# Complete Convergence for Moving Average Process of Martingale Differences

Authors: Wenzhi Yang; Shuhe Hu; Xuejun Wang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Under some simple conditions, by using some techniques such as truncated method for random variables (see e.g., Gut (2005)) and properties of martingale differences, we studied the moving process based on martingale differences and obtained complete convergence and complete moment convergence for this moving process. Our results extend some related ones.


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## Body

## 1. Introduction

Let{Yi,-∞<i<∞} be a doubly infinite sequence of random variables. Assume that {ai,-∞<i<∞} is an absolutely summable sequence of real numbers and
(1.1)Xn=∑i=-∞∞aiYi+n,n≥1
is the moving average process based on the sequence {Yi,-∞<i<∞}. As usual, Sn=∑k=1n‍Xk, n≥1, denotes the sequence of partial sums.For the moving average process{Xn,n≥1}, where {Yi,-∞<i<∞} is a sequence of independent identically distributed (i.i.d.) random variables, Ibragimov [1] established the central limit theorem, Burton and Dehling [2] obtained a large deviation principle, and Li et al. [3] gave the complete convergence result for {Xn,n≥1}. Zhang [4] and Li and Zhang [5] extended the complete convergence of moving average process for i.i.d. sequence to φ-mixing sequence and NA sequence, respectively. Theorems  A and  B are due to Zhang [4] and Kim et al. [6], respectively.Theorem A.
Suppose that{Yi,-∞<i<∞} is a sequence of identically distributed φ-mixing random variables with ∑m=1∞‍φ1/2(m)<∞ and {Xn,n≥1} is as in (1.1). Let h(x)>0(x>0) be a slowly varying function and 1≤p<2, r≥1. If Y1=0 and E[|Y1|rph(|Y1|p)]<∞, then
(1.2)∑n=1∞nr-2h(n)P(|Sn|≥ɛn1/p)<∞,∀ɛ>0.Theorem B.
Suppose that{Yi,-∞<i<∞} is a sequence of identically distributed φ-mixing random variables with EY1=0, EY12<∞ and ∑m=1∞‍φ1/2(m)<∞ and {Xn,n≥1} is as in (1.1). Let h(x)>0(x>0) be a slowly varying function and 1≤p<2, r>1. If E[|Y1|rph(|Y1|p)]<∞, then
(1.3)∑n=1∞nr-2-1/ph(n)E(|Sn|-ɛn1/p)+<∞,∀ɛ>0,
where x+=max{x,0}.Chen et al. [7] and Zhou [8] also studied the limit behavior of moving average process under φ-mixing assumption. Yang et al. [9] investigated the moving average process for AANA sequence. For more works on complete convergence, one can refer to [3–6, 10–13] and the references therein.Recall that the sequence{Xn,n≥1} is stochastically dominated by a nonnegative random variable X if
(1.4)sup⁡n≥1⁡P(|Xn|>t)≤CP(X>t)forsomepositiveconstantC,∀t≥0.Recently, Chen and Li [14] investigated the limit behavior of moving process under martingale difference sequences. They obtained the following theorems.Theorem C.
Letr≥1, 1≤p<2 and rp<2. Assume that {Xn,n≥1} is a moving average process defined in (1.1), where {Yi,ℱi,-∞<i<∞} is a martingale difference related to an increasing sequence of σ-fields ℱi and stochastically dominated by a nonnegative random variable Y. If E[Yrp+Ylog(1+Y)]<∞, then for every ɛ>0,
(1.5)∑n=1∞nr-2P(max⁡1≤k≤n⁡|Sk|>ɛn1/p)<∞.Theorem D.
Letr≥1, 1≤p<2, rp<2 and 0<q<2. Assume that {Xn,n≥1} is a moving average process defined in (1.1), where {Yi,ℱi,-∞<i<∞} is a martingale difference related to an increasing sequence of σ-fields ℱi and stochastically dominated by a nonnegative random variable Y. If
(1.6){E[Yrp+Ylog⁡(1+Y)]<∞,ifq<rp,E[Yrplog(1+Y)+Ylog2(1+Y)]<∞,ifq=rp,EYq<∞,ifq>rp,
then for every ɛ>0,
(1.7)∑n=1∞nr-2-q/pE(max⁡1≤k≤n⁡|Sk|-ɛn1/p)+q<∞,
where x+=x when x>0 and x+=0 when x≤0 and x+q=(x+)q.Inspired by Chen and Li [14], Chen et al. [7], Sung [13] and other papers above, we go on to investigate the limit behavior of moving process under martingale difference sequence and obtain some similar results of Theorems  C and  D, but we only need some simple conditions. Our results extend some results of Chen and Li [14] (see Remark 3.3 in Section 3). Two lemmas and two theorems are given in Sections 2 and 3, respectively. The proofs of theorems are presented in Section 4.For various results of martingales, one can refer to Chow [15], Hall and Heyde [16], Yu [17], Ghosal and Chandra [18], and so forth. As an application of moving average process based on martingale differences, we can refer to [19–22] and the references therein. Throughout the paper, I(A) is the indicator function of set A, x+=max{x,0} and C, C1, C2,… denote some positive constants not depending on n, which may be different in various places.


## 2. Two Lemmas

The following lemmas are our basic techniques to prove our results.Lemma 2.1 (cf. Hall and Heyde [16, Theorem  2.11]).
If{Xi,ℱi,1≤i≤n} is a martingale difference and p>0, then there exists a constant C depending only on p such that
(2.1)E(max⁡1≤k≤n|∑i=1k‍Xi|p)≤C{E(∑i=1n‍E(Xi2∣Fi-1))p/2+E(max⁡1≤i≤n|Xi|p)}.Lemma 2.2 (cf. Wu [23, Lemma  4.1.6]).
Let{Xn,n≥1} be a sequence of random variables, which is stochastically dominated by a nonnegative random variable X. Then for any a>0 and b>0, the following two statements hold:
(2.2)E[|Xn|aI(|Xn|≤b)]≤C1{E[XaI(X≤b)]+baP(X>b)},E[|Xn|aI(|Xn|>b)]≤C2E[XaI(X>b)],
where C1 and C2 are positive constants.


## 3. Main Results

Theorem 3.1.
Letr>1 and 1≤p<2. Assume that {Xn,n≥1} is a moving average processes defined in (1.1), where {Yi,ℱi,-∞<i<∞} is a martingale difference related to an increasing sequence of σ-fields ℱi and stochastically dominated by a nonnegative random variable Y. Let K be a constant. Suppose that EYrp<∞ for rp>1 and supiE(|Yi|rp∣ℱi-1)≤K almost surely (a.s.), if rp≥2. Then for every ɛ>0,
(3.1)∑n=1∞‍nr-2P(max⁡1≤k≤n|Sk|>ɛn1/p)<∞,(3.2)∑n=1∞‍nr-2P(supk≥n|Skk1/p|>ɛ)<∞.Theorem 3.2.
Let the conditions of Theorem3.1 hold. Then for every ɛ>0,
(3.3)∑n=1∞‍nr-2-1/pE(max⁡1≤k≤n|Sk|-ɛn1/p)+<∞,(3.4)∑n=1∞‍nr-2E(supk≥n|Skk1/p|-ɛ)+<∞.Remark 3.3.
Letℱ0⊂ℱ1⊂… be an increasing family of σ-algebras and {(Xn,ℱn),n≥1} be a sequence of martingale differences. Assume that for some p≥2,
(3.5)E(|Xn|p∣Fn-1)≤K,a.s.,
where K is a constant not depending on n, and other conditions are satisfied, Yu [17] investigated the complete convergence of weighted sums of martingale differences. On the other hand, under the condition
(3.6)sup⁡n,k⁡E(Xn,k2∣Fn,k-1)≤K,a.s.,
and other conditions, Ghosal and Chandra [18] obtained the complete convergence of martingale arrays. Thus, if rp≥2, our assumption supiE(|Yi|rp∣ℱi-1)≤K, a.s., is reasonable. Chen and Li [14] obtained Theorems  C and  D for the case 1≤rp<2. We go on to investigate this moving average process for the case rp>1, especially for the case rp≥2 and get the results of (3.1)–(3.4). If E[Yrp+Ylog(1+Y)]<∞ for r>1, 1≤p<2 and rp<2, result (3.1) follows from Theorem  C (see Theorem  1.1 of Chen and Li), but we can obtain results (3.1) and (3.2) under weaker condition EYrp<∞. On the other hand, comparing with the conditions of Theorem  D, our conditions of Theorem 3.2 are relatively simple.


## 4. The Proofs of Main Results

Proof of Theorem3.1.
First, we show that the moving average process (1.1) converges a.s. under the conditions of Theorem 3.1. Since rp>1, it has EY<∞, following from EYrp<∞. On the other hand, applying Lemma 2.2 with a=1 and b=1, one has
(4.1)E|Yi|≤1+C2E[YI(Y>1)]≤1+C2EY<∞,-∞<i<∞.
Consequently, we have by ∑i=-∞∞‍|ai|<∞ that
(4.2)∑i=-∞∞E|aiYi+n|≤C3∑i=-∞∞|ai|<∞,
which implies ∑i=-∞∞‍aiYi+n converges a.s.
Note that(4.3)Sn=∑k=1nXk=∑k=1n∑i=-∞∞aiYi+k=∑i=-∞∞ai‍∑k=i+1i+nYk.
Let
(4.4)Ynj=YjI(|Yj|≤n1/p)-E[YjI(|Yj|≤n1/p)∣Fj-1],-∞<j<∞.
Since Yj=YjI(|Yj|>n1/p)+Ynj+E[YjI(|Yj|≤n1/p)∣ℱj-1], we can see that
(4.5)∑n=1∞nr-2P(max⁡1≤k≤n|Sk|>ɛn1/p)∑n=1∞nr-2P≤∑n=1∞‍nr-2P(max⁡1≤k≤n⁡|∑i=-∞∞‍ai∑j=i+1i+k‍YjI(|Yj|>n1/p)|>ɛn1/p2)∑n=1∞nr-2P≤+∑n=1∞‍nr-2P(max⁡1≤k≤n⁡|∑i=-∞∞‍ai∑j=i+1i+k‍E[YjI(|Yj|≤n1/p)∣Fj-1]|>ɛn1/p4)∑n=1∞nr-2P≤+∑n=1∞‍nr-2P(max⁡1≤k≤n⁡|∑i=-∞∞‍ai∑j=i+1i+k‍Ynj|>ɛn1/p4)∑n=1∞nr-2P=:H+I+J.
ForH, by Markov’s inequality, Lemma 2.2, ∑i=-∞∞‍|ai|<∞ and EYrp<∞, one has
(4.6)H≤2ɛ∑n=1∞nr-2n-1/pE(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍YjI(|Yj|>n1/p)|)≤C1∑n=1∞nr-2n-1/p∑i=-∞∞‍|ai|E(max⁡1≤k≤n∑j=i+1i+k‍|Yj|I(|Yj|>n1/p))≤C2∑n=1∞nr-1-1/pE[YI(Y>n1/p)]=C2∑n=1∞nr-1-1/p∑m=n∞E[YI(m<Yp≤m+1)]=C2∑m=1∞E[YI(m<Yp≤m+1)]∑n=1mnr-1-1/p≤C3∑m=1∞mr-1/pE[YI(m<Yp≤m+1)]≤C4E|Y|rp<∞.
Meanwhile, by the martingale property, Lemma 2.2 and the proof of (4.6), it follows that
(4.7)I≤4ɛ∑n=1∞nr-2n-1/pE(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍E[YjI(|Yj|≤n1/p)∣Fj-1]|)=4ɛ∑n=1∞nr-2n-1/pE(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍E[YjI(|Yj|>n1/p)∣Fj-1]|)≤C1∑n=1∞nr-2n-1/p∑i=-∞∞‍|ai|∑j=i+1i+n‍E[|Yj|I(|Yj|>n1/p)]≤C2∑n=1∞nr-1-1/pE[YI(Y>n1/p)]≤C3E|Y|rp<∞.
Obviously, one can find that {Ynj,ℱj-1,-∞<j<∞} is a martingale difference. So, by Markov’s inequality, Hölder’s inequality, and Lemma 2.1, we get that for any q≥2,
(4.8)J≤(4ɛ)q∑n=1∞‍nr-2n-q/pE{max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍Ynj|q}≤(4ɛ)q∑n=1∞‍nr-2n-q/pE{∑i=-∞∞‍(|ai|1-1/q)(|ai|1/qmax⁡1≤k≤n|∑j=i+1i+k‍Ynj|)}q≤(4ɛ)q∑n=1∞‍nr-2n-q/p(∑i=-∞∞‍|ai|)q-1∑i=-∞∞‍|ai|E{max⁡1≤k≤n|∑j=i+1i+k‍Ynj|q}≤C1∑n=1∞‍nr-2n-q/p∑i=-∞∞‍|ai|E(∑j=i+1i+n‍E(Ynj2∣Fj-1))q/2+C1∑n=1∞‍nr-2n-q/p∑i=-∞∞‍|ai|∑j=i+1i+n‍E|Ynj|q≕C1J1+C1J2.
If rp≥2, then we take q large enough such that q>max{(r-1)/(1/p-1/2),rp}. From supjE(|Yj|rp∣ℱj-1)≤K, a.s. and Jensen’s inequality for conditional expectation, we have supjE(Yj2∣ℱj-1)≤K2/(rp), a.s. On the other hand,
(4.9)E(Ynj2∣Fj-1)=E[Yj2I(|Yj|≤n1/p)∣Fj-1]-[E(|Yj|I(|Yj|≤n1/p)∣Fj-1)]2≤E[Yj2I(|Yj|≤n1/p)∣Fj-1],a.s.,-∞<j<∞.
Consequently, we obtain by ∑i=-∞∞‍|ai|<∞ that
(4.10)J1≤C1∑n=1∞‍nr-2n-q/p∑i=-∞∞‍|ai|E(∑j=i+1i+n‍E[Yj2I(|Yj|≤n1/p)∣Fj-1])q/2≤C2∑n=1∞‍nr-2-q/p+q/2<∞,
following from the fact that q>(r-1)/(1/p-1/2). Meanwhile, by Cr inequality, Lemma 2.2 and ∑i=-∞∞‍|ai|<∞,
(4.11)J2≤C4∑n=1∞‍nr-2n-q/p∑i=-∞∞‍|ai|∑j=i+1i+n‍E[|Yj|qI(|Yj|≤n1/p)]≤C5∑n=1∞‍nr-1-q/pE[YqI(Y≤n1/p)]+C6∑n=1∞‍nr-1P(Y>n1/p)≤C5∑n=1∞‍nr-1-q/pE[YqI(Y≤n1/p)]+C6∑n=1∞‍nr-1-1/pE[YI(Y>n1/p)]=:C5J21+C6J22.
Since q>rp and EYrp<∞, one has
(4.12)J21=∑n=1∞‍nr-1-q/p∑i=1n‍E[YqI((i-1)1/p<Y≤i1/p)]=∑i=1∞‍E[YqI((i-1)1/p<Y≤i1/p)]∑n=i∞‍nr-1-q/p≤C1∑i=1∞E[YrpYq-rpI((i-1)1/p<Y≤i1/p)]ir-q/p≤C1EYrp<∞.
By the proof of (4.6),
(4.13)J22=∑n=1∞nr-1-1/pE[YI(Y>n1/p)]≤CEYrp<∞.
If rp<2, then we take q=2. Similar to the proofs of (4.8), and (4.11), it has
(4.14)J≤C1∑n=1∞‍nr-2n-2/p∑i=-∞∞‍|ai|∑j=i+1i+n‍EYnj2≤C2∑n=1∞‍nr-2n-2/p∑i=-∞∞‍|ai|∑j=i+1i+n‍E[Yj2I(|Yj|≤n1/p)]≤C3EYrp<∞,
following from q>rp, (4.12), and (4.13). Therefore, (3.1) follows from (4.5)–(4.13) and the inequality above.
Inspired by the proof of Theorem  12.1 of Gut [24], it can be checked that(4.15)∑n=1∞‍nr-2P(sup⁡k≥n|Skk1/p|>22/pɛ)=∑m=1∞‍∑n=2m-12m-1‍nr-2P(sup⁡k≥n|Skk1/p|>22/pɛ)≤22-r∑m=1∞‍P(sup⁡k≥2m-1|Skk1/p|>22/pɛ)∑n=2m-12m-1‍2m(r-2)≤22-r∑m=1∞‍2m(r-1)P(sup⁡k≥2m-1|Skk1/p|>22/pɛ)=22-r∑m=1∞‍2m(r-1)P(sup⁡l≥mmax⁡2l-1≤k<2l|Skk1/p|>22/pɛ)≤22-r∑m=1∞‍2m(r-1)∑l=m∞‍P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p)=22-r∑l=1∞‍P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p)∑m=1l‍2m(r-1)≤C1∑l=1∞‍2l(r-1)P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p):=D.
If r<2, then
(4.16)D=22-rC1∑l=1∞‍∑n=2l2l+1-1‍2(l+1)(r-2)P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p)≤22-rC1∑l=1∞‍∑n=2l2l+1-1‍nr-2P(max⁡1≤k≤n|Sk|>ɛn1/p)≤22-rC1∑n=1∞‍nr-2P(max⁡1≤k≤n|Sk|>ɛn1/p).
Otherwise,
(4.17)D=C1∑l=1∞‍∑n=2l2l+1-1‍2l(r-2)P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p)≤C1∑l=1∞‍∑n=2l2l+1-1‍nr-2P(max⁡1≤k≤n|Sk|>ɛn1/p)≤C1∑n=1∞‍nr-2P(max⁡1≤k≤n|Sk|>ɛn1/p).
Combining (3.1) with these inequalities above, we obtain (3.2) immediately.Proof of Theorem3.2.
For allɛ>0, it has
(4.18)∑n=1∞‍nr-2-1/pE(max⁡1≤k≤n|Sk|-ɛn1/p)+=∑n=1∞‍nr-2-1/p∫0∞P(max⁡1≤k≤n|Sk|-ɛn1/p>t)dt=∑n=1∞‍nr-2-1/p∫0n1/pP(max⁡1≤k≤n|Sk|-ɛn1/p>t)dt+∑n=1∞‍nr-2-1/p∫n1/p∞P(max⁡1≤k≤n|Sk|-ɛn1/p>t)dt≤∑n=1∞‍nr-2P(max⁡1≤k≤n|Sk|>ɛn1/p)+∑n=1∞‍nr-2-1/p∫n1/p∞‍P(max⁡1≤k≤n|Sk|>t)dt.
By Theorem 3.1, in order to proof (3.3), we only have to show that
(4.19)∑n=1∞nr-2-1/p∫n1/p∞P(max⁡1≤k≤n⁡|Sk|>t)dt‍<∞.
For t>0, denote
(4.20)Ytj=YjI(|Yj|≤t)-E[YjI(|Yj|≤t)∣Fj-1],-∞<j<∞.
Since Yj=YjI(|Yj|>t)+Ytj+E[YjI(|Yj|≤t)∣ℱj-1], it is easy to see that
(4.21)∑n=1∞‍nr-2-1/p∫n1/p∞‍P(max⁡1≤k≤n|Sk|>t)dt∑n=1∞‍nr-2-1/p≤∑n=1∞‍nr-2-1/p∫n1/p∞‍P(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍YjI(|Yj|>t)|>t2)dt∑n=1∞‍nr-2-1/p≤+∑n=1∞‍nr-2-1/p∫n1/p∞‍P(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍Ytj|>t4)dt∑n=1∞‍nr-2-1/p≤+∑n=1∞‍nr-2-1/p∫n1/p∞‍P(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍E[YjI(|Yj|≤t)∣Fj-1]|>t4)dt∑n=1∞‍nr-2-1/p=:I1+I2+I3.
By Markov’s inequality, Lemma 2.2 and EYrp<∞,
(4.22)I1≤2∑n=1∞‍nr-2-1/p∫n1/p∞‍t-1E(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍YjI(|Yj|>t)|)dt≤2∑n=1∞‍nr-2-1/p∫n1/p∞‍t-1∑i=-∞∞‍|ai|E(max⁡1≤k≤n∑j=i+1i+k‍|Yj|I(|Yj|>t))dt≤C1∑n=1∞‍nr-1-1/p∫n1/p∞‍t-1E[YI(Y>t)]dt=C1∑n=1∞‍nr-1-1/p∑m=n∞‍∫m1/p(m+1)1/p‍t-1E[YI(Y>m1/p)]dt≤C2∑n=1∞‍nr-1-1/p∑m=n∞‍m1/p-1-1/pE[YI(Y>m1/p)]=C2∑m=1∞‍m-1E[YI(Y>m1/p)]∑n=1m‍nr-1-1/p≤C3∑m=1∞‍mr-1-1/pE[YI(Y>m1/p)]≤C4EYrp<∞.
Since {Ytj,ℱj-1,-∞<j<∞} is a martingale difference, we have by Markov’s inequality, Hölder’s inequality, and Lemma 2.1 that for any q≥2,
(4.23)I2≤4q∑n=1∞‍nr-2-1/p∫n1/p∞‍t-qE(max⁡1≤k≤n|∑i=-∞∞‍ai∑j=i+1i+k‍Ytj|q)dt≤C∑n=1∞‍nr-2-1/p∫n1/p∞‍t-qE{∑i=-∞∞‍(|ai|1-1/q)(|ai|1/qmax⁡1≤k≤n|∑j=i+1i+k‍Ytj|)}qdt≤C∑n=1∞‍nr-2-1/p∫n1/p∞‍t-q(∑i=-∞∞‍|ai|)q-1∑i=-∞∞‍|ai|E{max⁡1≤k≤n|∑j=i+1i+k‍Ytj|q}dt≤C1∑n=1∞‍nr-2-1/p∫n1/p∞‍t-q∑i=-∞∞‍|ai|E(∑j=i+1i+n‍E(Ytj2∣Fj-1))q/2dt+C1∑n=1∞‍nr-2-1/p∫n1/p∞‍t-q∑i=-∞∞‍|ai|∑j=i+1i+n‍E|Ytj|qdt≕C1I21+C1I22.
If rp≥2, then we take large enough q such that q>max{(r-1)/(1/p-1/2),rp}. By supjE(|Yj|rp∣ℱj-1)≤K, a.s. and Jensen’s inequality for conditional expectation, it has supjE(Yj2∣ℱj-1)≤K2/(rp),   a.s.. Meanwhile,
(4.24)E(Ynj2∣Fj-1)=E[Yj2I(|Yj|≤n1/p)∣Fj-1]-[E(|Yj|I(|Yj|≤n1/p)∣Fj-1)]2≤E[Yj2I(|Yj|≤n1/p)∣Fj-1],a.s.,-∞<j<∞.
Thus, by ∑i=-∞∞‍|ai|<∞, one has that
(4.25)I21=C1∑n=1∞‍nr-2-1/p∫n1/p∞‍t-q∑i=-∞∞‍|ai|E(∑j=i+1i+n‍Znj)q/2dt≤C1∑n=1∞‍nr-2-1/p∫n1/p∞‍t-q∑i=-∞∞‍|ai|E(∑j=i+1i+n‍2E[Yj2I(|Yj|≤n1/p)∣Fj-1])q/2dt≤C2∑n=1∞‍nr-2-1/p+q/2∫n1/p∞‍t-qdt≤C3∑n=1∞‍nr-2-1/p+q/2⋅n(-q+1)/p=C3∑n=1∞‍nr-2+q/2-q/p<∞,
following from the fact that q>(r-1)/(1/p-1/2). We also have by Cr inequality and Lemma 2.2 that
(4.26)I22≤C2∑n=1∞‍nr-2-1/p∫n1/p∞‍t-q∑i=-∞∞‍|ai|∑j=i+1i+n‍E[|Yj|qI(|Yj|≤t)]dt≤C3∑n=1∞‍nr-1-1/p∑m=n∞‍∫m1/p(m+1)1/p‍t-qE[YqI(Y≤t)]dt+C4∑n=1∞‍nr-1-1/p∫n1/p∞‍P(Y>t)dt≕C2I22*+C3I22**.
Since q>rp and EYrp<∞, it follows that
(4.27)I22*≤C1∑n=1∞‍nr-1-1/p∑m=n∞‍m1/p-1-q/pE[YqI(Y≤(m+1)1/p)]=C1∑m=1∞‍m1/p-1-q/pE[YqI(Y≤(m+1)1/p)]∑n=1m‍nr-1-1/p≤C2∑m=1∞‍mr-1-q/pE[YqI(Y≤(m+1)1/p)]=C2∑m=1∞‍mr-1-q/p∑i=1m+1‍E[YqI((i-1)1/p<Y≤i1/p)]=C2∑m=1∞‍mr-1-q/pE[YrpYq-rpI(m1/p<Y≤(m+1)1/p)]+C2∑m=1∞‍mr-1-q/p∑i=1m‍E[YqI((i-1)1/p<Y≤i1/p)]≤2(q-rp)/pC2∑m=1∞‍m-1E[YrpI(m1/p<Y≤(m+1)1/p)]+C2∑i=1∞‍E[YqI((i-1)1/p<Y≤i1/p)]∑m=i∞‍mr-1-q/p≤2(q-rp)/pC2∑m=1∞‍m-1E[YrpI(m1/p<Y≤(m+1)1/p)]+C2∑i=1∞‍E[YrpYq-rpI((i-1)1/p<Y≤i1/p)]ir-q/p≤C3EYrp<∞.
From the proof of (4.22),
(4.28)I22**≤∑n=1∞‍nr-1-1/p∫n1/p∞‍t-1EYI(Y>t)dt≤CEYrp<∞.
If rp<2, then we take q=2. Similar to the proofs of (4.23) and (4.26), we get that
(4.29)I2≤C1∑n=1∞‍nr-2-1/p∫n1/p∞‍t-2∑i=-∞∞‍|ai|∑j=i+1i+n‍EYtj2dt≤C2EYrp<∞,
following from q>rp, (4.27) and (4.28). Consequently, by (4.18)–(4.28), Theorem 3.1 and inequality above, (3.3) holds true.
Now, we turn to prove (3.4). Similar to the proof of (3.2), we have that
(4.30)∑n=1∞‍nr-2E(sup⁡k≥n|Skk1/p|-ɛ22/p)+∑n=1∞‍nr-2E=∑n=1∞‍nr-2∫0∞‍P(sup⁡k≥n|Skk1/p|>ɛ22/p+t)dt∑n=1∞‍nr-2E=∑m=1∞‍∑n=2m-12m-1‍nr-2∫0∞‍P(sup⁡k≥n|Skk1/p|>ɛ22/p+t)dt∑n=1∞‍nr-2E≤22-r∑m=1∞‍∫0∞‍P(sup⁡k≥2m-1|Skk1/p|>ɛ22/p+t)dt∑n=2m-12m-1‍2m(r-2)∑n=1∞‍nr-2E≤22-r∑m=1∞‍2m(r-1)∫0∞‍P(sup⁡k≥2m-1|Skk1/p|>ɛ22/p+t)dt∑n=1∞‍nr-2E=22-r∑m=1∞‍2m(r-1)∫0∞‍P(sup⁡l≥mmax⁡2l-1≤k<2l|Skk1/p|>ɛ22/p+t)dt∑n=1∞‍nr-2E≤22-r∑m=1∞‍2m(r-1)∑l=m∞‍∫0∞‍P(max⁡1≤k≤2l|Sk|>(ɛ22/p+t)2(l-1)/p)dt∑n=1∞‍nr-2E=22-r∑l=1∞‍∫0∞‍P(max⁡1≤k≤2l|Sk|>(ɛ22/p+t)2(l-1)/p)dt∑m=1l‍2m(r-1)∑n=1∞‍nr-2E≤22-r∑l=1∞‍2l(r-1)∫0∞‍P(max⁡1≤k≤2l|Sk|>(ɛ22/p+t)2(l-1)/p)dt(lets=2(l-1)/pt)∑n=1∞‍nr-2E≤C1∑l=1∞‍2l(r-1-1/p)∫0∞‍P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p+s)ds:=F.
If r<2+1/p, then
(4.31)F=2(2+1/p-r)C1∑l=1∞‍∑n=2l2l+1-1‍2(l+1)(r-2-1/p)∫0∞‍P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p+s)ds≤2(2+1/p-r)C1∑l=1∞‍∑n=2l2l+1-1‍nr-2-1/p∫0∞‍P(max⁡1≤k≤n|Sk|>ɛn1/p+s)ds≤2(2+1/p-r)C1∑n=1∞‍nr-2-1/pE(max⁡1≤k≤n|Sk|-ɛn1/p)+<∞.
Otherwise,
(4.32)F=C1∑l=1∞‍∑n=2l2l+1-1‍2l(r-2-1/p)∫0∞‍P(max⁡1≤k≤2l|Sk|>ɛ2(l+1)/p+s)ds≤C1∑l=1∞‍∑n=2l2l+1-1‍nr-2-1/p∫0∞‍P(max⁡1≤k≤n|Sk|>ɛn1/p+s)ds≤C1∑n=1∞‍nr-2-1/pE(max⁡1≤k≤n|Sk|-ɛn1/p)+<∞.
Therefore, (3.4) holds true following from (3.3).



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*Source: 128492-2012-07-08.xml*