# Almost Sure Central Limit Theory for Self-Normalized Products of Sums of Partial Sums

Authors: Qunying Wu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

LetX,X1,X2,… be a sequence of independent and identically distributed random variables in the domain of attraction of a normal law. An almost sure limit theorem for the self-normalized products of sums of partial sums is established.


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## Body

## 1. Introduction

Let{X,Xn}n∈ℕ be a sequence of independent and identically distributed (i.i.d.) positive random variables with a nondegenerate distribution function and 𝔼X=μ>0. For each n≥1, the symbol Sn/Vn denotes self-normalized partial sums, where Sn=∑i=1nXi and Vn2=∑i=1n(Xi-μ)2. We say that the random variable X belongs to the domain of attraction of the normal law if there exist constants an>0, bn∈ℝ such that(1.1)Sn-bnan⟶dN,asn⟶∞,
here and in the sequel 𝒩 is a standard normal random variable. We say that {X,Xn}n∈ℕ satisfies the central limit theorem (CLT).It is known that (1.1) holds if and only if(1.2)lim⁡x→∞x2P(|X|>x)EX2I(|X|≤x)=0.
In contrast to the well-known classical central limit theorem, Gine et al. [1] obtained the following self-normalized version of the central limit theorem: (Sn-𝔼Sn)/Vn→d𝒩 as n→∞ if and only if (1.2) holds.Brosamler [2] and Schatte [3] obtained the following almost sure central limit theorem (ASCLT): let {Xn}n∈ℕ be i.i.d. random variables with mean 0, variance σ2>0, and partial sums Sn. Then(1.3)lim⁡n→∞1Dn∑k=1ndkI{Skσk<x}=Φ(x)a.s.∀x∈R,
with dk=1/k and Dn=∑k=1ndk; here and in the sequel I denotes an indicator function, and Φ(x) is the standard normal distribution function. Some ASCLT results for partial sums were obtained by Lacey and Philipp [4], Ibragimov and Lifshits [5], Miao [6], Berkes and Csáki [7], Hörmann [8], Wu [9, 10], and Ye and Wu [11]. Huang and Pang [12] and Zhang and Yang [13] obtained ASCLT results for self-normalized version. However, ASCLT results for self-normalized products of sums of partial sums have not been reported yet.Under mild moment conditions, ASCLT follows from the ordinary CLT, but in general the validity of ASCLT is a delicate question of a totally different character as CLT. The difference between CLT and ASCLT lies in the weight in ASCLT. The terminology of summation procedures (see e.g., Chandrasekharan and Minakshisundaram [14], page 35) shows that the larger the weight sequence {dk;k≥1} in (1.3) is, the stronger the relation becomes. By this argument, one should also expect to get stronger results if we use larger weights. And it would be of considerable interest to determine the optimal weights.On the other hand, by the Theorem  1 of Schatte [3], (1.3) fails for weight dk=1. The optimal weight sequence remains unknown.The purpose of this paper is to study and establish the ASCLT for self-normalized products of sums of partial sums of random variables in the domain of attraction of the normal law, we will show that the ASCLT holds under a fairly general growth condition ondk=k-1exp((lnk)α), 0≤α<1/2.In the following, we assume that{X,Xn}n∈ℕ is a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with 𝔼X=μ>0 and define Sk=∑i=1kXi,Vk=∑i=1k(Xi-μ)2,Tk=∑i=1kSi. Let bn,k=∑j=kn1/j,cn,k=2∑j=kn(j+1-k)/(j(j+1)), and dn,k=(n+1-k)/(n+1) for 1≤k≤n. I(A) denotes the indicator function of set A, and an~bn denotes an/bn→1,n→∞. The symbol c stands for a generic positive constant, which may differ from one place to another. Let(1.4)l(x)=E(X-μ)2I{|X-μ|≤x},b=inf⁡{x≥1;l(x)>0},ηj=inf⁡{s;s≥b+1,l(s)s2≤1j}forj≥1.By the definition ofηj, we have jl(ηj)≤ηj2 and jl(ηj-ε)>(ηj-ε)2 for any ε>0. It implies that(1.5)nl(ηn)~ηn2,asn→∞,ηn<n+1.Our theorem is formulated in a more general setting.Theorem 1.1.
Let{X,Xn}n∈ℕ be a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with  𝔼X=μ>0. Suppose
(1.6)l(ηn)~l(ηnln⁡n).
For0≤α<1/2, set
(1.7)dk=exp⁡(ln⁡αk)k,Dn=∑k=1ndk.
Then
(1.8)lim⁡n→∞1Dn∑k=1ndkI((∏j=1k(Tjj(j+1)μ/2))μ/Vk≤x)=F(x)a.s.,
for any x∈ℝ, where F is the distribution function of the random variable exp(10/3𝒩).By the terminology of summation procedures, we have the following corollary.Corollary 1.2.
Theorem1.1 remains valid if we replace the weight sequence  {dk}k∈ℕ by  {dk*}k∈ℕ such that  0≤dk*≤dk,  ∑k=1∞dk*=∞.Remark 1.3.
If𝔼X2=σ2<∞, thenXis in the domain of attraction of the normal law andl(x)→σ2,ηn2~σ2n, thus (1.6) holds. Therefore, the class of random variables in Theorems1.1is of very broad range.Remark 1.4.
Whether Theorem1.1holds for1/2≤α<1remains open.


## 2. Proofs

Furthermore, the following four lemmas will be useful in the proof, and the first is due to Csörgo et al. [15].Lemma 2.1.
LetX be a random variable with  𝔼X=μ, and denote  l(x)=𝔼(X-μ)2I{|X-μ|≤x}. The following statements are equivalent: (i)
l(x) is a slowly varying function at ∞;(ii)
X is in the domain of attraction of the normal law;(iii)
x2ℙ(|X-μ|>x)=o(l(x));(iv)
x𝔼(|X-μ|I(|X-μ|>x))=o(l(x));(v)
𝔼(|X-μ|αI(|X-μ|≤x))=o(xα-2l(x)) for α>2.Lemma 2.2.
Let{ξ,ξn}n∈ℕ be a sequence of uniformly bounded random variables. If there exist constants  c>0 and  δ>0 such that
(2.1)|Eξkξj|≤c(kj)δ,for1≤k<j,
then
(2.2)lim⁡n→∞1Dn∑k=1ndkξk=0a.s.,
where dk and Dn are defined by (1.7).Proof.
We can easily apply the similar arguments of (2.1) in Wu [16] to get Lemma 2.2, and we omit the details here.The following Lemma2.3 can be directly verified.Lemma 2.3.
(i)cn,i=2(bn,i-dn,i);
(ii)
∑i=1nbn,i2=2n-bn,1~2n;(iii)
∑i=1ndn,i2=n3-n6(n+1)~n3;(iv)
∑i=1ncn,i2=10n3-4bn,1+10n3(n+1)~10n3.For every1≤i≤k≤n, let(2.3)X̅ki=(Xi-μ)I(|Xi-μ|≤ηk),S̅k,i=∑j=1ick,jX̅kj,V̅k2=∑j=1kX̅kj2.Lemma 2.4.
Suppose that the assumptions of Theorem1.1 hold. Then
(2.4)lim⁡n→∞1Dn∑k=1ndkI{S̅k,k-ES̅k,k10kl(ηk)/3≤x}=Φ(x)a.s.foranyx∈R,(2.5)lim⁡n→∞1Dn∑k=1ndk(I(⋃i=1k(|Xi-μ|>ηk))-EI(⋃i=1k(|Xi-μ|>ηk)))=0a.s.,(2.6)lim⁡n→∞1Dn∑k=1ndk(f(V̅k2kl(ηk))-Ef(V̅k2kl(ηk)))=0a.s.,
where dk and Dn are defined by (1.7) and f is a nonnegative, bounded Lipschitz function.Proof.
By𝔼(X-μ)=0, Lemma 2.1(iv), we have
(2.7)|EX̅ni|≤E|X-μ|I(|X-μ|>ηn)=o(l(ηn))ηn.
Thus, by (1.5) and Lemma 2.3 (iv),
(2.8)Var⁡X̅ni=EX̅ni2-(EX̅ni)2~l(ηn),∑i=1nVar⁡(cn,iX̅ni)~10n3l(ηn):=Bn2.
By (1.5) and Lemma 2.3 (i),
(2.9)max⁡1≤i≤ncn,i≤2max⁡1≤i≤nbn,i≤2bn,1~2ln⁡⁡n,ln⁡⁡nmax⁡1≤i≤n|EX̅ni|Bn⟶0.
Thus by combining Lemma 2.3 (iv), (1.6), and (2.8), Lindeberg condition
(2.10)1Bn2∑i=1nE(cn,iX̅ni)2I(|cn,iX̅ni-Ecn,iX̅ni|>εBn)~cnl(ηn)∑i=1ncn,i2EX̅ni2I(|X̅ni-EX̅ni|>εBn/2ln⁡n)≤cnl(ηn)∑i=1ncn,i2EX̅ni2I(|X̅ni|>εBn/4ln⁡n)=cnl(ηn)∑i=1ncn,i2E(X-μ)2I(cηn/ln⁡n<|X-μ|≤ηn)=cl(ηn)-l(cηn/ln⁡n)l(ηn)⟶0,asn⟶∞
hold.
Hence, it follows that(2.11)S̅n,n-ES̅n,nBn⟶dN,asn⟶∞.
This implies that for any g(x), which is a nonnegative, bounded Lipschitz function,
(2.12)Eg(S̅n,n-ES̅n,nBn)⟶Eg(N),asn⟶∞.
Hence, we obtain
(2.13)lim⁡n→∞1Dn∑k=1ndkEg(S̅k,k-ES̅k,kBk)=Eg(N)
from the Toeplitz lemma.
On the other hand, note that (2.4) is equivalent to
(2.14)lim⁡n→∞1Dn∑k=1ndkg(S̅k,k-ES̅k,kBk)=Eg(N)a.s.,
from Theorem  7.1 of Billingsley [17] and Section  2 of Peligrad and Shao [18]. Hence, to prove (2.4), it suffices to prove
(2.15)lim⁡n→∞1Dn∑k=1ndk(g(S̅k,k-ES̅k,kBk)-Eg(S̅k,k-ES̅k,kBk))=0a.s.,
for any g(x) which is a nonnegative, bounded Lipschitz function.
Let(2.16)ξk=g(S̅k,k-ES̅k,kBk)-Eg(S̅k,k-ES̅k,kBk),fork≥1.
Clearly, there is a constantc>0 such that
(2.17)|g(x)|≤c,|g(x)-g(y)|≤c|x-y|foranyx,y∈R,|ξk|≤2c,foranyk.
For any1≤k<l, note that
(2.18)S̅l,l-S̅l,k=∑i=1lcl,iX̅li-∑i=1kcl,iX̅li=∑i=k+1lcl,iX̅li.
For any1≤k<j, note that g((S̅k,k-𝔼S̅k,k)/Bk) and g((S̅j,j-𝔼S̅j,j-(S̅j,k-𝔼S̅j,k))/Bj) are independent, g(x) is a nonnegative, bounded Lipschitz function, ∑i=1kck,i2~10k/3,cj,i2≤4bj,i2, ∑i=1kbk,i2~2k, and lnx≤4x1/4,x≥1. By the definition of ηj, we get
(2.19)|Eξkξj|=|Cov(g(S̅k,k-ES̅k,kBk),g(S̅j,j-ES̅j,jBj))|=|Cov(g(S̅k,k-ES̅k,kBk),g(S̅j,j-ES̅j,jBj)-g(S̅j,j-ES̅j,j-(S̅j,k-ES̅j,k)Bj))|≤cE|∑i=1kcj,i(X̅ji-EX̅ji)|jl(ηj)≤cE(∑i=1kcj,i(X̅ji-EX̅ji))2jl(ηj)≤c∑i=1kbj,i2EX̅ji2jl(ηj)≤c∑i=1k(bk,i+bj,k+1)2l(ηj)jl(ηj)=c∑i=1kbk,i2+∑i=1kbj,k+12j≤ck+kln⁡2(j/k)j≤c(kj)1/4.
By Lemma 2.2, (2.15) holds.
Now we prove (2.5). Let
(2.20)Zk=I(⋃i=1k(|Xi-μ|>ηk))-EI(⋃i=1k(|Xi-μ|>ηk))foranyk≥1.
It is known that I(A∪B)-I(B)≤I(A) for any sets A and B; then for 1≤k<j, by Lemma 2.1 (iii) and (1.5), we get
(2.21)P(|X-μ|>ηj)=o(1)l(ηj)ηj2=o(1)j.
Hence for 1≤k<j,
(2.22)|EZkZj|=|Cov(I(⋃i=1k(|Xi-μ|>ηk)),I(⋃i=1j(|Xi-μ|>ηj)))|=|Cov(I(⋃i=1k(|Xi-μ|>ηk)),I(⋃i=1j(|Xi-μ|>ηj))-I(⋃i=k+1j(|Xi-μ|>ηj)))|≤E|I(⋃i=1j(|Xi-μ|>ηj))-I(⋃i=k+1j(|Xi-μ|>ηj))|≤EI(⋃i=1k(|Xi-μ|>ηj))≤kP(|X-μ|>ηj)≤kj.
By Lemma 2.2, (2.5) holds.
Finally, we prove (2.6). Let
(2.23)ζk=f(V̅k2kl(ηk))-Ef(V̅k2kl(ηk))foranyk≥1.
For 1≤k<j,
(2.24)|Eζkζj|=|Cov(f(V̅k2kl(ηk)),f(V̅j2jl(ηj)))|=|Cov(f(V̅k2kl(ηk)),f(V̅j2jl(ηj))-f(V̅j2-∑i=1k(Xi-μ)2I(|Xi-μ|≤ηj)jl(ηj)))|≤cE(∑i=1k(Xi-μ)2I(|Xi-μ|≤ηj))jl(ηj)=ckE(X-μ)2I(|X-μ|≤ηj)jl(ηj)=ckl(ηj)jl(ηj)=ckj.
By Lemma 2.2, (2.6) holds. This completes the proof of Lemma 2.4.Proof of Theorem1.1.
LetZj=Tj/(j(j+1)μ/2); then (1.8) is equivalent to
(2.25)lim⁡n→∞1Dn∑k=1ndkI(3μ10Vk∑j=1kln⁡Zj≤x)=Φ(x),a.s.foranyx,
where Φ(x) is the distribution function of the standard normal random variable 𝒩.
Letq∈(4/3,2), then 𝔼|X|q<∞. Using Marcinkiewicz-Zygmund strong large number law, we have
(2.26)Sk-μk=o(k1/q)a.s.
Thus,
(2.27)|Zi-1|=|∑j=1iSj-i(i+1)μ/2|i(i+1)μ/2≤∑j=1i|Sj-μj|i(i+1)μ/2≤∑j=1ij1/qi(i+1)μ/2≤ci1/q+1i2=i1/q-1⟶0,a.s.
Hence by |ln(1+x)-x|=O(x2) for |x|<1/2, for any 0<ε<1,
(2.28)|11±εBk∑i=1kln⁡Zi-11±εBk∑i=1k(Zi-1)|≤c1kl(ηk)∑i=1k(Zi-1)2≤ckl(ηk)∑i=1ki2(1/q-1)≤c1k3/2-2/ql(ηk)⟶0a.s.k⟶∞,
from 3/2-2/q>0, l(x) is a slowly varying function at ∞, and ηk≤k+1.
Hence for almost every eventω and any δ>0, there exists k0=k0(ω,δ,x) such that for k>k0,
(2.29)I(μ1±εBk∑i=1k(Zi-1)≤x-δ)≤I(μ1±εBk∑i=1kln⁡Zi≤x)≤I(μ1±εBk∑i=1k(Zi-1)≤x+δ).
Note that under condition|Xj-μ|≤ηk,1≤j≤k,
(2.30)μ∑i=1k(Zi-1)=∑i=1k∑l=1iSl-μ∑l=1ili(i+1)/2=∑i=1k1i(i+1)/2∑l=1i∑j=1l(Xj-μ)=∑i=1k1i(i+1)/2∑j=1i∑l=ji(Xj-μ)=∑i=1k∑j=1i2(i+1-j)i(i+1)(Xj-μ)=∑j=1k∑i=jk2(i+1-j)i(i+1)X̅kj=∑j=1kck,jX̅kj=S̅k,k.
Thus, for any given0<ε<1, δ>0, combining (2.29), we have for k>k0(2.31)I(3μ∑i=1kln⁡Zi10Vk≤x)≤I(3μ∑i=1kln⁡Zi10(1+ε)kl(ηk)≤x)+I(V̅k2>(1+ε)kl(ηk))+I(⋃i=1k(|Xi-μ|>ηk))≤I(μ∑i=1k(Zi-1)1+εBk≤x+δ)+I(V̅k2>(1+ε)kl(ηk))+I(⋃i=1k(|Xi-μ|>ηk))≤I(S̅k,k1+εBk≤x+δ)+I(V̅k2>(1+ε)kl(ηk))+2I(⋃i=1k(|Xi-μ|>ηk))forx≥0,I(3μ∑i=1kln⁡Zi10Vk≤x)≤I(S̅k,k1-εBk≤x+δ)+I(V̅k2<(1-ε)kl(ηk))+2I(⋃i=1k(|Xi-μ|>ηk))forx<0,I(3μ∑i=1kln⁡Zi10Vk≤x)≥I(S̅k,k1-εBk≤x-δ)-I(V̅k2<(1-ε)kl(ηk))-2I(⋃i=1k(|Xi-μ|>ηk)),forx≥0,I(3μ∑i=1kln⁡Zi10Vk≤x)≥I(S̅k,k1+εBk≤x-δ)-I(V̅k2>(1+ε)kl(ηk))-2I(⋃i=1k(|Xi-μ|>ηk)),forx<0.
Therefore, to prove (2.25), it suffices to prove
(2.32)lim⁡n→∞1Dn∑k=1ndkI(S̅k,kBk≤1±εx±δ1)=Φ(1±εx±δ1)a.s.,(2.33)lim⁡n→∞1Dn∑k=1ndkI(⋃i=1k(|Xi-μ|>ηk))=0a.s.,(2.34)lim⁡n→∞1Dn∑k=1ndkI(V̅k2>(1+ε)kl(ηk))=0a.s.,(2.35)lim⁡n→∞1Dn∑k=1ndkI(V̅k2<(1-ε)kl(ηk))=0a.s,
for any 0<ε<1 and δ1>0.
Firstly, we prove (2.32). Let 0<β<1/2 and h(·) be a real function, such that for any given x∈ℝ,
(2.36)I(y≤1±εx±δ1-β)≤h(y)≤I(y≤1±εx±δ1+β).
By𝔼(Xi-μ)=0, Lemma 2.1 (iv) and (1.5), we have
(2.37)|ES̅k,k|=|E∑i=1kck,i(Xi-μ)I(|Xi-μ|≤ηk)|≤2∑i=1kbk,iE|Xi-μ|I(|Xi-μ|>ηk)=2∑i=1k∑j=ik1jE|X-μ|I(|X-μ|>ηk)=∑j=1k∑i=1j1jo(l(ηk))ηk=o(kl(ηk)).
This, combining with (2.4), (2.36) and the arbitrariness of β in (2.36), (2.32) holds.
By (2.5), (2.21) and the Toeplitz lemma,
(2.38)0≤1Dn∑k=1ndkI(⋃i=1k(|Xi-μ|>ηk))~1Dn∑k=1ndkEI(⋃i=1k(|Xi-μ|>ηk))≤1Dn∑k=1ndkkP(|X-μ|>ηk)⟶0a.s.
That is, (2.33) holds.
Now we prove (2.34). For any given ε>0, let f be a nonnegative, bounded Lipschitz function such that
(2.39)I(x>1+ε)≤f(x)≤I(x>1+ε2).
From𝔼V̅k2=kl(ηk), X̅ki is i.i.d., 𝔼(X̅ki2-𝔼X̅ki2)=0, Lemma 2.1 (v), and (1.5),
(2.40)P(V̅k2>(1+ε2)kl(ηk))=P(V̅k2-EV̅k2>ε2kl(ηk))≤cE(V̅k2-EV̅k2)2k2l2(ηk)≤cE|X-μ|4I(|X-μ|≤ηk)kl2(ηk)=o(1)ηk2kl(ηk)=o(1)⟶0.
Therefore, combining (2.6) and the Toeplitz lemma,
(2.41)0≤1Dn∑k=1ndkI(V̅k2>(1+ε)kl(ηk))≤1Dn∑k=1ndkf(V̅k2kl(ηk))~1Dn∑k=1ndkEf(V̅k2kl(ηk))≤1Dn∑k=1ndkEI(V̅k2>(1+ε2)kl(ηk))=1Dn∑k=1ndkP(V̅k2>(1+ε2)kl(ηk))⟶0a.s.
Hence, (2.34) holds. By similar methods used to prove (2.34), we can prove (2.35). This completes the proof of Theorem 1.1.



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