Yi - American Mathematical Society

Theor. Probability and Math. Statist. No. 77, 2008, Pages 165–176 S 0094-9000(09)00755-8 Article electronically published on January 21, 2009

Teor Imovr. ta Matem. Statist. Vip. 77, 2007

LIMITING BEHAVIOUR OF MOVING AVERAGE PROCESSES UNDER NEGATIVE ASSOCIATION ASSUMPTION UDC 519.21

P. CHEN, T.-C. HU, AND A. VOLODIN Abstract. Let {Yi , −∞ < i < ∞} be a doubly infinite sequence of identically distributed negatively associated random variables, and {ai , −∞ < i < ∞} an absolutely summable sequence of real numbers. In this paper, we prove the complete convergence and complete moment convergenceof the maximum partial sums of mov∞ ing average processes i=−∞ ai Yi+n , n ≥ 1 . We improve the results of Baek et al. (2003) and Li and Zhang (2005).

1. Introduction We assume that {Yi , −∞ < i < +∞} is a doubly infinite sequence of identically distributed random variables. Let {ai , −∞ < i < +∞} be an absolutely summable sequence of real numbers, and let Xn =

∞

ai Yi+n ,

n ≥ 1,

i=−∞

be the moving average process based on the sequence {Yi , −∞ < i < +∞}. For the moving average process {Xn , n ≥ 1}, many limiting results have been obtained. For example, under the independence assumption of the base sequence {Yi , −∞ < i < +∞}, Burton and Dehling [3] obtained a large deviation principle, and Li et al. [11] obtained the complete convergence result. Under different dependence assumptions of the base sequence {Yi , −∞ < i < +∞}, Zhang [17] obtained the complete convergence result when the base sequence {Yi , −∞ < i < +∞} consists of ϕ-mixing random variables, and Baek et al. [2] and Liang et al. [13] obtained the complete convergence result when the base sequence {Yi , −∞ < i < +∞} consists of negatively associated random variables. For the Banach space generalizations we refer to the papers Ahmed et al. [1], Chen et al. [5, 6]. Recall that a finite family of random variables {Yi , 1 ≤ i ≤ n} is said to be negatively associated (abbreviated to NA) if for any disjoint subsets A and B of {1, 2, . . . , n} and 2000 Mathematics Subject Classification. Primary 60F15. Key words and phrases. Complete convergence, complete moment convergence, moving average, negative association. The research of P. Chen was supported by the National Natural Science Foundation of China. The research of T.-C. Hu was partially supported by the National Science Council. The research of A. Volodin was partially supported by the National Sciences and Engineering Research Council of Canada. c 2009 American Mathematical Society

165

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166

P. CHEN, T.-C. HU, AND A. VOLODIN

any real coordinatewise nondecreasing functions f on RA and g on RB , Cov (f (Yi , i ∈ A), g(Yj , j ∈ B)) ≤ 0 whenever the covariance exists. An infinite family of random variables {Yi , < −∞ ≤ i < ∞} is NA if every finite subfamily is NA. This concept was introduced by Joag-Dev and Proschan [10]. In what follows, we let Sn = nk=1 Xk , n ≥ 1, be the partial sums of the sequence {Xi , i ≥ 1}, and {ai , −∞ < i < ∞} be an absolutely summable sequence of real numbers, ∞ that is, i=−∞ |ai | < ∞. Furthermore, for a real number x, let x+ = max{0, x}, and for any number q, we define xq+ = (x+ )q . As usual, C represents a positive constant although its value may change from one appearance to the next. First we discuss the previous results connected with complete convergence. The following was proved in Hsu and Robbins [9] and Erd¨ os [8]. Theorem A. Suppose that {Xn , n ≥ 1} is a sequence of independent identically distributed random variables. Then E X1 = 0, E |X1 |2 < ∞ if and only if ∞ P{|Sn | ≥ εn} < ∞ n=1

for all ε > 0. Hsu–Robbins–Erd¨ os result was generalized by Li et al. [11] for a moving average process based on a sequence of i.i.d. random variables {Yi , −∞ < i < +∞}. Theorem B. Suppose {Xn , n ≥ 1} is the moving average process based on a sequence of independent identically distributed {Yi , −∞ < i < ∞} with E Y1 = 0, random variables 1/t E |Y1 |2t < ∞, 1 ≤ t < 2. Then ∞ P{|S | ≥ εn } < ∞ for all ε > 0. n n=1 The result of Li et al. was generalized for a moving average process based on a sequence of NA random variables {Yi , −∞ < i < +∞} by Baek et al. [2] and Liang et al. [13]. If we omit some insignificant details connected with slowly varying functions and stochastic domination condition, their result could be formulated in the following way. Theorem C. Suppose that {Xn , n ≥ 1} is the moving average process based on a sequence of NA identically distributed {Yi , −∞ < i < ∞} with E Y1 = 0, ∞ random variables 2t 1/t E |Y1 | < ∞, 1 ≤ t < 2. Then n=1 P{|Sn | ≥ εn } < ∞ for all ε > 0. Next, we discuss the previous results connected with complete moment convergence. The notion was introduced in Chow [7], where the following result was also proved. Theorem D. Suppose that {Xn , n ≥ 1} is a sequence of independent and identically distributed random variables with E X1 = 0 and 1 ≤ p < 2, r > p. If E{|X1 |rp + |X1 | log(1 + |X1 |)} < ∞, then

∞ n=1

nr−2−1/p E |Sn | − εn1/p 0.

+

We refer the interested reader to the papers by Chen [4] and Rosalsky et al. [14] for the generalizations of Theorem D on the Banach space setting. Li and Zhang [12] extended Theorem D to the case of moving average process based on NA random variables. Their result can be formulated in the following way, where we again omit some insignificant details connected with slowly varying functions.


LIMITING BEHAVIOUR OF MOVING AVERAGE PROCESSES

167

Theorem E. Suppose {Xn , n ≥ 1} is the moving average process based on a sequence of independent identically distributed random variables with E Y1 = 0, E Yi2 < ∞. Let 1 ≤ p < 2, r > 1 + p/2. Then E |Y1 |rp < ∞ implies that ∞ nr−2−1/p E |Sn | − εn1/p < ∞ for all ε > 0. +

n=1

2. Formulation of the main results The purpose of this paper is to improve the results of Baek et al. [2] (stated above as Theorem C) to the maximum partial sums, and extend the results of Li and Zhang [12] (Theorem E) to the maximum partial sums of a moving average process based on a sequence of NA random variables {Yi , −∞ < i < +∞} under more optimal moment conditions. Our main results are as follows. Theorem 1. Let 1 ≤ p < 2, r ≥ 1, rp = 1. Suppose that {Xn , n ≥ 1} is the moving average process based on a sequence of independent identically distributed random variables {Yi , −∞ < i < ∞}. If E Y1 = 0 and E |Y1 |rp < ∞, then ∞ r−2 1/p n P max |Sk | > εn < ∞ for all ε > 0. 1≤k≤n

n=1

Theorem 2. Let q > 0, 1 ≤ p < 2, r ≥ 1, rp = 1. Suppose that {Xn , n ≥ 1} is the moving average process based on a sequence of independent identically distributed random variables {Yi , −∞ < i < ∞}. If E Y1 = 0 and E |Y1 |rp < ∞,

q < rp,

E |Y1 |rp log(1 + |Y1 |) < ∞, E |Y1 | < ∞, q

then

∞

q > rp,

nr−2−q/p E

n=1

q = rp,

q max |Sk | − εn1/p

1≤k≤n

0.

+

3. Two technical lemmas The following lemma plays a crucial role in our proofs; see Shao [16, Theorem 2]. Lemma 1. Let q ≥ 2, and let {Xj , 1 ≤ i ≤ n} be a sequence of NA random variables with zero mean and E |Xj |q < ∞ for every 1 ≤ j ≤ n. Then ⎧ ⎫ q q/2 n n ⎨ ⎬ k Xi ≤ C E |Xj |2 + E |Xj |q . E max ⎩ ⎭ 1≤k≤n j=1

j=1

j=1

The next lemma is pure technical. Lemma 2. Let Y be a random variable with E |Y |rp ψ(1 + |Y |) < ∞, where r ≥ 1, p ≥ 1, and ψ(x) = 1 or ψ(x) = log(x), x ≥ 1. Then ∞ r−1 (i) P |Y | > n1/p ≤ C E |Y |rp . n=1 n (ii) For q > rp, ∞ nr−1− q/p ψ(n) E |Y |q I |Y | ≤ n1/p ≤ C E |Y |rp ψ(|Y |). n=1


168


(iii) For s ≥ 1, v > 0, sp > v and E |Y |sp ψ(1 + |Y |) < ∞, ∞

ns−1−

v/p

ψ(n) E |Y |v I |Y | > n1/p ≤ C E |Y |sp ψ(1 + |Y |).

n=1

Proof. First, we mention that statement (i) is well known, so we will prove only (ii) and (iii). Note that the function ψ has the following properties: (a) for any m ≥ 1, m

nu−1 ψ(n) ≤ Cmu ψ(m) if u > 0

n=1

and

∞

nu−1 ψ(n) ≤ Cmu ψ(m) if u < 0;

n=m p

(b) for any p > 0, ψ(|x| ) = Cψ(|x|) ≤ Cψ(1 + |x|). (ii) Since r − q/p < 0, we have ∞

nr−1−q/p ψ(n) E |Y |q I |Y | ≤ n1/p

n=1

=C =C ≤C ≤C ≤C ≤C

∞

n

nr−1−q/p ψ(n)

n=1 ∞ m=1 ∞ m=1 ∞ m=1 ∞ m=1 ∞

E |Y |q I{m − 1 < |Y |p ≤ m}

m=1

E |Y |q I{m − 1 < |Y |p ≤ m}

∞

nr−1−q/p ψ(n)

n=m

mr−q/p ψ(m) E |Y |q I{m − 1 < |Y |p ≤ m}

(by (a))

E mr−q/p ψ(m)|Y |q I{m − 1 < |Y |p ≤ m} E(|Y |p )r−q/p ψ(1 + |Y |)|Y |q I{m − 1 < |Y |p ≤ m} E |Y |rp ψ(1 + |Y |)I{m − 1 < |Y |p ≤ m}

m=1

≤ C E |Y |rp ψ(1 + |Y |). (iii) We have ∞

ns−1−v/p ψ(n) E |Y |v I |Y | > n1/p

n=1

=C =C ≤C

∞

ns−1−v/p ψ(n)

n=1 ∞ m=1 ∞

∞

E |Y |v I{m < |Y |p ≤ m + 1}

m=n

E |Y | I{m < |Y | ≤ m + 1} v

p

m

ns−1−v/p ψ(n)

n=1

ms−v/p ψ(m) E |Y |v I{m < |Y |p ≤ m + 1}

m=1


(by (a))


=C ≤C

∞

169

E ms−v/p ψ(m)|Y |v I{m < |Y |p ≤ m + 1}

m=1 ∞

E(|Y |p )s−v/p ψ(|Y |p )|Y |v I{m < |Y |p ≤ m + 1}

m=1

≤ C E |Y |sp ψ(1 + |Y |)

(by (b)).

4. Proof of the main results First we prove Theorem 1. Proof. Let (1) Ynj = −n1/p I Yj < −n1/p + Yj I |Yj | ≤ n1/p + n1/p I Yj > n1/p , (2)

(1)

and let Ynj = Yj − Ynj be monotone truncations of Yj , −∞ < j < ∞. Then, by Joag-Dev and Proschan [10], for any n ≥ 1, (1) (1) (2) Ynj − E Ynj , −∞ < j < ∞ and Ynj , −∞ < j < ∞ are two sequences of NA random variables. Note that n

Xk =

n ∞

ai Yi+k =

k=1 i=−∞

k=1

∞ i=−∞

ai

i+n

Yj

j=i+1

and ∞ ∞ i+k i+k (1) (1) n−1/p max E ai Ynj ≤ n−1/p |ai | max E Ynj 1≤k≤n 1≤k≤n i=−∞

j=i+1

i=−∞

j=i+1

≤ Cn−1/p n E Y1 I |Y1 | ≤ n1/p + n1/p P |Y1 | > n1/p ≤ Cn1−1/p E |Y1 |I |Y1 | > n1/p + Cn P |Y1 | > n1/p ≤ C E |Y1 |p I |Y1 | > n1/p + Cn P |Y1 | > n1/p → 0 as n → ∞. Hence, for any ε > 0 there exists n large enough that n

−1/p

∞ i+k (1) max E ai Ynj < ε/4.

1≤k≤n

i=−∞

j=i+1

Therefore, in order to prove Theorem 1, it is enough to prove that ⎫ ⎧ ∞ ∞ i+k ⎨ ⎬ (1) (1) Ynj − E Ynj > εn1/p 4 < ∞ I := nr−2 P max ai ⎭ ⎩1≤k≤n n=1

i=−∞

and J :=

∞ n=1

j=i+1

⎫ ∞ i+k ⎬ (2) nr−2 P max ai Ynj > εn1/p 2 < ∞. ⎭ ⎩1≤k≤n ⎧ ⎨

i=−∞

j=i+1


170


For J, by Markov’s inequality, we have ∞ i+k ∞ (2) r−2 −1/p J ≤C n n E max ai Ynj 1≤k≤n

n=1

≤C ≤C

∞ n=1 ∞

i=−∞

j=i+1

nr−1−1/p E |Y1 |I |Y1 | > n1/p + n1/p P |Y1 | > n1/p nr−1−1/p E |Y1 |I |Y1 | > n1/p + C E |Y1 |rp

(by Lemma 2(i))

n=1

≤ C E |Y1 |rp < ∞

(by Lemma 2(iii) with ψ(x) = 1, v = 1, and s = r).

For I, fix any q ≥ 2 (to be specified later). Then ∞ i+k q ∞ (1) (1) r−2− q/p I≤C n E max ai Ynj − E Ynj 1≤k≤n

n=1

i=−∞

j=i+1

(by Markov’s inequality)

∞ ∞ r−2−q/p n E |ai | max ≤C n=1

i=−∞

q i+k (1) (1) Ynj − E Ynj 1≤k≤n

∞ ∞ r−2−q/p ≤C n E |ai |1− n=1

≤C

∞

i=−∞

nr−2−q/p

n=1

∞

i=−∞

j=i+1

1/q

q i+k (1) (1) 1/q max |ai | Ynj − E Ynj 1≤k≤n j=i+1

q−1 ∞ q i+k (1) (1) Ynj − E Ynj |ai | |ai | E max 1≤k≤n i=−∞

j=i+1

(by H¨ older’s inequality) ≤C

∞ n=1

∞ i=−∞

∞

q i+k (1) (1) Ynj − E Ynj |ai | E max 1≤k≤n j=i+1

i=−∞ |ai | < ∞)

(since ≤C

n

r−2−q/p

∞

n

r−2−q/p

n=1

∞

i+n 2 q/2 (1) (1) |ai | E Ynj − E Ynj

i=−∞

j=i+1 i+n

+

q (1) (1) E Ynj − E Ynj

j=i+1

(by Lemma 1) ≤C

∞

nr−2−q/p

n=1

≤C

∞

∞

q/2 |ai | n E Y12 I |Y1 | ≤ n1/p + n2/p P |Y1 | > n1/p

i=−∞

n

r−2−q/p

q/2 n E Y12 I |Y1 | ≤ n1/p + n2/p P |Y1 | > n1/p

n=1

(since

+ n E |Y1 |q I |Y1 | ≤ n1/p + nq/p P |Y1 | > n1/p

q 1/p q/p 1/p + n P |Y1 | > n + n E |Y1 | I |Y1 | ≤ n

∞ i=−∞

|ai | < ∞).



171

We consider two separate cases. If rp < 2, let q = 2. We have I≤C ≤C

∞ n=1 ∞

nr−1−2/p E |Y1 |2 I |Y1 | ≤ n1/p + n2/p P |Y1 | > n1/p nr−1−2/p E |Y1 |2 I |Y1 | ≤ n1/p + C E |Y1 |rp

(by Lemma 2(i))

n=1

≤ C E |Y1 |rp < ∞

(by Lemma 2(ii) with q = 2 and ψ(x) = 1).

If rp ≥ 2, let q > 2p(r − 1)/(2 − p) ≥ 2. Note that in this case E |Y1 |2 I |Y1 | ≤ n1/p ≤ E |Y1 |2 ≤ (E |Y1 |rp )2/(rp) < ∞ and by Markov’s inequality,

n2/p P |Y1 | > n1/p ≤ E |Y1 |2 < ∞.

Hence I≤C

∞

nr−2−q/p

q/2 n E |Y1 |2 I |Y1 | ≤ n1/p + n2/p P |Y1 | > n1/p

n=1

≤C

∞

q 1/p q/p 1/p + n E |Y1 | I |Y1 | ≤ n + n P |Y1 | > n

n=1 ∞

+C

nr−1−q/p E |Y1 |q I |Y1 | ≤ n1/p

∞

nr−2−q/p+q/2 + C

n=1

nr−1 P |Y1 | > n1/p

n=1

≤ C + C E |Y1 |rp < ∞

by Lemma 2(i) and (ii) with ψ(x) = 1. Next, we prove Theorem 2. Proof. For every ε > 0, ∞

nr−2−q/p E

n=1

=

=

∞ n=1 ∞

1≤k≤n

P

n

nq/p

r−2−q/p

+

n

+

n=1

P

n=1 ∞ r−2 n=1 ∞

dt

0

nr−2−q/p P

>t

1/q

∞

1≤k≤n

max |Sk | > εn

1/p

1≤k≤n

r−2−q/p

∞

P nq/p

dt

max |Sk | − εn1/p > t1/q

nq/p

n

max |Sk | − εn

1/p

1≤k≤n

P

max |Sk | − εn1/p > t1/q

1≤k≤n

0

+

∞

nr−2−q/p

n=1 ∞

≤

q max |Sk | − εn1/p

max |Sk | > t

1/q

1≤k≤n


dt.

dt

172


Hence, by Theorem 1, in order to prove Theorem 2, it is enough to show that ∞ ∞ nr−2−q/p P max |Sk | > t1/q dt < ∞. Let (t,1)

Yj

1≤k≤n

nq/p

n=1

= −t1/q I Yj < −t1/q + Yi I |Yj | ≤ t1/q + t1/q I Yj > t1/q ,

(t,2)

(t,1)

and let Yj = Yj − Yj be monotone truncations of Yj , −∞ < j < ∞. Then by Joag-Dev and Proschan [10], for any t > 0, (t,1) (t,1) Yj − E Yj , −∞ < j < ∞ and

(t,2)

Yj

, −∞ < j < ∞

are two sequences of NA random variables. Note that n

Xk =

sup t

−1/q

t≥nq/p

ai Yi+k =

k=1 i=−∞

k=1

and

∞ n

∞ i=−∞

ai

i+n

Yj

j=i+1

∞ i+k (t,1) max E ai Yj 1≤k≤n i=−∞

≤ sup t

∞

−1/q

t≥nq/p

i=−∞

j=i+1

i+k (t,1) |ai | max E Yj 1≤k≤n

j=i+1

≤ C sup t−1/q n E Y1 I |Y1 | ≤ t1/q + t1/q P |Y1 | > t1/q t≥nq/p

≤ C sup t−1/q n E |Y1 |I |Y1 | > t1/q + t1/q P |Y1 | > t1/q t≥nq/p

−1/q 1/q 1/q ≤ C nq/p n E |Y1 |I |Y1 | > nq/p + Cn P |Y1 | > nq/p ≤ C E n1−1/p |Y1 |I |Y1 | > n1/p + Cn P |Y1 | > n1/p ≤ C E |Y1 |p I |Y1 | > n1/p + Cn P |Y1 | > n1/p → 0 as n → ∞. Hence for n large enough we have ∞ i+k (t,1) −1/q sup t max E ai Yj < 1/4. 1≤k≤n

t≥nq/p

i=−∞

j=i+1

Therefore, in order to prove Theorem 2, it is enough to show that ∞ ∞ i+k ∞ (t,1) (t,1) r−2−q/p 1/q I := Yj 4 dt < ∞ n P max ai − E Yj > t 1≤k≤n

nq/p

n=1

and J :=

∞ n=1

nr−2−q/p

i=−∞

j=i+1

⎫ i+k ⎬ ∞ (t,2) 1/q 2 dt < ∞. P max ai Yj > t ⎭ ⎩1≤k≤n q/p

∞

n

⎧ ⎨

i=−∞

j=i+1



173

We first show that J < ∞. By Markov’s inequality, J ≤C ≤C ≤C =C ≤C =C

∞ n=1 ∞ n=1 ∞ n=1 ∞ n=1 ∞

nr−2−q/p

i+k ∞ (t,2) t−1/q E max ai Yj dt 1≤k≤n q/p

∞

n

nr−1−q/p

i=−∞

j=i+1

t−1/q E |Y1 |I |Y1 | > t1/q + t1/q P |Y1 | > t1/q dt

∞

nq/p

nr−1−q/p

t−1/q E |Y1 |I |Y1 | > t1/q dt

∞

nq/p

n

nr−1−q/p

n=1 ∞

∞

r−1−q/p

m=n ∞

(m+1)q/p

t−1/q E |Y1 |I |Y1 | > t1/q dt

mq/p

mq/p−1−1/p E |Y1 |I |Y1 | > m1/p

m=n

m mq/p−1−1/p E |Y1 |I |Y1 | > m1/p nr−1−q/p

m=1

n=1

⎧ ∞ r−1−1/p ⎪ if q < rp, E |Y1 |I |Y1 | > m1/p ⎨C m=1 m ∞ r−1−1/p 1/p ≤ C m=1 m if q = rp, log(1 + m) E |Y1 |I |Y1 | > m ⎪ ⎩ ∞ q/p−1−1/p 1/p if q > rp C m=1 m E |Y1 |I |Y1 | > m ⎧ rp ⎪ if q < rp, ⎨C E |Y1 | ≤ C E |Y1 |rp log(1 + |Y1 |) if q = rp, (by Lemma 2(iii) with v = 1) ⎪ ⎩ q if q > rp C E |Y1 | t + n E |Y1 | I |Y1 | ≤ t dt s/2 t−s/q ns/2 E |Y1 |2 I |Y1 | ≤ t1/q + t2/q |Y1 | > t1/q

∞

nr−2−q/p

nq/p

n=1

(since

s/2 |ai | ns/2 E |Y1 |2 I |Y1 | ≤ t1/q + t2/q P |Y1 | > t1/q

∞

∞ i=−∞

+ n E |Y1 |s I |Y1 | ≤ t1/q + ts/q P |Y1 | > t1/q dt

|ai | < ∞).

Consider two separate cases. If max{q, rp} < 2, let s = 2. We have

I≤C ≤C

∞ n=1 ∞

+C

=C

nq/p

nq/p

n

r−1−q/p

n=1 ∞ r−1−q/p

n

n=1 ∞

+C

t−2/q E |Y1 |2 I |Y1 | ≤ t1/q dt

∞

nr−1−q/p

n=1 ∞

t−2/q E |Y1 |2 I |Y1 | ≤ t1/q + t2/q P |Y1 | > t1/q dt

∞

nr−1−q/p

P |Y1 | > t1/q dt

∞

nq/p

∞ m=n+1

≤C

t−2/q E |Y1 |2 I |Y1 | ≤ t1/q dt

(m−1)q/p

nr−1−q/p E |Y1 |q I |Y1 | > n1/p

n=1 ∞

mq/p

nr−1−q/p

n=1

E |Y1 |2 I |Y1 | ≤ m1/q

∞ m=n+1

mq/p

t−2/q dt + C E |Y1 |rp

(m−1)q/p

(by Lemma 2(iii) with s = r and v = q) ≤C

∞ n=1

n

r−1−q/p

∞

(m − 1)q/p−1−2/p E |Y1 |2 I |Y1 | ≤ m1/p + C E |Y1 |rp

m=n+1



≤C =C

∞ n=1 ∞

mq/p−1−2/p E |Y1 |2 I |Y1 | ≤ m1/p + C E |Y1 |rp

∞

nr−1−q/p

175

m=n

m mq/p−1−2/p E |Y1 |2 I |Y1 | ≤ m1/p ) nr−1−q/p + C E |Y1 |rp

m=1

n=1

⎧ ∞ r−1−2/p ⎪ E |Y1 |2 I |Y1 | ≤ m1/p + C E |Y1 |rp if q < rp, ⎨C m=1 m r−1−2/p 2 1/p rp ≤ C ∞ + C E |Y m log(1 + m) E |Y | I |Y | ≤ m | if q = rp, 1 1 1 m=1 ⎪ ⎩ ∞ q/p−1−2/p 2 1/p rp + C E |Y1 | E |Y1 | I |Y1 | ≤ m if q > rp C m=1 m ⎧ rp ⎪ if q < rp, ⎨C E |Y1 | rp rp ≤ C E |Y1 | log(1 + |Y1 |) + C E |Y1 | (by Lemma 2(ii)) if q = rp, ⎪ ⎩ if q > rp C E |Y1 |q + C E |Y1 |rp < ∞. If max{q, rp} ≥ 2, let s > max{q, 2p(r − 1)/(2 − p)}. Note that in this case E |Y1 |2 I |Y1 | ≤ t1/q ≤ E |Y1 |2 < ∞ and by Markov’s inequality,

t2/q P |Y1 | > t1/q ≤ E |Y1 |2 < ∞.

Hence I≤C

∞

nr−2−q/p+s/2

n=1 ∞

+C +C

n=1 ∞

nr−1−q/p nr−1−q/p

∞

t−s/q dt

nq/p ∞

t−s/q E |Y1 |s I |Y1 | ≤ t1/q dt

nq/p ∞

P |Y1 | > t1/q dt

q/p

≤C

n n=1 ∞ r−2−s/p+s/2

∞

n=1

n=1

n

+C

n

r−1−q/p

+ C E |Y1 |rp ≤ C + C E |Y1 |

rp

+C

∞ n=1

n

t−s/q E |Y1 |s I |Y1 | ≤ t1/q dt

nq/p

r−1−q/p

∞

∞

t−s/q E |Y1 |s I |Y1 | ≤ t1/q dt

nq/p

< ∞.

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Received 18/AUG/2006 Originally published in English
