# On Complete Convergence and Strong Law for Weighted Sums of i.i.d. Random Variables

Authors: Pingyan Chen; Xiaofang Ma; Soo Hak Sung
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We improve and generalize the result of Stout (1974, Theorem 4.1.3). In particular, the sharp moment conditions are obtained and some well-known results can be obtained as special cases of the main result. The method of the proof is completely different from that in Stout. We also improve and generalize Li et al. (1995) strong law for weighted sums of i.i.d. random variables.


---

## Body

## 1. Introduction and the Main Result

Stout [1] obtained the following celebrated result.Theorem A (see Theorem  4.1.3 in [1] or p. 1556 in Stout [2]).
Let{X,Xn≥1} be a sequence of independent and identically distributed random variables with EX=0 and E|X|2/α<∞ for some 0<α≤1. Suppose that {ank,1≤k≤n,n≥1} is a sequence of constants with
(1)max⁡1≤k≤n⁡|ank|≤Kn-α
for some constant 0<K<∞ and
(2)lim⁡n→∞log⁡n∑k=1nank2=0.
Then
(3)∑n=1∞P{|∑k=1nankXk|>ε}<∞,∀ε>0.Formula (3) is called complete convergence and this concept was introduced by Hsu and Robbins [3]. Sung [4] and Cheng and Wang [5] extended Theorem A to random elements taking values in a Banach space. Wu [6] and Sung [7] extended Theorem A to negatively dependent random variables. It should be pointed out that they all used some exponential inequalities to prove their result and hence the proofs are similar to that of Theorem A except for more computational complexity.When1/2<α≤1, (1) clearly implies (2). Set ank=n-α for 1≤k≤n and n≥1. Then (3) reduces to
(4)∑n=1∞P{|∑k=1nXk|>εnα}<∞,∀ε>0,
which is equivalent to E|X|2/α<∞ and EX=0 by Katz [8] and Baum and Katz [9]. Hence, the moment conditions of Theorem A are sharp when 1/2<α≤1.Next, we consider the case of0<α≤1/2. Lai [10] showed that
(5)∑n=1∞P{|∑k=1nXk|>εnlog⁡n}<∞
for ε>2 provided EX=0, EX2=1, and EX4/log⁡2|X|<∞. Hence, for any ε>0,
(6)∑n=1∞P{|∑k=1nXk|>εcnnlog⁡n}<∞
if cn→∞. Set ank=(cnnlog⁡n)-1 for 1≤k≤n and n≥1. Then (1) and (2) hold for α=1/2. Note that the moment conditions are weaker than those of Theorem A when α=1/2. Hence, the moment conditions of Theorem A may not be optimal for a special case of weighted sums. But it is not known whether the moment conditions of Theorem A are not optimal when 0<α≤1/2.In this paper, we will discuss the optimized moment conditions of Theorem A when0<α≤1/2. We obtain a more generalized complete convergence result for weighted sums which includes the result of Lai [10]. Our method used is completely different from those in Lai [10] and Stout [1].Li et al. [11] obtained the following celebrated result.Theorem B (see Theorem  3.1 in [11]).
Let{X,Xn≥1} be a sequence of independent and identically distributed random variables with EX=0 and E|X|1/α<∞ for some 0<α≤1. Suppose that {ank,1≤k≤n,n≥1} is a sequence of constants with (1) for some constant 0<K<∞ and
(7)lim⁡n→∞⁡nγ∑k=1nank2=0,
where γ>2α-1 if 1/2≤α≤1 and γ>0 if 0<α<1/2. Then
(8)lim⁡n→∞∑k=1n‍ankXk=0a.s.Theorem B has been extended and improved by many authors. Jing and Liang [12] extended and improved to negatively associated (NA) random variables, Budsaba et al. [13] to certain types of U-statistics bases on this kind of weighted sums of NA random variables, and Thanh and Yin [14] to the random weighted sums. In particular, under the condition
(9)limsup⁡n→∞log⁡n∑k=1nank2<∞,
Jing and Liang [12] showed that
(10)limsup⁡n→∞|∑k=1nankXk|<∞a.s.
when 0<α<1/2.Is it possible to find the sharp bound of (10)? In this paper, we will give a definite answer to the question under more general case.Throughout this paper,c represents a positive constant which may vary in different places, [x] denotes the integer part of x, and f(x)~g(x) means f(x)/g(x)→1 as x→∞. It proves convenient to define log⁡x=max⁡{1,ln⁡x}, where ln⁡x denotes the natural logarithm.Now, we are ready to state the main results, and the proofs will be given in the next section.Theorem 1.
Letr>0,0<α≤1/2, g(x) be an increasing and regular varying function at infinity with index α and let h(x) be the inverse function of g(x). Let {X,Xn≥1} be a sequence of independent and identically distributed random variables with EX=0, EX2=1, and Ehr(|X|)|X|2/log⁡h(|X|)<∞. Suppose that {ank,1≤k≤n,n≥1} is a sequence of constants such that
(11)max⁡1≤k≤n⁡|ank|≤Kg-1(n)
for some constant K>0 and
(12)lim⁡n→∞log⁡n∑k=1nank2=ρ
for some constant 0≤ρ<∞. Then
(13)∑n=1∞nr-1P{|∑k=1nankXk|>ε}{<∞ifε>2rρ,=∞if0<ε<2rρ.Remark 2.
Recall that a measurable functionf(x) is said to be regularly varying at infinity with index β if it is positive on [0,∞) and
(14)lim⁡x→∞⁡f(λx)f(x)=λβ,∀λ>0.
We refer to Bingham et al. [15] for other equivalent definitions and for detailed and comprehensive study of properties of regularly varying functions. For example, if β>-1,
(15)∑k=1nf(k)≤cβnf(n),
and if β<-1,
(16)∑k=n∞f(k)≤Cβnf(n),
where cβ>0 and Cβ>0 are constants depending only on β.Remark 3.
Whenr=1 and g(x)=xα, the last moment condition of Theorem 1 is reduced to E|X|1/α+2/log⁡|X|<∞. When 0<α≤1/2, 1/α+2≤2/α and so the moment conditions in Theorem 1 are strictly weaker than those in Theorem A.Remark 4.
Letank=g-1(n) if 1≤k≤[g2(n)/log⁡n] and ank=0 if k>[g2(n)/log⁡n]. Then, it is easy to show that
(17)lim⁡n→∞log⁡n∑k=1n|ank|2=1.
Hence, (11) and (12) hold. Under the moment conditions of Theorem 1,
(18)∑n=1∞nr-1P{|∑k=1[g2(n)/log⁡n]Xk|>εg(n)}<∞
for all ε>2r. On the other hand, it is easy to show that if the above formula holds for some ε>0, then Ehr(|X|)|X|2/log⁡h(|X|)<∞ by the similar argument as in Lai [10]. Thus, the moment conditions of Theorem 1 are sharp in the sense that the moment conditions on X cannot be weakened.Remark 5.
Letg(x)=xlog⁡x and ank=g-1(n) for 1≤k≤n and n≥1. Then, by Theorem 1, the moment conditions EX=0, EX2=1, and E(X2/log⁡|X|)r+1<∞ imply that
(19)∑n=1∞nr-1P{|∑k=1nXk|>εnlog⁡n}{<∞ifε>2r,=∞if0<ε<2r.
So, the sufficient part of Theorem 3 in Lai [10] is a special case of Theorem 1.
By Theorem1 and Borel-Cantelli lemma, we have the following corollary.Corollary 6.
Under the conditions of Theorem1, let r=1, {Xnk,1≤k≤n,n≥1} be an array of independent random variables with the same distribution as X. Then
(20)limsup⁡n→∞|∑k=1nankXnk|=2ρa.s.
In particular, the moment conditions EX=0, EX2=1, and EX4/log⁡2|X|<∞ imply that
(21)limsup⁡n→∞|∑k=1nXnk|2nlog⁡n=1a.s.Remark 7.
Formula (21) is called the law of single logarithm which is due to Hu and Weber [16]. They proved it under the strong moment condition EX4<∞. Qi [17] and Li et al. [18] independently proved that (21) is equivalent to conditions EX=0, EX2=1, and EX4/log⁡2|X|<∞. In particular, Li et al. [18] gave a version of random elements taking values in a Banach space.Theorem 8.
Let0<α<1/2, g(x) be an increasing and regular varying function at infinity with index α and let h(x) be the inverse function of g(x). Let {X,Xn≥1} be a sequence of independent and identically distributed random variables with EX=0, EX2=1, and Eh(|X|)<∞. Suppose that {ank,1≤k≤n,n≥1} is a sequence of constants such that (11) holds for some constant K>0 and
(22)limsup⁡n→∞log⁡n∑k=1nank2=ρ
for some constant 0≤ρ<∞. Then
(23)limsup⁡n→∞|∑k=1nankXk|≤2ρa.s.Remark 9.
For anyp∈(0,1), set
(24)ank=(4nπ)1/4(log⁡n)-1/2(nk)pk(1-p)n-k.
Then, by Embrechts and Maejima [19],
(25)max⁡1≤k≤n⁡ank~1π1/4(p(1-p))1/2·1n1/4(log⁡n)1/2,log⁡n∑k=1nank2~2-1(p(1-p))-1/2
as n→∞. Hence, from Theorem 8, EX=0,EX2=1,E(|X|2/log⁡|X|)2<∞ imply that
(26)limsup⁡n→∞(4nπ)1/4(log⁡n)-1/2|∑k=1n(nk)pk(1-p)n-kXk|≤(p(1-p))-1/4a.s.
In fact, Lai [20] has proved that
(27)limsup⁡n→∞(4nπ)1/4(log⁡n)-1/2|∑k=1n(nk)pk(1-p)n-kXk|=(p(1-p))-1/4a.s.
if and only if EX=0,EX2=1,E(|X|2/log⁡|X|)2<∞. Therefore, both the upper bound and the moment condition of Theorem 8 are sharp.Remark 10.
Letα,β>0 and 1/α+1/β=1/2. Let {bnk,1≤k≤n,n≥1} be a sequence of constants with limsup⁡n→∞n-1∑k=1n|bnk|α<∞. Set ank=(nlog⁡n)-1bnk. Then, it is easy to show that
(28)max⁡1≤k≤n⁡|ank|≤K(n1/βlog⁡n)-1
for some K>0 and
(29)ρ=limsup⁡n→∞log⁡n∑k=1n|ank|2=limsup⁡n→∞n-1∑k=1n|bnk|2≤limsup⁡n→∞(n-1∑k=1n‍|bnk|α)2/α<∞
by Hölder’s inequality. Hence, from Theorem 8, EX=0,EX2=1, and E(|X|β/(log⁡|X|)β/2)<∞ imply that
(30)limsup⁡n→∞|∑k=1nbnkXk|nlog⁡n=limsup⁡n→∞|∑k=1nankXk|≤2ρa.s.
So, Theorem 1.4 in Chen and Gan [21] is a corollary of Theorem 8.Remark 11.
Whenα=1/2, Theorem 8 also holds, but the proof is completely different. So, we will discuss it in the other paper.


## 2. Proofs of the Main Results

The main idea in the proofs of the main results is to use the following invariance principle (see Sakhanenko [22–24]), which is a powerful tool in the field of limit theory (e.g., see Csörgő et al. [25], Jiang and Zhang [26], Chen and Gan [21], and Chen and Wang [27]).Lemma 12.
Let{ξk,1≤k≤n} be a sequence of independent random variables with Eξk=0 and Eξk2<∞ for 1≤k≤n. Then, there exists a sequence of independent normal random variables {ηk,1≤k≤n} with Eηk=0 and Eηk2=Eξk2 such that, for all q>2 and y>0,
(31)P{|∑k=1nξk-∑k=1nηk|>y}≤By-q∑k=1nE|ξk|q,
where B is a positive constant depending only on q.Proof of Theorem1.
For1≤k≤n and n≥1, we let
(32)Ynk=XkI(|Xk|>g(n))-EXkI(|Xk|>g(n)),Znk=Xk-Ynk.
We first prove that
(33)∑n=1∞nr-1P{|∑k=1nankXk|>ε}<∞,∀ε>2rρ.
For ε>2rρ, let ε=ε1+ε2, where ε1>0 and ε2>2rρ. Then
(34)∑n=1∞nr-1P{|∑k=1nankXk|>ε}≤∑n=1∞nr-1P{|∑k=1n‍ankYnk|>ε1}+∑n=1∞nr-1P{|∑k=1nankZnk|>ε2}∶=I1+I2.
For I1, we have, by Markov’s inequality, (12), (15), and a standard computation, that
(35)I1≤c∑n=1∞nr-1∑k=1nank2EX2I(|X|>g(n))≤c∑n=1∞nr-1log⁡nEX2I(|X|>g(n))≤cEhr(|X|)|X|2log⁡h(|X|)<∞.
For I2, we will use Lemma 12. By Lemma 12, there exists an array {ηnk,1≤k≤n,n≥1} of rowwise independent normal random variables with Eηnk=0,  Eηnk2=E|ankZnk|2 such that, for all q>2 and y>0,
(36)P{|∑k=1nankZnk-∑k=1nηnk|>y}≤By-q∑k=1n|ank|qE|X|qI(|X|≤g(n)).
For ε2>2rρ, let ε2=ε3+ε4, where ε3>0 and ε4>2rρ. Then
(37)I2≤∑n=1∞nr-1P{|∑k=1nankZnk-∑k=1nηnk|>ε3}+∑n=1∞nr-1P{|∑k=1nηnk|>ε4}∶=I21+I22.
Take q>2 such that r-1-α(q-2)<-1. Then, we have, by (36), (11), (12), (16), and a standard computation, that
(38)I21≤c∑n=1∞nr-1∑k=1n|ank|qE|X|qI(|X|≤g(n))≤c∑n=1∞nr-1g-(q-2)(n)log⁡nE|X|qI(|X|≤g(n))≤cEhr(|X|)|X|2log⁡h(|X|)<∞.
We finally prove that I22<∞. To do this, let δ>1 be sufficiently close to 1 such that ε4>δ2rρ. Then ∑k=1nank2≤ε42/(2rδlog⁡n) for all large enough n. Let N be a standard normal random variable. It is well known that P{|N|>x}~2/πx-1e-x2/2. Since EX2=1, EZnk2=E{XI(|X|≤g(n))-EXI(|X|≤g(n))}2≤1 for all 1≤k≤n and n≥1. Hence, for all large enough n,
(39)P{|∑k=1nηnk|>ε4}≤P{|N|>ε4∑k=1nank2}≤c∑k=1n‍ank2exp⁡⁡{-ε422∑k=1nank2}≤c(log⁡n)-1/2n-rδ,
which gives I22<∞.
Next, we prove that(40)∑n=1∞nr-1P{|∑k=1nankXk|>ε}=∞,∀0<ε<2rρ.
For 0<ε<2rρ, let ε=-ε5-ε6+ε7, where ε5>0,ε6>0 and 0<ε7<2rρ. Then
(41)∑n=1∞nr-1P{|∑k=1nηnk|>ε7}≤∑n=1∞nr-1P{|∑k=1naniYnk|>ε5}+∑n=1∞nr-1P{|∑k=1nankZnk-∑k=1nηnk|>ε6}+∑n=1∞nr-1P{|∑k=1nankXk|>ε}.
The first series on the right hand side converges by I1<∞. The second series converges by I21<∞. But by using P{|N|>x}~2/πx-1e-x2/2, it is easy to show that the series on the left-hand side diverges. Hence, the last series on the right-hand side also diverges. That is, (40) holds.Proof of Theorem8.
Letδ>0 be given. Let
(42)Xnk′=XkI(|ankXk|≤n-Δ)
for Δ>0 to be specified below and let
(43)Tn′=∑k=1nankXnk′.
Let
(44)Xnk′′=XkI(|ankXk|>δN)
for an integer N to be specified below and let
(45)Tn′′=∑k=1nankXnk′′.
Let
(46)Xnk′′′=Xk-Xnk′-Xnk′′=XkI(n-Δ<|ankXk|≤δN)
and let
(47)Tn′′′=∑k=1nankXnk′′′.
Since δ>0 is arbitrary, to prove (22), it is enough to show that
(48)limsup⁡n→∞|Tn′-ETn′|≤2ρa.s.ETn′⟶0Tn′′⟶0a.s.limsup⁡n→∞|Tn′′′|≤δa.s.
Using the same argument as in the proof of Theorem 1, we have that, for any ε>2ρ with ε=ε1+ε2, where ε1>2ρ and ε2>0,
(49)∑n=1∞P{|Tn′-ETn′|>ε}≤∑n=1∞(P{|∑k=1kηnk|>ε1}kkkk+P{|Tn′-ETn′-∑k=1kηnk|>ε2})≤c∑n=1∞‍∑k=1nank2exp⁡{-ε122∑k=1nank2}+c∑n=1∞‍∑k=1nE|ankXnk′|q≤c∑n=1∞‍exp⁡{-ε122∑k=1nank2}+c∑n=1∞‍n1-qΔ<∞
when q>2 is large enough. Hence, by Borel-Cantelli lemma, limsup⁡n→∞|Tn′-ETn′|≤2ρa.s. Taking r∈(2,1/α), we get, by EX=0, (11), and (22), that
(50)|ETn′|≤∑k=1n‍E|ankXk|I(|ankXk|>n-Δ)≤n(r-1)Δ∑k=1n‍E|ankXk|rI(|ankXk|>n-Δ)≤cn(r-1)Δ∑k=1n‍|ank|r≤cn(r-1)Δ(g(n))-(r-2);
Hence, ETn′→0 when Δ is small enough. By Eh(|X|)<∞,
(51)∑k=1∞‍P{|X|>(δNK)g(k)}<∞
which implies that the series ∑k=1∞‍|Xk|I(|Xk|>(δ/NK)g(k)) converges almost surely by Borel-Cantelli lemma. Hence,
(52)|Tn′′|≤Kg-1(n)∑k=1n‍|Xk|I(|Xk|>(δNK)g(n))≤Kg-1(n)∑k=1∞‍|Xk|I(|Xk|>(δNK)g(k))⟶0a.s.
To prove limsup⁡n→∞|Tn′′′|≤δa.s., by Borel-Cantelli lemma, it is enough to show that
(53)∑n=1∞‍P{|Tn′′′|>δ}<∞.
Since Xnk′′′=XkI(n-Δ<|ankXk|≤δ/N), |Tn′′′|=|∑k=1nankXnk′′′|>δ implies that there must exist at least N indices k such that Xnk′′′≠0. Hence, by Bonferroni’s inequality (see, e.g., Lemma  4.1.2 in [1]) and Markov’s inequality,
(54)P{|Tn′′′|>δ}≤P{Xnk′′′there exist at  leastNKKKindicesksuch thatXnk′′′≠0}≤∑1≤k1<k2<⋯<kN≤n∏j=1N‍P{|ankjXkj|>n-Δ}≤(∑k=1n‍P{|ankXk|>n-Δ})N≤(npΔ∑k=1n‍|ank|pE|X|p)N≤cnpΔN(g(n))-N(p-2),
where p∈(2,1/α). Thus, choosing Δ sufficiently small and N sufficiently large such that N[α(p-2)-pΔ]>1, we have
(55)∑n=1∞‍P{|Tn′′′|>δ}≤c∑n=1∞‍npΔN[g(n)]-N(p-2)<∞,
since the function xpΔN[g(x)]-N(p-2) is regularly varying at infinity with index -N[α(p-2)-pΔ]<-1. This completes the proof.



---

*Source: 251435-2014-06-25.xml*