apply the usual proof of the converse half of the Lindeberg-Feller theorem in qf notation. Theorem 0.1 below expresses one very nice resulting ANC equivalence. In Section 3 we show how the standard weak and strong bootstrap results follow. In Section 4 we obtain a universal uniform studentized CLT, as well as its bootstrap version; the key step in the formulation is to Winsorize a slowly growing fraction of observations. (Trimming such slowly growing fractions is a key part of S. Csorgo , Haeusler and .)
1.
Winsorization and Truncation
Weak Negligibility. Let X n1, ... , X nn be independent with df's Fn1, ... , . Let B > 0 be , X()n] to be the smallest dosed and symmetric interval to given. Define X()n by requiring which t; == I:~ Fnkl n assigns probability at least 1 - Oln. Let Pn(x) == I:~ P(IXnkl > x)/n denote the average tail probability, and then let K n denote the qf of the df 1 - r, (.). Note the quantile relationship X()n = K n(1- Oln). For any E > 0, let P~k == P(IXnkl > E). Now the maximum J'v[n == [max1S:k~n IXnkl] satisfies 1 - exp( - I:~ P~k) ::; P(Mn > E)
=1-
n~ (1 - P~k) ::; I:~ P~k'
[The equality is simple, and the bounds follow from it using first 1 - x ::; exp( -x) and then
ni A k == [UI Ak]C; see Loeve(1955, p. 316).] This gives the standard result that (1.1)
M; -+p 0
if and only if
nPn(E) = I:~ P~k -+ 0 for all
E
> O.
Suppose 0 < E ::; 1 and 0 > 0 are fixed, and that we are considering all n exceeding some Since K n(1- ()In) = inf{x : 1- Pn(x) ~ 1- Bin} = inf{x: Fn(x) ::; ()In}, we have
n€,()'
if and only if Thus (1.2) gives (1.3)
u; -+p 0
if and only if
.T()n
= K n (1 - B) -+ 0
for all () > 0
°
o
Weak negligibility in the CLT context. Now let ./Ynk denote Xnk Winsorized outside [-X()on, X()on] (often, 00 == 1); and let iink and a;k denote the resulting means and variances, and then set tln == I:~ iinkln and a; == I:~ a~kln. Applying the previous paragraph to the rv's IXnkllJn an, whose () In-th quantile is X()n/ vn an, gives the equivalence (
1.4 )
a.•
()
= max
L1" lVl n
l~k~n
IXnkl r:::; _ -+p 0 V
n a.,
Of an d on lv if y 1·
1
(b).o V
X()n_ -'-{ r:::;. n (Tn
0 f or au> II {} 0 .
When (1.4)(a)(b) holds then [maxklXnk - tlnkl]/vnan -+p 0 also (as all iink E [-X()on,X()on), and so the triangle inequality establishes this). If M n --7p 0, then the terms are called weakly negligible. If max, P(IXnkl]/vn an > c) -+ 0 for all E > 0, then the terms are called uniformly asymptotically neglible (abbreviated as uan). Let == vn[./Yn - iin]/an. . For 0 < o ::; 00 let X~k denote x.; Winsorized outside XBn]; and let ii~k and a~k denote the resulting means and stan~ard deviations,_ and define both == I:~ ii~kln == I:~(a~k)2In. X~k/n, let Z()n == third rplit.l"::l.l moments 13
z;
o
Proof.
Set
eo = 1.
Consider (i). (Let a
= b c mean that la bl S; c.] Observe that P(L~ x., =1= L~ .'Y~k)) .
O.
Let n(a n 1\ a~) ---t
0 00.
Then uniformly in F n :
9/vn(an I\a~) ---t O.
---t O.
11 ---t O.
Also, every non degenerate df F is eventually in all further F n if we also require (an V a~) ---t O. If all Fn are Bernoullif lfl "!"}, then n must be huge before o-n(a n)
Example 4.1
> O.
0
Proof. That IWzn - 11 ::; 9 in/ Vii o-~ is immediate from the Berry-Esseen theorem. Maximizing lI
