# The Almost Sure Local Central Limit Theorem for the Negatively Associated Sequences

Authors: Yuanying Jiang; Qunying Wu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

In this paper, the almost sure central limit theorem is established for sequences of negatively associated random variables:limn→∞(1/logn)∑k=1n(I(ak≤Sk<bk)/k)P(ak≤Sk<bk)=1, almost surely. This is the local almost sure central limit theorem for negatively associated sequences similar to results by Csáki et al. (1993). The results extend those on almost sure local central limit theorems from the i.i.d. case to the stationary negatively associated sequences.


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## Body

## 1. Introduction

Definition 1.
Random variablesX1,X2,…,Xn,n≥2 are said to be negatively associated (NA) if for every pair of disjoint subsets A1 and A2 of {1,2,…,n},
(1)Cov(f1(Xi,i∈A1),f2(Xj,j∈A2))≤0,
where f1 and f2 are increasing for every variable (or decreasing for every variable) such that this covariance exists. A sequence of random variables {Xi,i≥1} is said to be NA if every finite subfamily of {Xi,i≥1} is NA.Obviously, if{Xi,i≥1} is a sequence of NA random variables and {fi,i≥1} is a sequence of nondecreasing (or nonincreasing) functions, then {fi(Xi),i≥1} is also a sequence of NA random variables.This definition was introduced by the Joag-Dev and Proschan [1]. Statistical test depends greatly on sampling, and the random sampling without replacement from a finite population is NA, but it is not independent. NA sampling has wide applications such as those in multivariate statistical analysis and reliability theory. Because of the wide applications of NA sampling, the notions of NA random variables have received more and more attention recently. We refer to Joag-Dev and Proschan [1] for fundamental properties, Shao [2] for the moment inequalities, and Wu and Jiang [3] for Chover’s law of the iterated logarithm.Assume that{Xn,n≥1} is a strictly stationary sequence of NA random variables with EX1=0,0<EX12<∞. Define Sn=∑j=1nXj,
(2)σn2:=VarSn,(3)σ2:=VarX1+2∑j=2∞‍Cov(X1,Xj).(1)
Newman [4] and Matuła [5] showed that NA stationary sequences satisfy the central limit theorem (CLT) under σ2>0, that is,
(4)supx∈R|P(Snσn<x)-Φ(x)|=o(1).(2)
Applying Matuła [6] and Wu’s [7] methods, we can easily show that NA sequences satisfy the almost sure central limit theorem (ASCLT), that is,
(5)limn→∞1logn∑k=1n‍1kI{Sk≤xσk1/2}=Φ(x)a.s.∀x∈ℝ,
where Φ(x) is the standard normal distribution function and I{A} denotes the indicator of the event A.The ASCLT was stimulated by Brosamler [8] and Schatte [9]. Both were concerned with the partial sum of independent and identically distributed (i.i.d.) random variables with more than the second moment. The ASCLT was extensively studied in the past two decades and an interesting direction of the study is to prove it for dependent variables. There are some results for weakly dependent variables such as α,ρ,ϕ-mixing and associated random variables. Among those results, we refer to Peligrad and Shao [10], Matuła [11], and Wu [7].More general version of ASCLT was proved by Csáki et al. [12]. The following theorem is due to them.Theorem A.
Let{Xn,n≥1} be a sequence of i.i.d. random variables with E|X1|3<∞, let EX1=0. ak,bk satisfy
(6)-∞≤ak≤0≤bk≤∞,k=1,2,…,
and assume that
(7)∑k=1n‍logkk3/2P(ak≤Sk<bk)=O(logn),asn⟶∞,
and then
(8)limn→∞1logn∑k=1n‍I{ak≤Sk<bk}kP(ak≤Sk<bk)=1a.s.This result may be called almost sure local central limit theorem, while (5) may be called almost sure global central limit theorem. Hurelbaatar [13] extended (8) to the case of ρ-mixing sequences and Weng et al. [14] derived an almost sure local central limit theorem for the product of partial sums of a sequence of i.i.d. positive random variables under some regular conditions. For more details, we refer to Berkes and Csáki [15] and Földes [16].Our concern in this paper is to give a common generalization of (8) to the case of NA sequences.In the next section we present the exact results, postponing some technical lemmas and the proofs to Section3.


## 2. Main Results

Assume in the following that{Xn,n≥1} is a strictly stationary sequence of NA random variables with EX1=0,0<EX12<∞. We consider the limit behavior of the logarithmic average
(9)limn→∞1logn∑k=1n‍I{ak≤Sk<bk}kP(ak≤Sk<bk)
with -∞≤ak≤0≤bk≤∞, where the terms in the sum above are defined to be 1 if their denominator happens to be 0.More precisely let{an,n≥1} and {bn,n≥1} be two sequences of real numbers and put
(10)pk:=P(ak≤Sk<bk),αk:={I{ak≤Sk<bk}pk,ifpk≠0,1,ifpk=0.
So we need to investigate the limit behavior of
(11)μn:=∑k=1n‍αkk
under certain conditions.In our considerations, we will need the following Cox-Grimmett coefficient which describes the covariance structure of the sequence(12)u(n):=supk∈N∑j:|j-k|≥n‍|Cov(Xj,Xk)|,n∈N∪{0}.
We remark that for a stationary sequence of NA random variables
(13)u(n)=-2∑k=n+1∞‍Cov(X1,Xk),n∈N.
By Lemma 8 of Newman [4], we have u(0)<∞ and limn→∞u(n)=0.In the following,ξn~ηn denotes ξn/ηn→1,n→∞. ξn=O(ηn) denotes that there exists a constant c>0 such that ξn≤cηn for sufficiently large n. The symbols c,c1,c2,…, stand for generic positive constants which may differ from one place to another.Theorem 2.
Let{Xn,n≥1} be a strictly stationary sequence of NA random variables with EX1=0,E|X1|3<∞ and let σ2>0.ak,bk satisfy (6). Assume that
(14)∑n=1∞‍u(n)<∞,
and for some β>1,
(15)∑1≤k≤npk≠0‍(logk)1/3kpk=O((logn)2(loglogn)-β).
Then we have
(16)limn→∞μnlogn=1,a.s.,
where μn is defined by (11).Remark 3.
Letak=-∞ and bk=xσk1/2 in (6). By the central limit theorem (4), we have pk=P(Sk/σk1/2<x)→Φ(x), obviously (15) satisfies; then (16) becomes (5), which is the almost sure global central limit theorem. Thus the almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem.Remark 4.
The condition (15) is satisfied with a wide range of pk; for example, if
(17)pk=0orpk≥c(loglogk)β(logk)2/3
holds, then the condition (15) is satisfied. In fact, letting 0<δ<1, we have
(18)∑1≤k≤npk≠0‍(logk)1/3kpk≤c∑1≤k≤n‍(loglogk)-β(logk)δ(logk)1-δk≤c(loglogn)-β(logn)δ∑1≤k≤n‍(logk)1-δk≤c(loglogn)-β(logn)2=O((logn)2(loglogn)-β).In the given theorem below, we strengthen the condition (6) on ak and bk. Meanwhile, as a compensation, we do not need to impose restricting condition (15) on pk.Theorem 5.
Let{Xn,n≥1} be a strictly stationary sequence of NA random variables with EX1=0,E|X1|3<∞, and σ2>0, and let ak and bk satisfy
(19)-c1k1/2-α≤ak≤-c2k1/2-α,c3k1/2-α≤bk≤c4k1/2-α,
where 0<α<1/7. Assume that (14) hold, and then we have (16).


## 3. Proofs

The following lemmas play important roles in the proof of our theorems. The proofs are given in the Appendix.Lemma 6.
Assume that{ξn,n≥1} are random variables such that
(20)ξk≥0,Eξk=1,k=1,2,…
and furthermore there exists dk≥0 such that Dn=∑k=1ndk↑∞,Dn/Dn-1→1, and
(21)Var(∑k=1n‍dkξk)≤cDn2(logDn)-β,
with some β>1 and positive constant c, and then
(22)limn→∞1Dn∑k=1n‍dkξk=1a.s.Remark 7.
Letdk=1/k in (21), and then Dn=∑k=1n‍1/k~logn. Thus, if
(23)Var(∑k=1n‍1kξk)≤c(logn)2(loglogn)-β,
with some β>1 and positive constant c, then
(24)limn→∞1logn∑k=1n‍1kξk=1a.s.The following Lemma8 is obvious.Lemma 8.
Assume that the nonnegative random sequence{ξn,n≥1} satisfies (24) and the sequence {ηn,n≥1} is such that, for any ε>0, there exists a k0=k0(ε,ω) for which
(25)(1-ε)ξk≤ηk≤(1+ε)ξk,k>k0.
Then we have also
(26)limn→∞1logn∑k=1n‍1kηk=1a.s.The following Lemma9 is an easy corollary to the Corollary 2.2 in Matuła [5] under strictly stationary condition, which studies the rate of convergence in the CLT under negative dependence. It was also studied in Pan [17]. Of course this is the Berry-Esseen inequality for the NA sequence.Lemma 9.
Let{Xj,j∈N} be a strictly stationary sequence of NA random variables with EX1=0,E|X1|3<∞,σ2>0 satisfying (14). Then one has
(27)supx∈R|P(Snσn<x)-Φ(x)|=O(n-1/5).Lemma 10.
If the conditions of Lemma9 hold, ak and bk satisfy (19). Then one has
(28)c1′k-α≤pk≤c2′k-α(k≥k0),
where α is as (19).Lemma 11.
If the conditions of Lemma9 hold, ak and bk satisfy (19), and α is as (19). Assume that εl=l3α/2, and then the following asymptotic relations hold:
(29)∑1≤k<l≤nk<l-lα‍1klplP(|Slα|≥εl)=O(logn),(30)∑1≤k<l≤nk<l-lα‍1klpl1(l-k-lα)1/5=O(logn),(31)∑1≤k<l≤nk<l-lα‍1klpl|Φ(al-bk-εl(l-k-lα)1/2)-Φ(all1/2)|=O(logn),(32)∑1≤k<l≤nk<l-lα‍1klpl|(Φ(bl-ak+εl(l-k-lα)1/2)-Φ(bll1/2))|=O(logn).The main point in our proof is to verify the condition (23). We use global central limit theorem with remainders and the following elementary inequalities:
(33)|Φ(x)-Φ(y)|≤c|x-y|,foreveryx,y∈ℝ
with some constant c. Moreover, for each k>0, there exists c1=c1(k), such that
(34)|Φ(x)-Φ(y)|≥c1|x-y|,foreveryx,y∈ℝ,|x|+|y|≤k.Let{Xn,n≥1} be a strictly stationary sequence of NA random variables with σ2>0; we can immediately get σn2~nσ2, that is,
(35)c1n≤Var(Sn)=σn2≤c2n
for some constant c1,c2>0 and sufficiently large n.Proof of Theorem2.
First assume that(36)bk-ak≤ck1/2,k=1,2,…
with some constant c. Let 1≤k<l and εk=k1/2(logk)1/3.
If eitherpk=0 or pl=0, then obviously Cov(αk,αl)=0, and so we may assume that pkpl≠0; then, we have
(37)Cov(αk,αl)=1plpk(P(ak≤Sk<bk,al≤Sl<bl)-plpk)≤1plpk(P(al-bk≤Sl-Sk<bl-ak,ak≤Sk<bk)-plpk)≤1plpk(P(al-bk≤Sl-Sk<bl-ak)×P(ak≤Sk<bk)-plpk)=1pl(P(al-bk≤Sl-Sk<bl-ak)-pl)≤1pl(P(al-bk-εk<Sl<bl-ak+εk)-pl+P(|Sk|≥εk))≤1pl(P(al-bk-εk≤Sl<al)+P(bl≤Sl<bl-ak+εk)+P(|Sk|≥εk)).
Applying Lemma9, (33), (35), and (36) and noting that εk=k1/2(logk)1/3, we obtain
(38)P(al-bk-εk≤Sl<al)+P(bl≤Sl<bl-ak+εk)≤(Φ(alσl)-Φ(al-bk-εkσl))+(Φ(bl-ak+εkσl)-Φ(blσl))+c1l1/5≤bk+εkσl+-ak+εkσl+c1l1/5=bk-akσl+2εkσl+c1l1/5≤c(k1/2l1/2+εkl1/2+1l1/5)≤c(k1/2(logk)1/3l1/2+1l1/5).
By the condition of (15), we have
(39)∑l=1n‍∑k=1l‍1klplk1/2(logk)1/3l1/2≤∑l=1n‍1l3/2pl∑k=1l‍(logk)1/3k1/2≤c∑l=1n‍(logl)1/3l3/2pl∑k=1l‍1k1/2≤c∑l=1n‍(logl)1/3l3/2pll1/2≤c∑l=1n‍(logl)1/3lpl=O((logn)2(loglogn)-β),∑l=1n‍∑k=1l-1‍1klpl1l1/5=∑l=1n‍1l6/5pl∑k=1l-1‍1k≤c∑l=1n‍1l6/5pllogl≤c∑l=1n‍(logl)1/3lpl=O((logn)2(loglogn)-β).
By Chebyshev’s inequality and the condition of (15) and (35), we obtain
(40)∑l=1n‍∑k=1l‍1klplP(|Sk|≥εk)≤∑l=1n‍1lpl∑k=1l‍Var(Sk)kεk2≤c∑l=1n‍1lpl∑k=1l‍1k(logk)2/3≤c∑l=1n‍(logl)1/3lpl=O((logn)2(loglogn)-β).
Hence (37)–(40) imply that(41)∑l=1n‍∑k=1l-1‍Cov(αk,αl)kl=O((logn)2(loglogn)-β).
ButVar(αk)=0 if pk=0 and
(42)Var(αk)=1-pkpk≤1pkifpk≠0.
Thus(43)∑k=1n‍Var(αk)k2≤∑1≤k≤npk≠0‍1k2pk≤∑1≤k≤npk≠0‍(logk)1/3kpk=O((logn)2(loglogn)-β).
Equations (41) and (43) together imply that
(44)Var(∑k=1n‍αkk)=O((logn)2(loglogn)-β),asn→∞.
Hence applying Remark7, our theorem is proved under the restricting condition (36).
Now we drop the restricting condition (36). Fix x>0 and define
(45)a~k=max(ak,-xσk1/2),b~k=min(bk,xσk1/2),p~k=P(a~k≤Sk<b~k),
where σ is defined by (3).
Clearlyp~k≤pk, and so assuming p~k≠0; then, also we have pk≠0, and thus
(46)αk=1pkI{ak≤Sk<bk}=1pk(I{a~k≤Sk<b~k}+I{ak≤Sk<a~k}+I{b~k≤Sk<bk})≤1p~kI{a~k≤Sk<b~k}+1pk(I{ak≤Sk<a~k}+I{b~k≤Sk<bk})≤1p~kI{a~k≤Sk<b~k}+I{Sk<-xσk1/2}P(-xσk1/2≤Sk<0)+I{Sk≥xσk1/2}P(0≤Sk<xσk1/2).
By (35) and the central limit theorem for NA random variables (4), that is,
(47)supx∈R|P(Snσn<x)-Φ(x)|=o(1),
we obtain(48)limk→∞P(-xσk1/2≤Sk<0)=limk→∞P(-xσk≤Sk<0)=Φ(0)-Φ(-x),limk→∞P(0≤Sk<xσk1/2)=limk→∞P(0≤Sk<xσk)=Φ(x)-Φ(0).
Applying the almost sure central limit theorem for NA random variables (5), that is,
(49)limn→∞1logn∑k=1n‍1kI{Sk≤xσk1/2}=Φ(x)a.s.∀x∈ℝ,
Lemma 8, and (48), we have
(50)limn→∞1logn∑k=1n‍I{Sk<-xσk1/2}kP(-xσk1/2≤Sk<0)=Φ(-x)Φ(0)-Φ(-x)a.s.,limn→∞1logn∑k=1n‍I{Sk>xσk1/2}kP(0≤Sk<xσk1/2)=1-Φ(x)Φ(x)-Φ(0)a.s.
Since a~k and b~k satisfy (36), we get
(51)limn→∞1logn∑k=1n‍α~kk=1a.s.,
where
(52)α~k={I{a~k≤Sk<b~k}p~k,ifp~k≠0,1,ifp~k=0.
Equations (46) and (50)–(51) together imply that(53)limsupn→∞1logn∑k=1n‍αkk≤1+21-Φ(x)Φ(x)-Φ(0)a.s.
On the other hand, if p~k≠0, then we have
(54)1pkI{ak≤Sk<bk}≥1p~kI{a~k≤Sk<b~k}(1-pk-p~kpk)≥α~k(1-P(Sk<-σxk1/2)+P(Sk>σxk1/2)min{P(0≤Sk<σxk1/2),P(-σxk1/2≤Sk<0)}),
and by the central limit theorem,
(55)limk→∞P(Sk<-σxk1/2)+P(Sk>σxk1/2)min{P(0≤Sk<σxk1/2),P(-σxk1/2≤Sk<0)}=1-21-Φ(x)Φ(x)-Φ(0).
Applying Lemma 8, (51) and (54) imply that
(56)liminfn→∞1logn∑k=1n‍αkk≥1-21-Φ(x)Φ(x)-Φ(0)a.s.,
and hence
(57)1+21-Φ(x)Φ(x)-Φ(0)≥limsupn→∞1logn∑k=1n‍αkk≥liminfn→∞1logn∑k=1n‍αkk≥1-21-Φ(x)Φ(x)-Φ(0)a.s.
By the arbitrariness of x, let x→∞ in (57); we have
(58)1≥limsupn→∞1logn∑k=1n‍αkk≥liminfn→∞1logn∑k=1n‍αkk≥1a.s.
Thus
(59)limn→∞1logn∑k=1n‍αkk=1a.s.
This completes the proof of Theorem 2.Proof of Theorem5.
Letk<l-lα, 1≤k<l, and εl=l3α/2; we have
(60)Cov(αk,αl)≤1plpk(P(al-bk≤Sl-Sk+lα+Sk+lα-Sk<bl-ak,ak≤Sk<bk)-plpk)≤1plpk(P(al-bk≤Sl-Sk+lα+Sk+lα-Sk<bl-ak)P(ak≤Sk<bk)-plpk)≤1pl(P(al-bk-εl<Sl-Sk+lα<bl-ak+εl)-pl+P(|Sk+lα-Sk|≥εl))≤1pl(P(al-bk-εl<Sl-k-lα<bl-ak+εl)-pl+P(|Slα|≥εl)).
Applying Lemma9, (33), and (35), we obtain
(61)P(al-bk-εl<Sl-k-lα<bl-ak+εl)-pl≤(Φ(bl-ak+εl(l-k-lα)1/2)-Φ(bll1/2))+(Φ(all1/2)-Φ(al-bk-εl(l-k-lα)1/2))+c(1l1/5+1(l-k-lα)1/5).
Hence Lemma 11, (60), and (61) imply that
(62)∑1≤k<l≤nk<l-lα‍Cov(αk,αl)kl=O(logn).
On the other hand,
(63)∑1≤k<l≤nk>l-lα‍Cov(αk,αl)kl=O(logn),
because l-lα<k<l,lγ-(l-lα)γ→0 as l→∞ for γ<1,α<1.
ButVar(αk)=0 if pk=0 and
(64)Var(αk)=1-pkpk≤1pkifpk≠0.
Then
(65)∑k=1n‍Var(αk)k2≤∑1≤k≤npk≠0‍1k2pk≤∑k=1n‍1k5/2-α=O(logn).
Noting that
(66)Var(∑k=1n‍αkk)=∑k=1n‍Var(αk)k2+2∑1≤k<l≤nk>l-lα‍Cov(αk,αl)kl+2∑1≤k<l≤nk<l-lα‍Cov(αk,αl)kl,
thus (62)–(66)  imply that
(67)Var(∑k=1n‍αkk)=O(logn),asn⟶∞.
Hence applying Remark 7, we have(68)limn→∞1logn∑k=1n‍αkk=1a.s.
This completes the proof of Theorem 5.



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