Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 140548, 10 pages
doi:10.1155/2008/140548
Research Article
On Chung-Teicher Type Strong Law of Large
Numbers for ρ∗-Mixing Random Variables
Anna Kuczmaszewska
Department of Applied Mathematics, Lublin University of Technology, Nadbystrzycka 38 D,
20-618 Lublin, Poland
Correspondence should be addressed to Anna Kuczmaszewska, a.kuczmaszewska@pollub.pl
Received 23 December 2007; Revised 05 March 2008; Accepted 20 March 2008
Recommended by Stevo Stevic
In this paper the classical strong laws of large number of Kolmogorov, Chung, and Teicher for
independent random variables were generalized on the case of ρ∗ -mixing sequence. The main result
was applied to obtain a Marcinkiewicz SLLN.
Copyright q 2008 Anna Kuczmaszewska. 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.
1. Introduction
Let {Xn , n ≥ 1} be a sequence of random variables defined on the probability space Ω, F, P with value in a real space R and let Sn ni1 Xi . We say that the sequence {Xn , n ≥ 1} satisfies
the strong law of large numbers SLLN if there exists some increasing sequence {bn , n ≥ 1}
and some sequence {an , n ≥ 1} such that
n i1
Xi − ai
−→ 0 a.s. as n −→ ∞.
bn
1.1
In this paper, we consider the strong law of large numbers for sequences of dependent random
variables which are said to be ρ∗ -mixing. To introduce the concept of ρ∗ -mixing sequence, we
need the maximal correlation coefficient defined as follows:
covX, Y ρ∗ k sup
,
1.2
sup
√
S,T
X∈L2 FS , Y ∈L2 FT Var X · Var Y
where S, T are the finite subsets of positive integers such that dist S, T infx∈S,y∈T |x − y| ≥ k
and FW is the σ-field generated by the random variables {Xi , i ∈ W ⊂ N}.
2
Discrete Dynamics in Nature and Society
Definition 1.1. A sequence of random variables {Xn , n ≥ 1} is said to be a ρ∗ -mixing sequence
if
lim ρ∗ n < 1.
1.3
n→∞
ρ∗ -mixing random variables were investigated by many authors. Various moment
inequalities for sums and maximum of partial sums can be found in papers by Bradley
1, Bryc and Smoleński 2, Peligrad 3, Peligrad and Gut 4, and Utev and Peligrad 5.
These inequalities are used in many papers concerning the problems of invariance principle
Utev and Peligrad 5, CLT Peligrad 3, or complete convergence for some stochastically
dominated sequence of ρ∗ -mixing random variables Cai 6, and for an array of rowwise ρ∗ mixing random variables Zhu 7. They will be also important in our further consideration.
The aim of this paper is to give some sufficient conditions for SLLN for a sequence of
∗
ρ -mixing random variables without assumptions of identical distribution and stochastical
domination. The result presented in this paper is obtained by using the maximal type
inequality and the following strong law of large numbers proved by Fazekas and Klesov 8.
Theorem 1.2 Fazekas and Klesov 8. Let {bn , n ≥ 1} be a nondecreasing, unbounded sequence of
positive numbers. Let {αn , n ≥ 1} be nonnegative numbers. Let r be a fixed positive number. Assume
that for each n ≥ 1:
n
r E max Si ≤
αi .
1≤i≤n
1.4
i1
If
∞
αi
r < ∞,
b
i1 i
1.5
then
lim
Sn
n→∞ bn
0 a.s.
1.6
Using this theorem, we are going to show that classical Kolmogorov, Chung, and
Teicher’s strong law of large numbers for independent random variables {Xn , n ≥ 1} Chung
9 and Teicher 10 can be generalized to the case of ρ∗ -mixing sequences.
In our further consideration, we need the following result.
Lemma 1.3 Utev and Peligrad 5. Let {Xn , n ≥ 1} be a ρ∗ -mixing sequence with EXn 0,
E|Xn |q < ∞, n ≥ 1, q ≥ 2. Then, there exists a positive constant c such that
q
q/2 k
n
n
q
2
,
E Xi EXi
E max Xi ≤ c
1≤k≤n i1
i1
i1
∀ n ≥ 1.
Let C denote a constant which is not necessary the same in its each appearance.
1.7
Anna Kuczmaszewska
3
2. The main result
Let {ϕn , n ≥ 1} be a sequence of nonnegative, even, continuous and nondecreasing on 0, ∞
functions ϕn : R→R with limx→∞ ϕn x ∞ and such that for all n ≥ 1 and 1 < p ≤ 2:
a
ϕn x/x or b
ϕn x/xp ϕn x/x ,
as x −→ ∞.
2.1
Theorem 2.1. Let {Xn , n ≥ 1} be a sequence of ρ∗ -mixing random variables and let {bn , n ≥ 1} be an
increasing sequence of positive real numbers. Let 1 < p ≤ 2.
Assume that {ϕn , n ≥ 1} satisfies a in 2.1 with
p p ∞
i−1
ϕj Xj ϕi Xi −p
p
2.2
bi E
bj E
A
< ∞,
p p p p ϕi bi ϕi Xi ϕj bj ϕj Xj i2
j1
∞
P Xn ≥ an < ∞,
B
2.3
n1
for some sequence {an , n ≥ 1} of positive numbers such that
2p ∞
ϕn min Xn , an
C
E
< ∞,
2p 2p ϕn bn ϕn Xn n1
2.4
or {ϕn , n ≥ 1} satisfies b in 2.1 with
∞
i−1
ϕj Xj ϕi Xi −p
p
bi E
bj E
A1
< ∞,
ϕi bi ϕi Xi ϕj bj ϕj Xj i2
2.5
j1
and B for some sequence {an , n ≥ 1} of positive numbers such that
2
∞
ϕn min Xn , an
E
C1
< ∞.
ϕ2n bn ϕ2n Xn 2.6
n1
Then,
bn−1
n
Xi − E Xi I Xi < bi
−→ 0 a.s. n −→ ∞.
i1
Proof. Let Xi Xi I|Xi | < bi , Xi∗ Xi − EXi , Sn Then,
∞
P Xn / Xn
n
∗
i1 Xi ,
and Sn 2.7
n
i1 Xi .
n1
∞
∞
2r Xn ≥ ϕ2r
P Xn ≥ bn ≤
P 2ϕ2r
Xn n
n bn ϕn
n1
∞ ≤C
E
n1
n1
Xn Xn ϕ2r
ϕ2r
n
n
X
·
I
|X
·
I
<
a
|
≥
a
E
n
n
n
n
2r 2r ϕ2r
ϕ2r
Xn Xn n bn ϕn
n bn ϕn
2r ∞
∞
ϕn min Xn , an
≤C
P Xn ≥ an E
<∞
2r Xn ϕ2r
n bn ϕn
n1
n1
2.8
4
Discrete Dynamics in Nature and Society
for r p in the case a or r 1 in the case b. Hence, the sequences {Xn , n ≥ 1} and {Xn ,
n ≥ 1} are equivalent in Khinchin’s sense.
Thus, we need only to show that
Sn
−→ 0
bn
a.s. n −→ ∞.
2.9
By Lemma 1.3, for q 4, we have
2 n
n
n
n
i−1
4
4
2
4
2
2
∗
∗
∗
∗
∗
E max Si ≤ C
≤ C 2 EXi 2 EXi
EXi EXi
EXj
1≤i≤n
i1
i1
≤C
4
EX1
n
4
EXi
i2
i2
i1
2
EXi
i−1
j1
j1
2
EXj
.
2.10
By Theorem 1.2 applied with
α1 EX1 ,
4
αi EXi EXi
4
2
i−1
j1
EXj
2
for i 2, 3, . . . , n,
2.11
we see that in order to show 2.9 it is enough to prove that
∞
αi
< ∞,
4
i1 bi
2.12
which holds if
∞ EX 4
i
i1
Put I :
∞
i1 P |Xi |
∞ EX 4
i
i1
bi4
≤
bi4
< ∞,
∞ EX 2 i−1
i
i2
bi4
j1
EXj < ∞.
2
2.13
≥ ai . Then, we have
4
∞ E X I X < min a , b
i
i
i i
bi4
i1
I ≤
∞ E X 2p I X < min a , b
i
i
i i
i1
2p
bi
I.
2.14
Moreover, we note that
a
a
· Ia ≤ b ≤ 2 ·
b
ab
∀ a, b > 0.
2.15
Hence, in case a we get, by B and C,
∞ EX 4
i
i1
bi4
≤
∞
i1
E
2p ∞
ϕi min Xi , ai
Xi I < ∞,
I ≤ 2 E
I Xi < min ai , bi
2p 2p 2p ϕi bi
ϕi bi ϕi Xi i1
2.16
2p ϕi
Anna Kuczmaszewska
5
while in case b we get, by B and C1 ,
2
∞ EX 4
∞
ϕi min Xi , ai
i
≤2 E
I < ∞.
4
ϕ2i bi ϕ2i Xi i1 bi
i1
Using the fact that
p
2 2
−2p p bi−4 E Xi E Xj ≤ bi EXi EXj 2.17
∀j < i
2.18
both in either case a
∞ EX 2 i−1
i
bi4
i2
j1
EXj
2
i−1
p p ∞ EX p ∞
i−1
ϕj Xj p
ϕi Xi −p
p
i
≤
E Xj ≤ 4 b i E
bj E
< ∞,
p p p p 2p
ϕi bi ϕi Xi ϕj bj ϕj Xj bi
i2
i2
j1
j1
2.19
or case b
∞ EX 2 i−1
i
i2
bi4
j1
EXj
2
i−1
∞ EX p ∞
i−1
p
ϕj Xj ϕi Xi −p
p
i
≤
E Xj ≤ 4 b i E
bj E
∞.
2p
ϕi bi ϕi Xi ϕj bj ϕj Xj bi
i2
i2
j1
j1
2.20
Thus, we have established 2.9 and consequently 2.7.
Corollary 2.2. Let {Xn , n ≥ 1} be a sequence of ρ∗ -mixing random variables satisfying the condition
rt ∞
Xi
2.21
E
rt < ∞
ir Xi i1
with r p for 0 < t < 1 and all 1 < p ≤ 2 , or r 2/t for 1 ≤ t < 2 .
Then,
n
n−1/t
Xi − E Xi I Xi < i1/t
−→ 0 a.s. as n −→ ∞,
2.22
i1
Proof. Let an bn n1/t for any 0 < t < 2. Then, the assumption B of Theorem 2.1 is fulfilled.
Indeed we see
∞
B :
P Xn ≥ an
n1
∞
∞
∞
rt rt
P Xn ≥ n1/t ≤ C P 2Xn ≥ nr Xn < C ·
E
n1
n1
by 2.21 with r p for 0 < t < 1 or r 2/t for 1 ≤ t < 2.
n1
rt Xn rt < ∞
nr Xn 2.23
6
Discrete Dynamics in Nature and Society
Let now 0 < t < 1. Then, for ϕn x xt , x > 0, n ≥ 1, the conditions A and C with
1 < p ≤ 2 are fulfilled:
p p pt 2
∞
i−1
∞
ϕj Xj ϕi Xi Xi
−p
p
A :
bi E p p bj E p p ≤
E
< ∞,
pt
ϕi bi ϕi Xi j1
ϕj bj ϕj Xj ip Xi i2
i1
2.24
by 2.21 with r p.
2p ∞ 2pt ∞
ϕn min Xn , an
min Xn , an
C :
E
E
2pt
2p 2p n2p Xn ϕn bn ϕn Xn n1
n1
∞
≤
E
n1
∞
n1
2pt
Xn 1/t
2pt I Xn ≥ n
n2p Xn E
2pt
Xn 1/t pt pt pt I Xn < n
Xn np /Xn · np Xn ∞
∞
≤ C P Xn ≥ n1/t E
n1
n1
pt Xn pt < ∞
np Xn 2.25
by 2.23 and 2.21 with r p, 1 < p ≤ 2.
Thus, by Theorem 2.1, we have
n−1/t
n
Xi − E Xi I Xi < i1/t
−→ 0 a.s. as n −→ ∞
2.26
i1
for 0 < t < 1.
Now, we need to show that 2.26 also holds for 1 ≤ t < 2.
Let ϕn x x2 . Then, for p 2, by the similar calculations as in case 0 < t < 1, we get
2 2
∞
i−1
∞
ϕj Xj ϕi Xi Xi
−p
p
bi E
bj E
E
<∞
A1 :
≤
2
2/t
ϕ
ϕ
b
ϕ
X
b
ϕ
X
i i
i
i
j
j
j
j
i Xi i2
j1
i1
2.27
by 2.21 with r 2/t and
2
∞
ϕn min Xn , an
C1 :
E
ϕ2n bn ϕ2n Xn n1
∞
n1
E
4 2
∞
∞
min Xn , an
Xn
1/t
E
≤ C P Xn ≥ n
2 < ∞
4
n2/t Xn n4/t Xn n1
n1
2.28
by 2.23 and 2.21 with r 2/t.
Therefore, by Theorem 2.1, we get 2.26 for 1 ≤ t < 2.
This completes the proof of Corollary 2.2.
Anna Kuczmaszewska
7
Corollary 2.3. Let {X, Xn , n ≥ 1} be a sequence of ρ∗ -mixing random variables with E|X|t < ∞,
0 < t < 2. Let
P Xi > x ≤ CP |X| > x
2.29
for all x > 0, i ≥ 1 and some positive constant C.
Moreover, when 1 ≤ t < 2, let EX 0. Then,
n−1/t
n
Xi −→ 0
a.s. as n −→ ∞.
2.30
i1
Proof. We first note that 2.29 implies
s E Xi I Xi < a ≤ C E |X|s I |X| < a as P |X| ≥ a
2.31
for any a > 0 and s > 0.
Put now an bn n1/t for any 0 < t < 2.
It is easy to see that E|X|t < ∞, 2.29 and 2.31 with s rt imply convergence of the
series:
rt ∞
Xn E
2.32
rt
nr Xn n1
for r p, 1 < p ≤ 2 in the case 0 < t < 1 and r 2/t in the case 1 ≤ t < 2.
We have
rt rt
rt
∞
∞
∞
Xn Xn Xn 1/t
1/t
E
E
E
rt rt I Xn < n
rt I Xn ≥ n
nr Xn nr Xn nr Xn n1
n1
n1
rt
∞
∞
Xn
1/t
≤
<
n
E
I
X
P Xn ≥ n1/t
n
r
n
n1
n1
rt
∞
∞
|X| 1/t
1/t
E
I |X| < n
P |X| ≥ n
≤C
nr
n1
n1
≤C
∞ E |X|ttr−1 I |X| < n1/t
n1
n11/t · tr−1
t
E|X|
< ∞.
2.33
Because of Corollary 2.2, this proves that 2.26 holds for any 0 < t < 2.
To complete the proof we should show that
n
1/t −→ 0 as n −→ ∞
E Xi I Xi < i
i1
−1/t n
for any 0 < t < 2.
2.34
8
Discrete Dynamics in Nature and Society
For 0 < t < 1 and I1 : n−1/t ni1 i1/t P |X| ≥ i1/t , we have
n
n
1/t −1/t n E Xi I Xi < i
≤ n−1/t E Xi I Xi < i1/t
i1
i1
−1/t
≤C n
n
E |X|I |X| < i1/t I1
i1
−1/t
≤C n
n
E |X|I |X| < n1/t I1
2.35
i1
≤ C n1−1/t E |X|I |X| < n1/t I1
C n
1−1/t
n
t
E |X|I j − 1 ≤ |X| < j I1 .
j1
But
∞
P |X| ≥ n1/t ≤ CE|X|t < ∞,
2.36
n1
so by Kronecker’s lemma we get
I1 : n−1/t
n
i1/t P |X| ≥ i1/t −→ 0
as n −→ ∞.
2.37
i1
Moreover, we note
∞
∞
j 1−1/t E |X|I j − 1 ≤ |X|t < j ≤
E |X|I j − 1 ≤ |X|t < j E|X|t < ∞,
j1
2.38
j1
and by Kronecker’s lemma
n1−1/t
n
E |X|I j − 1 ≤ |X|t < j −→ 0
as n −→ ∞,
2.39
j1
which together with 2.35 and 2.37 gives 2.34 for 0 < t < 1.
Let 1 ≤ t < 2. First, we will show that
n
1/t
lim P Xi > εn
0.
n→∞
i1 To achieve this, we put Yi Xi I|Xi | < n1/t for 1 ≤ i ≤ n.
Because of EXi 0 and E|X|t < ∞, we have
n
n
−1/t n EYi ≤ n−1/t E Xi I Xi ≥ n1/t ≤ CE |X|t I |X| ≥ n1/t −→ 0
i1
i1
2.40
as n −→ ∞.
2.41
Anna Kuczmaszewska
9
Therefore, for large enough n, we obtain
n
n
n
1/t
1/t
1/t
≤
.
P Xi > εn
P Xi ≥ n
P Yi − EYi > εn
i1 i1
i1
Hence, we need only to show that
n
P Xi ≥ n1/t −→ 0
as n −→ ∞,
i1
n
1/t
−→ 0
Y − EYi > εn
P i1 i
2.42
2.43
as n −→ ∞.
By E|X|t < ∞, we get
∞
n
∞
1
P Xi ≥ n1/t ≤ C P |X| ≥ n1/t ≤ CE|X|t < ∞,
n i1
n1
n1
2.44
2.45
which implies 2.43.
By Lemma 1.3 and 2.23 with s 2,
∞
n
1 1/t
P Yi − EYi > εn
n i1
n1
≤C
∞
n−12/t
n1
≤C
∞
n
i1
−12/t
n
n1
≤C
∞
EYi2
n
n
∞
2 1/t
−12/t
2/t
1/t
E X I |X| < n
n
n P |X| ≥ n
i1
−2/t
n
n1
n1
i1
n
∞
E X 2 I j − 1 ≤ |X|t < j P |X| ≥ n1/t
j1
n1
2.46
∞
∞
2 t
−2/t
t
≤C
E X I k − 1 ≤ |X| < k
n
E|X|
k1
≤C
∞
nk
k
−2/t1
E X I k − 1 ≤ |X|t < k E|X|t
2
k1
≤C
∞
kP k − 1 ≤ |X|t < k E|X|t
< CE|X|t < ∞
k1
which gives 2.44.
Thus, we have established that 2.40 holds true. Equations 2.40 and 2.26 imply 2.34
which completes the proof of Corollary 2.3.
Corollary 2.3 gives the Marcinkiewicz SLLN for ρ∗ -mixing random variables {Xn , n ≥
1} stochastically dominated by a random variable X for 0 < t < 2. The identical result was
obtained by Cai 6 as a consequence of complete convergence theorem. Both of them, for
0 < t < 1, are a special case of more general result of Fazekas and Tómács 11 obtained for
stochastically dominated random variables without assuming any kind of dependence.
10
Discrete Dynamics in Nature and Society
Acknowledgment
The author thanks the referees for their comments and suggestions which allowed to improve
this paper.
References
1 R. C. Bradley, “On the spectral density and asymptotic normality of weakly dependent random
fields,” Journal of Theoretical Probability, vol. 5, no. 2, pp. 355–373, 1992.
2 W. Bryc and W. Smoleński, “Moment conditions for almost sure convergence of weakly correlated
random variables,” Proceedings of the American Mathematical Society, vol. 119, no. 2, pp. 629–635, 1993.
3 M. Peligrad, “Maximum of partial sums and an invariance principle for a class of weak dependent
random variables,” Proceedings of the American Mathematical Society, vol. 126, no. 4, pp. 1181–1189,
1998.
4 M. Peligrad and A. Gut, “Almost-sure results for a class of dependent random variables,” Journal of
Theoretical Probability, vol. 12, no. 1, pp. 87–104, 1999.
5 S. Utev and M. Peligrad, “Maximal inequalities and an invariance principle for a class of weakly
dependent random variables,” Journal of Theoretical Probability, vol. 16, no. 1, pp. 101–115, 2003.
6 G.-H. Cai, “Strong law of large numbers for ρ∗ -mixing sequences with different distributions,” Discrete
Dynamics in Nature and Society, vol. 2006, Article ID 27648, 7 pages, 2006.
7 M.-H. Zhu, “Strong laws of large numbers for arrays of rowwise ρ∗ -mixing random variables,” Discrete
Dynamics in Nature and Society, vol. 2007, Article ID 74296, 6 pages, 2007.
8 I. Fazekas and O. Klesov, “A general approach to the strong laws of large numbers,” Theory of
Probability and Its Applications, vol. 45, no. 3, pp. 436–449, 2000.
9 K.-L. Chung, “Note on some strong laws of large numbers,” American Journal of Mathematics, vol. 69,
no. 1, pp. 189–192, 1947.
10 H. Teicher, “Some new conditions for the strong law,” Proceedings of the National Academy of Sciences of
the United States of America, vol. 59, no. 3, pp. 705–707, 1968.
11 I. Fazekas and T. Tómács, “Strong laws of large numbers for pairwise independent random variables
with multidimensional indices,” Publicationes Mathematicae Debrecen, vol. 53, no. 1-2, pp. 149–161,
1998.