Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 630608, 13 pages
doi:10.1155/2010/630608
Research Article
Complete Convergence for Weighted Sums of
ρ∗ -Mixing Random Variables
Soo Hak Sung
Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea
Correspondence should be addressed to Soo Hak Sung, sungsh@pcu.ac.kr
Received 2 August 2009; Accepted 24 March 2010
Academic Editor: Leonid Shaikhet
Copyright q 2010 Soo Hak Sung. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We obtain the complete convergence for weighted sums of ρ∗ -mixing random variables. Our result
extends the result of Peligrad and Gut 1999 on unweighted average to a weighted average under
a mild condition of weights. Our result also generalizes and sharpens the result of An and Yuan
2008.
1. Introduction
In many stochastic models, the assumption that random variables are independent is not
plausible. So it is of interest to extend the concept of independence to dependence cases. One
of these dependence structures is ρ∗ -mixing.
Let {Xn , n ≥ 1} be a sequence of random variables defined on a probability
space Ω, F, P , and let Fnm denote the σ-algebra generated by the random variables
Xn , Xn1 , . . . , Xm . For any S ⊂ N, define FS σXi , i ∈ S. Given two σ-algebras A, B in
F, put
ρA, B sup{corrX, Y ; X ∈ L2 A, Y ∈ L2 B},
1.1
where corrX, Y EXY − EXEY / varXvarY . Define the ρ∗ -mixing coefficients by
ρn∗ sup ρFS , FT ; S, T ⊂ N with distS, T ≥ n .
1.2
2
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∗
≤ ρn∗ ≤ ρ0∗ 1. The sequence {Xn , n ≥ 1} is called ρ∗ -mixing or ρ-mixing
Obviously, 0 ≤ ρn1
if there exists k ∈ N such that ρk∗ < 1. Note that if {Xn , n ≥ 1} is a sequence of independent
random variables, then ρn∗ 0 for all n ≥ 1.
A number of limit results 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 An and Yuan
8, Cai 9, Gan 5, Kuczmaszewska 10, Peligrad and Gut 3, and Zhu 11 for complete
convergence.
The concept of complete convergence of a sequence of random variables was
introduced by Hsu and Robbins 12. A sequence {Xn , n ≥ 1} of random variables converges
completely to the constant θ if
∞
P |Xn − θ| > < ∞ ∀ > 0.
n1
1.3
In view of the Borel-Cantelli lemma, this implies that Xn → θ almost surely. Therefore, the
complete convergence is a very important tool in establishing almost sure convergence of
summation of random variables as well as weighted sums of random variables. Hsu and
Robbins 12 proved that the sequence of arithmetic means of independent and identically
distributed random variables converges completely to the expected value if the variance of
the summands is finite. Erdös 13 proved the converse. The result of Hsu-Robbins-Erdös
is a fundamental theorem in probability theory and has been generalized and extended in
several directions by many authors. One of the most important generalizations is Baum and
Katz 14 strong law of large numbers.
Theorem 1.1 Baum and Katz 14. Let p ≥ 1/α and 1/2 < α ≤ 1. Let {Xn , n ≥ 1} be
a sequence of independent and identically distributed random variables with EX1 0. Then the
following statements are equivalent:
i E|X1 |p < ∞;
ii
∞
n1
npα−2 P max1≤j≤n |
j
i1
Xi | > nα < ∞ for all > 0,
Peligrad and Gut [3] extended the result of Baum and Katz [14] to ρ∗ -mixing random variables.
Theorem 1.2 Peligrad and Gut 3. Let p > 1/α and 1/2 < α ≤ 1. Let {Xn , n ≥ 1} be a sequence
of identically distributed ρ∗ -mixing random variables with EX1 0. Then the following statements
are equivalent:
i E|X1 |p < ∞;
ii
∞
n1
npα−2 P max1≤j≤n |
j
i1
Xi | > nα < ∞ for all > 0.
Cai [9] complemented Theorem 1.2 when p 1/α.
Recently, An and Yuan [8] obtained a complete convergence result for weighted sums of
identically distributed ρ∗ -mixing random variables.
Discrete Dynamics in Nature and Society
3
Theorem 1.3 An and Yuan 8. Let p > 1/α and 1/2 < α ≤ 1. Let {Xn , n ≥ 1} be a sequence of
identically distributed ρ∗ -mixing random variables with EX1 0. Assume that {ani , 1 ≤ i ≤ n, n ≥ 1}
is an array of real numbers satisfying
n
|ani |p O nδ
for some 0 < δ < 1,
i1
#Ank # 1 ≤ i ≤ n : |ani |p > k 1−1 ≥ ne−1/k
∀k ≥ 1, n ≥ 1.
1.4
1.5
Then the following statements are equivalent:
i E|X1 |p < ∞;
ii
∞
n1
npα−2 P max1≤j≤n |
j
i1
ani Xi | > nα < ∞ for all > 0.
Note that the result of An and Yuan 8 is not an extension of Peligrad and Gut’s 3
result, since condition 1.4 does not hold for the array with ani 1, 1 ≤ i ≤ n, n ≥ 1. An
and Yuan 8 proved the implication i⇒ii under condition 1.4, and proved the converse
under conditions 1.4 and 1.5. However, the array satisfying both 1.4 and 1.5 does not
exist. Noting that #Ank /k 1 ≤ ni1 |ani |p ≤ Onδ , we have that ne−1/k ≤ #Ank ≤ k 1Onδ . 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
identically distributed ρ∗ -mixing random variables. Our result extends the result of Peligrad
and Gut 3, and generalizes and sharpens the result of An and Yuan 8.
Throughout this paper, the symbol C denotes a positive constant which is not
necessarily the same one in each appearance, x denotes the integer part of x, and a ∧ b min{a, b}.
2. Main Result
To prove our main result, we need the following lemma which is a Rosenthal-type inequality
for ρ∗ -mixing random variables.
Lemma 2.1 Utev and Peligrad 4. Let {Xn , n ≥ 1} be a sequence of ρ∗ -mixing random variables
with EXn 0 and E|Xn |r < ∞ for some r ≥ 2 and all n ≥ 1. Then there exists a constant D Dr, k, ρk∗ depending only on r, k, and ρk∗ such that for any n ≥ 1,
⎧
j
⎞
r/2 ⎫
n
n
⎨
⎬
r
E⎝max Xi ⎠ ≤ D
,
E|Xi |r EXi2
⎩ i1
⎭
1≤j≤n
i1
i1
⎛
where ρk∗ < 1.
Now we state the main result of this paper.
2.1
4
Discrete Dynamics in Nature and Society
Theorem 2.2. Let p > 1/α and 1/2 < α ≤ 1. Let {Xn , n ≥ 1} be a sequence of identically distributed
ρ∗ -mixing random variables with EX1 0. Assume that {ani , 1 ≤ i ≤ n, n ≥ 1} is an array of real
numbers satisfying
n
|ani |q On
for some q > p.
2.2
i1
If E|X1 |p < ∞, then
∞
npα−2 P
n1
j
α
< ∞ ∀ > 0.
max ani Xi > n
1≤j≤n
i1
2.3
Conversely, if 2.3 holds for any array {ani } satisfying 2.2, then E|X1 |p < ∞.
To prove Theorem 2.2, we first prove the following lemma which is the sufficiency of
Theorem 2.2 when the array is bounded.
Lemma 2.3. Let {Xn , n ≥ 1} be a sequence of identically distributed ρ∗ -mixing random variables with
EX1 0 and E|X1 |p < ∞ for some p > 1/α and 1/2 < α ≤ 1. Assume that {ani , 1 ≤ i ≤ n, n ≥ 1} is
an array of real numbers satisfying |ani | ≤ 1 for 1 ≤ i ≤ n and n ≥ 1. Then 2.3 holds.
Proof. For 1 ≤ i ≤ n and n ≥ 1, define Xni
Xi I|Xi | ≤ nα . Since EXi 0 and
have that
n
i1
|ani | ≤ n, we
j
j
−α
α n max ani EXni n max ani EXi I|Xi | > n 1≤j≤n
1≤j≤n
i1
i1
−α
≤ n−α
n
|ani |E|X1 |I|X1 | > nα i1
2.4
≤ n1−α E|X1 |I|X1 | > nα ≤ n1−pα E|X1 |p I|X1 | > nα −→ 0
as n → ∞. Hence for n large enough, we have
j
n max ani EXni < .
2
1≤j≤n
i1
−α
2.5
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It follows that
∞
pα−2
n
P
n1
j
α
max ani Xi > n
1≤j≤n
i1
∞
∞
n
≤
npα−2 P |Xi | > nα npα−2 P
n1
i1
n1
∞
∞
≤
npα−1 P |X1 | > nα C npα−2 P
n1
j
α
max ani Xni > n
1≤j≤n
i1
n1
2.6
j
nα
max ani Xni − EXni >
1≤j≤n
2
i1
: I CJ.
pα−1
Noting that ∞
P |X1 | > nα ≤ CE|X1 |p < ∞, we have I < ∞. Thus, it remains to show
n1 n
that J < ∞.
We have by Markov’s inequality and Lemma 2.1 that for any r ≥ 2,
j
r
r ∞
2
pα−rα−2
J≤
n
Emax ani Xni − EXni 1≤j≤n
n1
i1
⎧
⎫
r/2
∞
n
n
⎨ ⎬
2
r
≤ C npα−rα−2
a2ni EXni
|ani |r EXni
⎩ i1
⎭
n1
i1
≤C
2.7
∞
∞
r/2
npα−rα−2r/2 E|X1 |2 I|X1 | ≤ nα C npα−rα−1 E|X1 |r I|X1 | ≤ nα n1
n1
: CJ1 CJ2 .
In the last inequality, we used the fact that |ani | ≤ 1 for 1 ≤ i ≤ n and n ≥ 1.
If p ≥ 2, then we take large enough r such that r > max{pα − 1/α − 1/2, p}. Since
r > pα − 1/α − 1/2, we get
J1 ≤ C
∞
n1
npα−rα−2r/2 < ∞.
2.8
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Discrete Dynamics in Nature and Society
Since r > p, we also get
J2 n
∞
npα−rα−1 E|X1 |r I i − 1α < |X1 | ≤ iα
n1
i1
∞
∞
E|X1 |r I i − 1α < |X1 | ≤ iα
npα−rα−1
ni
i1
≤C
2.9
∞
E|X1 |r I i − 1α < |X1 | ≤ iα ipα−rα
i1
≤ CE|X1 |p < ∞.
If p < 2, then we take r 2. Since r > p, 2.9 still holds, and so J1 J2 < ∞.
We next prove the sufficiency of Theorem 2.2 when the array is unbounded.
Lemma 2.4. Let {Xn , n ≥ 1} be a sequence of identically distributed ρ∗ -mixing random variables with
EX1 0 and E|X1 |p < ∞ for some p > 1/α and 1/2 < α ≤ 1. Assume that {ani , 1 ≤ i ≤ n, n ≥ 1} is
an array of real numbers satisfying ani 0 or |ani | > 1, and
n
|ani |q ≤ n for some q > p.
i1
2.10
Then 2.3 holds.
Proof. If p < 2, then we can take δ > 0 such that p < p δ < min{2, q}. Since ani 0 or |ani | > 1,
we have that ni1 |ani |pδ ≤ ni1 |ani |q ≤ n. Thus we may assume that 2.10 holds for some
p < q < 2 when p < 2.
j
Let Snj i1 ani Xi I|ani Xi | ≤ nα for 1 ≤ j ≤ n and n ≥ 1. In view of EXi 0, we get
j
−α
α n maxESnj n max ani EXi I|ani Xi | > n 1≤j≤n
1≤j≤n
i1
−α
≤ n−α
n
E|ani Xi |I|ani Xi | > nα i1
≤ n−pα
n
E|ani Xi |p I|ani Xi | > nα i1
≤ n−pα
n
|ani |p E|X1 |p
i1
−pα
≤n
n
p/q
|ani |
q
i1
≤ n1−pα E|X1 |p −→ 0,
n1−p/q E|X1 |p
2.11
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7
since pα > 1. Hence for n large enough, we have that n−α max1≤j≤n |ESnj | < /2. It follows that
∞
pα−2
n
P
n1
j
α
max ani Xi > n
1≤j≤n
i1
∞
∞
pα−2
α
pα−2
α
≤
n
P max|ani Xi | > n n
P maxSnj > n
1≤i≤n
n1
≤
n1
1≤j≤n
2.12
∞
∞
n
nα npα−2 P |ani Xi | > nα C npα−2 P maxSnj − ESnj >
1≤j≤n
2
n1
i1
n1
: I CJ.
For 1 ≤ j ≤ n − 1 and n ≥ 2, let
−1/q
Inj 1 ≤ i ≤ n : n1/q j 1
< |ani | ≤ n1/q j −1/q .
Then {Inj , 1 ≤ j ≤ n − 1} are disjoint,
1 ≤ k ≤ n − 1, since
n≥
|ani |q 0}
{1≤i≤n:ani /
n−1
j1
Inj {1 ≤ i ≤ n : ani /
0}, and
2.13
k
j1
n−1 k
k
1
n #Inj ≥
#Inj .
|ani |q ≥ n
j 1
k 1 j1
j1 i∈Inj
j1
For convenience of notation, let t 1/α − 1/q. Since ani 0 or |ani | > 1, and
we have a11 0. It follows that
I
∞
n−1 npα−2
P |ani Xi | > nα n2
≤
≤
j1 i∈Inj
∞
n−1 npα−2 P |X1 |t > nj t/q #Inj
n2
j1
∞
n−1
npα−2 #Inj
P k < |X1 |t ≤ k 1
n2
j1
#Inj ≤ k 1 for
k≥nj t/q 2.14
n
i1
|ani |q ≤ n,
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Discrete Dynamics in Nature and Society
≤
q/t
∞
∞ n−1∧k1/n
npα−2 P k < |X1 |t ≤ k 1
#Inj
n2
j1
kn
∞
∞ ≤
npα−2 P k < |X1 |t ≤ k 1 n ∧
n2
kn
1t/q
n
∞
≤
npα−2
P k < |X1 |t ≤ k 1
n1
kn
∞
npα−1
n1
∞
k1
n
k1
n
q/t q/t 1
1
P k < |X1 |t ≤ k 1
kn1t/q 1
: I1 I2 .
2.15
Since pα − 2 − q/t −αq − p − 1 < −1, we obtain
I1 ≤ C
∞
pα−2−q/t
n
1t/q
n
n1
≤C
P k < |X1 |t ≤ k 1 kq/t
kn
∞ P k < |X1 |t ≤ k 1 kq/t
k1
≤C
k
npα−2−q/t
nkq/qt 2.16
∞ P k < |X1 |t ≤ k 1 kq/t−qαq−p/qt
k1
≤ CE|X1 |p < ∞.
We also obtain
I2 ≤
∞ kq/tq
P k < |X1 |t ≤ k 1
npα−1
k1
n1
∞ ≤ C P k < |X1 |t ≤ k 1 kpα−1/q ≤ CE|X1 |p < ∞.
k1
From I1 < ∞ and I2 < ∞, we have I < ∞. Thus, it remains to show that J < ∞.
2.17
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9
We have by Markov’s inequality and Lemma 2.1 that for any r ≥ 2,
J≤C
∞
n1
≤C
∞
r
npα−rα−2 EmaxSnj − ESnj 1≤j≤n
pα−rα−2
n
r/2
n
2
α
E|ani Xi | I|ani Xi | ≤ n n1
C
i1
∞
npα−rα−2
n1
2.18
n
E|ani Xi |r I|ani Xi | ≤ nα i1
: J1 J2 .
Observe that for r ≥ q and n > m,
n≥
n−1 j1
n−1
n−1
−r/q
1
#Inj ≥ nm 1r/q−1
#Inj .
j 1
|ani |q ≥ n
j
1
jm
j1
i∈Inj
2.19
n−1
j −r/q #Inj ≤ Cm−r/q−1 for r ≥ q and n > m.
For J1 and J2 , we proceed with two cases.
i If p ≥ 2, then we take r large enough such that r > max{pα − 1/α − 1/2, q}. Then
we obtain that
So
jm
r/2
∞
n
2
pα−rα−2
J1 ≤ C n
|ani |
n1
i1
r/2
∞
n
q
pα−rα−2
≤C n
|ani |
n1
≤C
i1
∞
npα−rα−2r/2 < ∞.
n1
The second inequality follows by the fact that ani 0 or |ani | > 1.
2.20
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Discrete Dynamics in Nature and Society
Noting that a11 0, we also obtain that
J2 ∞
n−1 npα−rα−2
E|ani Xi |r I|ani Xi | ≤ nα n2
≤
j1 i∈Inj
∞
n−1
t/q npα−rα−2r/q j −r/q #Inj E|X1 |r I |X1 |t ≤ n j 1
n2
≤
j1
∞
n−1
npα−rα−2r/q j −r/q #Inj
n2
j1
E|X1 |r I k < |X1 |t ≤ k 1
0≤k≤nj1t/q 2.21
∞
n−1
2n
npα−rα−2r/q j −r/q #Inj E|X1 |r I k < |X1 |t ≤ k 1
n2
j1
k0
t/q
nj1
∞
n−1
npα−rα−2r/q j −r/q #Inj
n2
j1
E|X1 |r I k < |X1 |t ≤ k 1
k2n1
: J3 J4 .
Since pα − rα − 2 r/q < qα − rα − 2 r/q −r − qα − 1/q − 1 < −1 and q > p, we have that
J3 ∞
n−1
2n
npα−rα−2r/q E|X1 |r I k < |X1 |t ≤ k 1
j −r/q #Inj
n2
≤C
∞
2n
npα−rα−2r/q E|X1 |r I k < |X1 |t ≤ k 1
n2
≤C
k0
∞
∞
E|X1 |r I k < |X1 |t ≤ k 1
npα−rα−2r/q
k1
≤C
nk/2
∞
E|X1 |r I k < |X1 |t ≤ k 1 kpα−rα−1r/q
k1
≤C
j1
k0
∞ P k < |X1 |t ≤ k 1 kpα−1
k1
≤ CE|X1 |p < ∞.
2.22
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11
Since 1/t 1/q − α 0 and pα − 2 − q/t −αq − p − 1 < −1, we also have that
J4 ≤
nqt/q
∞
npα−rα−2r/q
E|X1 |r I k < |X1 |t ≤ k 1
n2
k2n1
n−1
j −r/q #Inj
jk/nq/t −1
−r/q−1
q/t k
−1
n
nqt/q
∞
≤ C npα−rα−2r/q
E|X1 |r I k < |X1 |t ≤ k 1
n2
≤C
k2n1
∞
E|X1 |r I k < |X1 |t ≤ k 1 k−r−q/t
k5
≤C
k/2
2.23
npα−2−q/t
nkq/qt ∞
E|X1 |r I k < |X1 |t ≤ k 1 k−r−q/t−α−1/qq−p
k5
≤ CE|X1 |p < ∞.
From J3 < ∞ and J4 < ∞, we have J2 < ∞.
ii If p < 2, then we take r 2. As noted above, we may assume that p < q < 2. Since
r > q, as in the case p ≥ 2, we have J1 J2 ≤ CE|X1 |p < ∞.
We now prove Theorem 2.2 by using Lemmas 2.3 and 2.4.
Proof of Theorem 2.2.
Sufficiency. Without loss of generality, we may assume that
n ≥ 1, let
An {1 ≤ i ≤ n : |ani | ≤ 1},
n
i1
|ani |q ≤ n for some q > p. For
Bn {1 ≤ i ≤ n : |ani | > 1},
2.24
and let ani ani if i ∈ An , ani 0 otherwise, and ani ani if i ∈ Bn , ani 0 otherwise. Then
j
j
j
max ani Xi ≤ max ani Xi max ani Xi .
1≤j≤n
1≤j≤n
1≤j≤n
i1
i1
i1
2.25
It follows that
∞
n1
pα−2
n
P
j
j
∞
nα
α
pα−2
≤
max ani Xi > n
n
P max ani Xi >
1≤j≤n
1≤j≤n
2
i1
n1
i1
∞
pα−2
n
n1
P
j
nα
max ani Xi >
1≤j≤n
2
i1
: I J.
By Lemma 2.3, we have I < ∞. By Lemma 2.4, we have J < ∞. Hence 2.3 holds.
2.26
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Discrete Dynamics in Nature and Society
Necessity. Choose, for each n ≥ 1, an1 · · · ann 1. Then {ani } satisfies 2.2. By 2.3, we
obtain that
∞
npα−2 P
n1
j
α
max Xi > n
<∞
1≤j≤n
i1
∀ > 0,
2.27
which implies that
∞
npα−2 P maxXj > nα < ∞
1≤j≤n
n1
∀ > 0.
2.28
Observe that
∞>
2i
∞ npα−2 P maxXj > nα
1≤j≤n
i1 n2i−1 1
⎧∞
i α
pα−2 i−1
⎪
⎪
i−1
⎪
if pα ≥ 2,
2 P max Xj > 2
2
⎪
⎪
⎨ i1
1≤j≤2i−1
≥
⎪
∞
⎪
i α
i pα−2 i−1
⎪
⎪
⎪
if 1 < pα < 2,
2 P max Xj > 2
2
⎩
1≤j≤2i−1
i1
2.29
⎧∞ i α
⎪
⎪
⎪
P max Xj > 2
if pα ≥ 2,
⎪
⎪
⎨ i1
1≤j≤2i−1
≥
⎪
∞
⎪
i α
⎪
pα−2
⎪
⎪2
if 1 < pα < 2.
P max Xj > 2
⎩
1≤j≤2i−1
i1
Hence we have that for any > 0, P max1≤j≤2i−1 |Xj | > 2i α → 0 as i → ∞, and so
P max1≤j≤n |Xj | > nα → 0 as n → ∞. The rest of the proof is same as that of Peligrad
and Gut 3 and is omitted.
Remark 2.5. Taking ani 1 for 1 ≤ i ≤ n and n ≥ 1, we can immediately get Theorem 1.2
from Theorem 2.2. If the array {ani } satisfies 1.4, then it satisfies 2.2: taking q such that
p < q < p/δ, we have
n
n
|ani |q ≤ max|ani |q−p |ani |p ≤ Cnδq−p/p nδ ≤ Cn.
i1
1≤i≤n
i1
2.30
So the implication i⇒ii of Theorem 1.3 follows from Theorem 2.2. As noted after
Theorem 1.3, the implication ii⇒i of Theorem 1.3 is not true. Therefore, our result extends
the result of Peligrad and Gut 3 to a weighted average, and generalizes and sharpens the
result of An and Yuan 8.
Discrete Dynamics in Nature and Society
13
Acknowledgments
The author is grateful to the editor Leonid Shaikhet and the referees for the helpful comments
and suggestions that considerably improved the presentation of this paper. This work was
supported by the Korea Science and Engineering Foundation KOSEF Grant funded by the
Korea government MOST no. R01-2007-000-20053-0.
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