# Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums ofρ~-Mixing Random Variables

Authors: Chongfeng Lan
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The equivalent conditions of complete convergence are established for
weighted sums ofρ~-mixing random variables with different distributions. Our results extend and improve the Baum and Katz complete convergence theorem. As  an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequence of ρ~-mixing random variables is obtained.


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## Body

## 1. Introduction

Let(Ω,F,P) be a probability space. The random variables we deal with are all defined on (Ω,F,P). Let {Xn,n≥1} be a sequence of random variables. For each nonempty set S⊂N, write Fs=σ(Xi,i∈S). Given σ-algebras B, R in F, let
(1)ρ(B,R)=sup⁡{|corr(X,Y)|;X∈L2(B),Y∈L2(R)},
where corr(X,Y)=(EXY-EXEY)/(Var⁡XVar⁡Y)1/2. Define the ρ~-mixing coefficients by
(2)ρ~(n)=sup⁡ρ(FS,FT),
where (for a given positive integer N) this sup is taken over all pairs of nonempty finite subsets S, T of N such that dist (S,T)≥n.Obviously0≤ρ~(n+1)≤ρ~(n)≤1, n≥0, and ρ~(0)=1 except in the trivial case where all of the random variables Xi are degenerate.Definition 1.
A sequence of random variables is said to be aρ~-mixing sequence of random variables if there exists k∈N such that ρ~(k)<1.
Note that if{Xn,n≥1} is a sequence of independent random variables, then ρ~(n)=0 for all n≥1. ρ~-mixing is similar to ρ-mixing, but both are quite different. ρ(k) is defined by (2) with index sets restricted to subsets S of [1,n] and subsets of T of [n+k,∞), n, k∈N. On the other hand, ρ-mixing sequence assumes the condition ρ(k)→0, but ρ~-mixing sequence assumes the condition that there exists k∈N such that ρ(k)<1; from this point of view, ρ~-mixing is weaker than ρ-mixing.The concept ofρ~-mixing random variables was introduced by Bradley [1] and a number of limit theories for ρ~-mixing sequences of random variables have been established by many authors. We refer to Bradley [1] for the central limit theorem, Bryc and Smoleński [2], Peligrad and Gut [3], and Utev and Peligrad [4] for moment inequalities, Gan [5], Kuczmaszewska [6], and Wu and Jiang [7] for almost sure convergence, and Cai [8], Zhu [9], An and Yuan [10], Zhou et al. [11], Shen and Hu [12], Guo and Zhu [13], Wang et al. [14], and Sung [15, 16] for complete convergence.A sequence{Xn,n≥1} of random variablesconverges completely to the constant C if
(3)∑n=1∞‍P(|Xn-C|>ϵ)<∞foranyϵ>0.In view of the Borel-Cantelli lemma, this implies thatXn→C almost surely. Hence, complete convergence is one of the most important problems in probability theory. Since the concept of complete convergence was introduced by Hsu and Robbins [17], there have been many authors who devoted the study to complete convergence for independent and identically distributed random variables. One of the most important results is Baum and Katz theorem [18]. The theorem was further generalized and extended in different ways. Katz [19] and Chow [20] formed the following generalization with a normalization of Marcinkiewicz-Zygmund type theorem for the strong law of large numbers.Theorem 2 (see [21]).
Letαp≥1, α>1/2, and let {Xn,n≥1} be a sequence of independent and identically distributed random variables. If p≥1, assume that EX1=0. Then the following statements are equivalent:(i)
E|X1|p<∞;(ii)
∑n=1∞nαp-2P(max⁡1≤j≤n|∑i=1jXi|>ϵnα)<∞ for all ϵ>0.In many stochastic models, the assumption of independence among random variables is not plausible. So it is necessary to extend the concept of independence to dependence cases. Peligrad and Gut [3] extended this result from independent and identically distributed case to the case of ρ~-mixing random variables with identical distribution. But they did not prove whether the result of Theorem 2 of the case αp=1 holds for ρ~-mixing sequence. In practical applications it is difficult to check the independence of a sample or the samples are not independent observations. Therefore, in recent investigations limit theorems are very often considered for sequences of dependent random variables. Recently, a number of limit theorems for dependent random variables have been established by many authors. We can refer to Sung [22], Wu and Jiang [23], Wu [24], and Shen [25].Let{Xn,n≥1} be a sequence of identically distributed random variables and let {ank,1≤k≤n,n≥1} be an array of constants. The strong convergence results for weighted sums ∑k=1n‍ankXk have been studied by many authors; see, for example, Cuzick [26], Choi and Sung [27], Bai and Cheng [28], Chen and Gan [29], and so forth. Many useful linear statistics are weighted sums. Examples include least squares estimators, nonparametric regression function estimators, and jackknife estimates.Inspired by Theorem 2.1 of Kuczmaszewska [30], our main purpose in this work is to extend the complete convergence for weighted sums ∑k=1n‍ankXk of independent and identically distributed random variables to the case of ρ~-mixing random variables. However, our proven methods are different from the ones by Kuczmaszewska [30]; by applying inequality (13) of Lemma 10 our proof is much simpler than the one by Kuczmaszewska. Our proof of necessary condition (using Lemma 10) is original. We provide sufficient and necessary conditions of complete convergence for weighted sums of ρ~-mixing random variables with different distributions. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of ρ~-mixing random variables are obtained. In addition, our main results extend and improve the corresponding results of Peligrad and Gut [3].Throughout this paper, the symbolC denotes a positive constant which is not necessarily the same one in each appearance, an=O(bn) will mean an≤C(bn) for sufficiently large n, an≪bn will mean an=O(bn), and I(A) is the indicator function of event A.


## 2. Main Results

Now we state our main results of this paper. The proofs will be given in Section3.Theorem 3.
LetX be a random variable and let {Xn,n≥1} be a sequence of ρ~-mixing random variables satisfying the condition
(4)1n∑k=1n‍P(|Xk|>x)=CP(|X|>x)
for all x>0, all n≥1, and some positive constant C. Let {ank,1≤k≤n,n≥1} be a sequence of real numbers such that
(5)|ank|≍n-α,∀1≤k≤n,n≥1,
where a≍b means a=O(b) and b=O(a). Let αp≥1, α>1/2, and if α≤1, assume that EXn=0, n≥1. Then the following statements are equivalent: (i)
E|X|p<∞,(ii)
∑n=1∞nαp-2P(max⁡1≤k≤n|∑i=1kaniXi|>ϵ)<∞forallϵ>0.Remark 4.
When proving the limit theorem ofρ~-mixing random variables with different distributions, many authors apply the condition of {Xn,n≥1} being stochastically dominated by X, that is, for some constant C>0, P(|Xn|≥x)≤CP(|X|≥x), for all x≥0, n≥1, which implies that (1/n)∑k=1n‍P(|Xk|>x)≤CP(|X|>x), but the converse is not true. Hence our condition of (4) is weaker than the condition of stochastic dominance.When{Xn,n≥1} is a sequence of ρ~-mixing identically distributed random variables and ani=n-α, for all 1≤i≤n, n≥1, then Theorem 3 becomes Baum and Katz complete convergence theorem as follows.Corollary 5.
Let{Xn,n≥1} be a sequence of ρ~-mixing identically distributed random variables. Let αp≥1, α>1/2, and if α≤1, assume that EXn=0, n≥1. Then the following statements are equivalent:(i)
E|X1|p<∞;(ii)
∑n=1∞nαp-2P(max⁡1≤j≤n|∑i=1jXi|>ϵnα)<∞ for all ϵ>0.Remark 6.
Corollary5 not only generalizes Theorem 2 to ρ~-mixing case, but also extends Theorem 2 of Peligrad and Gut [3] to the case αp=1. Therefore, Corollary 5 improves and extends the well-known Baum and Katz theorem.An and Yuan [10, Theorem 2] presented a Marcinkiewicz-Zygmund type strong law of large numbers for ρ~-mixing sequence. We find that the proof of their Theorem 2 is wrong because the theorem is based on Theorem 1 [10]. However, the author thinks that their proofs of Theorem 1 have a little problem, since condition (1.2) does not hold for the array with {ani,1≤i≤n}. An and Yuan [10, Theorem 1] proved the implication (i) ⇒ (ii) under condition (1.3) and proved the converse under conditions (1.2) and (1.3). However, the array satisfying both (1.2) and (1.3) does not exist. Noting that ♯Ank/(k+1)≤∑i=1n‍|ani|p≤O(nδ), we have that ne-1/k≤♯Ank≤(k+1)O(nδ). But this does not hold when k is fixed and n is large enough. In this paper, we obtain a new complete convergence result for weighted sums of ρ~-mixing random variables without assumption of identical distribution. Our result generalizes and sharpens the result of An and Yuan [10]. The following corollary provides the Marcinkiewicz-Zygmund type strong law of large numbers of ρ~-mixing random variables without assumption of identical distribution.Corollary 7.
LetX be a random variable and let {Xn,n≥1} be a sequence of ρ~-mixing random variables satisfying the condition
(6)1n∑k=1n‍P(|Xk|>x)=CP(|X|>x)
for all x>0, all n≥1, and some positive constant C.  E|X|p<∞ for some 0<p<2 and if 1≤p<2, assume that EXn=0, n≥1. Then, for any ϵ>0,
(7)∑n=1∞‍n-1P(max⁡1≤j≤n|∑i=1j‍Xi|>ϵn1/p)<∞,1n1/p|∑i=1n‍Xi|⟶0a.s.,n⟶∞.


## 3. Proof of Main Results

The following lemmas are useful for the proof of the main results.Lemma 8 (see [4]).
SupposeK is a positive integer, 0≤r<1, and q≥2. Then there exists a constant D=D(K,r,q) such that the following statement holds.
If{Xi,i≥1} is a sequence of random variables such that ρ~(K)≤r and EXn=0 and E|Xi|q<∞ for all i≥1, then, for every n≥1,
(8)E(max⁡1≤i≤n|∑j=1i‍Xj|q)≤D[∑i=1n‍E|Xi|q+(∑i=1n‍EXi2)q/2].Lemma 9 (see [30]).
Let{Xn,n≥1} be a sequence of random variables which is weakly mean dominated by a random variable X; that is, for all x≥0 and some positive constant C>0,
(9)1n∑k=1n‍P(|Xk|>x)≤CP(|X|>x).
Then for anyu>0, t>0, and n≥1, the following three statements hold:
(10)IfE|X|u<∞,then1n∑k=1n‍E|Xk|u≤CE|X|u,(11)1n∑k=1n‍E|Xk|uI(|Xk|≤t)≤C[E|X|uI(|X|≤t)+tuP(|X|>t)],(12)1n∑k=1n‍E|Xk|uI(|Xk|>t)≤CE|X|uI(|X|>t).Lemma 10.
Let{Xn,n≥1} be a sequence of ρ~-mixing random variables. Then there exists a positive constant C such that, for any x≥0 and all n≥1,
(13)(12-P(max⁡1≤k≤n|Xk|>x))∑k=1n‍P(|Xk|>x)≤(C2+1)P(max⁡1≤k≤n|Xk|>x).Proof of Lemma10.
Since{max⁡1≤k≤n|Xk|>x}=⋃k=1n‍{|Xk|>x,max⁡1≤j≤k-1|Xj|≤x}, we have
(14)∑k=1n‍P(|Xk|>x)=∑k=1n‍P(|Xk|>x,max⁡1≤j≤k-1|Xj|≤x)+∑k=1n‍P(|Xk|>x,max⁡1≤j≤k-1|Xj|>x)=P(max⁡1≤k≤n|Xk|>x)+∑k=1n‍P(|Xj|>x,max⁡1≤j≤k-1|Xj|>x).
Note that(15)∑k=1n‍P(|Xk|>x,max⁡1≤j≤k-1|Xj|>x)=∑k=1n‍E(I(|Xk|>x)I(max⁡1≤j≤k-1|Xj|>x))≤E(∑k=1n‍I(|Xk|>x)-E(I(|Xk|>x)))×I(max⁡1≤j≤n|Xj|>x)+∑k=1n‍P(|Xk|>x)P(max⁡1≤j≤n|Xj|>x)≜J1+J2.
Obviously, by Lemma8, we get
(16)E(|∑k=1n‍Xk|q)≤C[∑k=1n‍E|Xk|q+(∑k=1n‍EXk2)q/2].
Combining with the Cauchy-Schwarz inequality and (16), we obtain
(17)J1=E(∑k=1n‍I(|Xk|>x)-E(I(|Xk|>x)))×I(max⁡1≤j≤n|Xj|>x)≤[E(∑k=1n‍I(|Xk|>x)-E(I(|Xk|>x)))2×P(max⁡1≤j≤n|Xj|>x)(∑k=1n‍I(|Xk|>x)-E(I(|Xk|>x)))2]1/2≤[C∑k=1n‍P(|Xk|>x)P(max⁡1≤j≤n|Xj|>x)]1/2≤12∑k=1n‍P(|Xk|>x)+C2P(max⁡1≤k≤n|Xk|>x).
Now, we substitute (17) into (15) and then into (14), which implies that (13) holds.Consequently, we prove our main results.Proof of Theorem3.
First, we prove that(i)⇒(ii).
Note thatani=ani+-ani-, where ani+=max⁡{0,ani} and ani-=max⁡{0,-ani}. To prove (ii) it suffices to show that
(18)∑n=1∞nαp-2P(max⁡1≤k≤n|∑i=1k‍ani±Xi|>ϵ)<∞,∀ϵ>0.
Thus, without loss of generality, we may assume thatani>0 for all 1≤i≤n, n≥1.
For fixedn≥1, denote that
(19)Xni=XiI(|Xi|≤nα),1≤i≤n.
Firstly, we show that(20)max⁡1≤k≤n|∑i=1k‍E(aniXni)|⟶0,n⟶∞.
If1/2<α≤1, by EXn=0, (i), (5), (12) of Lemma 9, and αp≥1, we have
(21)max⁡1≤k≤n|∑i=1k‍E(aniXni)|≤∑i=1n‍E|aniXni|≪n-α∑i=1n‍E|Xni|=n-α∑i=1n‍E|Xi|I(|Xi|>nα)≤n1-αE|X|(|X|nα)p-1I(|X|>nα)≪n1-αpE|X|pI(|X|>nα)⟶0,n⟶∞.
Ifα>1, p≥1,   by (5), (11) of Lemma 9, Markov inequality, and E|X|<∞ from (i), we get
(22)max⁡1≤k≤n|∑i=1k‍E(aniXni)|≪n-α∑i=1n‍E|Xi|I(|Xi|≤nα)≪n1-α[E|X|I(|X|≤nα)+nαP(|X|>nα)]≪n1-αE|X|⟶0,n⟶∞.
Ifα>1, 0<p<1, by αp≥1, we can get lim⁡n→∞nP(|X|>nα)=0, and thus
(23)max⁡1≤k≤n|∑i=1k‍E(aniXni)|≪n-α∑i=1n‍E|Xi|I(|Xi|≤nα)≪n1-α[E|X|I(|X|≤nα)+nαP(|X|>nα)]≪n1-α∑i=1n‍E|X|I((i-1)α<|X|≤iα)+nP(|X|>nα).
Note that, ifαp≥1, we have
(24)∑i=1∞‍i1-αE|X|I((i-1)α<|X|≤iα)≤∑i=1∞‍i1-αpE|X|pI((i-1)α<|X|≤iα)≤∑i=1∞‍E|X|pI((i-1)α<|X|≤iα)=E|X|p<∞.
Hence, by Kronecker lemma and (23), we obtain
(25)max⁡1≤k≤n|∑i=1k‍E(aniXni)|≪n1-α∑i=1n‍E|X|I((i-1)α<|X|≤iα)⟶0,n⟶∞.
From (21), (22), and (25) we can get (20) immediately. Hence, for all n sufficiently large and any ϵ>0, we have
(26)max⁡1≤k≤n|∑i=1k‍E(aniXni)|<ϵ2.
It is easy to check that for alln sufficiently large
(27){max⁡1≤k≤n|∑i=1k‍aniXi|>ϵ}⊂⋃i=1n{|aniXi|>ϵ}∪{max⁡1≤k≤n|∑i=1k‍aniXni|>ϵ}⊂⋃i=1n{|aniXi|>ϵ}∪{max⁡1≤k≤n|∑i=1k‍(aniXni-EaniXni)|>ϵ2}≜An∪Bn,
which implies that for all n sufficiently large
(28)P(max⁡1≤k≤n|∑i=1k‍aniXi|>ϵ)≤P(An)+P(Bn).
Therefore, in order to prove (ii), we only need to prove that(29)∑n=1∞‍nαp-2P(An)<∞,(30)∑n=1∞‍nαp-2P(Bn)<∞.
By (4), (5), and αp≥1, we can get that
(31)∑n=1∞‍nαp-2P(An)≤∑n=1∞‍nαp-2∑i=1n‍P(|aniXi|>ϵ)≤∑n=1∞‍nαp-2∑i=1n‍P(|Xi|>ϵani-1≥Cϵnα)≪∑n=1∞‍nαp-1P(|X|>Cϵnα)=∑n=1∞‍nαp-1∑i=n∞‍P(Cϵiα<|X|≤Cϵ(i+1)α)=∑i=1∞‍∑n=1i‍nαp-1P(Cϵiα<|X|≤Cϵ(i+1)α)≤∑i=1∞‍iαpP(Cϵiα<|X|≤Cϵ(i+1)α)≪E|X|p<∞.
That is, (29) holds. Thus, it remains to prove (30).
Since{Xn,n≥1} is a sequence of ρ~-mixing random variables and Xni=XiI(|Xi|≤nα), 1≤i≤n, thus {Xni-EXni,1≤i≤n} is still a sequence of ρ~-mixing random variables with E(Xni-EXni)=0. By Markov inequality and Lemma 8, let q≥2. Then,
(32)∑n=1∞‍nαp-2P(Bn)=∑n=1∞‍nαp-2P(max⁡1≤k≤n|∑i=1k‍(aniXni-EaniXni)|>ϵ2)≪∑n=1∞‍nαp-2E(max⁡1≤k≤n|∑i=1k‍(aniXni-EaniXni)|q)≪∑n=1∞‍nαp-2-αqE(max⁡1≤k≤n|∑i=1k‍(Xni-EXni)|q)≪∑n=1∞‍nαp-2-αq[(∑i=1n‍EXni2)q/2+∑i=1n‍E|Xni|q]≜I1+I2.
Whenp≥2, taking q>max⁡{(αp-1)/(α-1/2),2}, then αp-2-αq+q/2<-1, and by Jensen inequality and (11) of Lemma 9, we have
(33)I1=∑n=1∞‍nαp-2-αq(∑i=1n‍EXni2)q/2≪∑n=1∞‍nαp-2-αq+q/2[EX2I(|X|≤nα)+n2αP(|X|>nα)]q/2≪∑n=1∞‍nαp-2-αq+q/2[EX2]q/2≪∑n=1∞‍nαp-2-αq+q/2<∞.
Takingq>p, we have by (11) of Lemma 9 that
(34)I2=∑n=1∞‍nαp-2-αq∑i=1n‍E|Xni|q≪∑n=1∞‍nαp-2-αq+1[E|X|qI(|X|≤nα)+nαqP(|X|>nα)]≪∑n=1∞‍nαp-1-αqE|X|qI(|X|≤nα)+∑n=1∞‍nαp-1P(|X|>nα)≪∑n=1∞‍nαp-1-αq∑i=1n‍E|X|qI((i-1)α<|X|≤iα)+E|X|p≪∑i=1∞‍∑n=i∞‍nαp-1-αqE|X|qI((i-1)α<|X|≤iα)≪∑i=1∞‍iαp-αqE|X|qI((i-1)α<|X|≤iα)=∑i=1∞‍iαp-αqE|X|p|X|q-pI((i-1)α<|X|≤iα)≤∑i=1∞‍E|X|pI((i-1)α<|X|≤iα)=E|X|p<∞.
Whenp<2, then, taking q=2, by (32), we get
(35)∑n=1∞‍nαp-2P(Bn)≪∑n=1∞‍nαp-2-2α∑i=1n‍E|Xni|2.
Similarly to the proof of inequality (34), we obtain
(36)∑n=1∞‍nαp-2-2α∑i=1n‍E|Xni|2<∞,
which implies that
(37)∑n=1∞‍nαp-2P(Bn)<∞.
Now, we prove the converse. To prove that (ii) implies (i), it suffices to show that(38)∑n=1∞‍nαp-1P(|X|>ϵnα)<∞,∀ϵ>0.
Noting that(39)max⁡1≤i≤n|aniXi|≤max⁡1≤i≤n|∑j=1i‍anjXj|+max⁡1≤i≤n|∑j=1i-1‍anjXj|,
then from (ii) and (5), we have
(40)∑n=1∞‍nαp-2P(max⁡1≤i≤n|Xi|>ϵnα)≪∑n=1∞‍nαp-2P(max⁡1≤i≤n|aniXi|>ϵ)≪∑n=1∞‍nαp-2P(max⁡1≤i≤n|∑j=1i‍anjXj|>ϵ)<∞.
Combining with the condition ofαp≥1,
(41)P(max⁡1≤i≤n|Xi|>ϵnα)⟶0,n⟶∞.
Thus, for sufficiently largen,
(42)P(max⁡1≤i≤n|Xi|>ϵnα)<12.
Therefore, by applying Lemma10, it is easy to see that
(43)∑i=1n‍P(|Xi|>ϵnα)≪P(max⁡1≤i≤n|Xi|>ϵnα),
which, together with the conditions of (4) and (40), gives
(44)∑n=1∞‍nαp-1P(|X|>ϵnα)≪∑n=1∞‍nαp-2∑i=1n‍P(|Xi|>ϵnα)<∞,
which implies that (i) holds. This completes the proof of Theorem 3.Proof of Corollary7.
Takingα=1/p and ani=n-α, for all 1≤i≤n,n≥1, in Theorem 3, we can get (7) immediately; thus
(45)∞>∑n=1∞‍n-1P(max⁡1≤j≤n|∑i=1j‍Xi|>ϵn1/p)=∑i=0∞‍∑n=2i2i+1-1‍n-1P(max⁡1≤j≤n|∑i=1j‍Xi|>ϵn1/p)≥12∑i=1∞‍P(max⁡1≤j≤2i|∑i=1j‍Xi|>ϵ2(i+1)/p).
It follows from Borel-Cantelli lemma that(46)P(max⁡1≤j≤2i|∑i=1j‍Xi|>ϵ2(i+1)/p,i.o.)=0.
Hence,(47)lim⁡i→∞12(i+1)/pmax⁡1≤j≤2i|∑i=1j‍Xi|=0a.s.
For all positive integersn, there exists a nonnegative integer i0 such that 2i0-1≤n<2i0. We have by (47) that
(48)max⁡2i0-1≤n≤2i01n1/p|∑i=1n‍Xi|≤12(i0-1)/pmax⁡1≤j≤2i0|∑i=1j‍Xi|≤22/p2(i0+1)/pmax⁡1≤j≤2i0|∑i=1j‍Xi|⟶0a.s.,i0⟶∞,
which implies that
(49)1n1/p|∑i=1n‍Xi|⟶0a.s.,n⟶∞.
The proof of Corollary7 is completed.



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