Abstract For a sequence of dependent square-integrable random variables and a sequence of positive constants {b n , n ≥ 1}, conditions are provided under which the series n i=1 (X i − E X i )/b i converges almost surely as n → ∞ and {X n , n ≥ 1} obeys the strong law of large numbers lim n→∞ n i=1 (X i − E X i )/b n = 0 almost surely. The hypotheses stipulate that two series converge, where the convergence of the first series involves the growth rates of {Var X n , n ≥ 1} and {b n , n ≥ 1} and the convergence of the second series involves the growth rate of {sup n≥1 |Cov (X n , X n+k )|, k ≥ 1}

Andrei Volodin

Andrew Rosalsky

Tien-Chung Hu

CiteSeerX

Statistics and Probability Letters 78 (2008) 1999–2005
www.elsevier.com/locate/stapro
On convergence properties of sums of dependent random variables
under second moment and covariance restrictions
Tien-Chung Hua,∗, Andrew Rosalskyb, Andrei Volodinc
a Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, ROC
b Department of Statistics, University of Florida, Gainesville, FL 32611, USA
c Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
Received 23 April 2007; accepted 9 January 2008
Available online 3 February 2008
Abstract
For a sequence of dependent square-integrable random variables and a sequence of positive constants {bn, n ≥ 1}, conditions
are provided under which the series
∑n
i=1(Xi − E Xi )/bi converges almost surely as n → ∞ and {Xn, n ≥ 1} obeys the strong
law of large numbers limn→∞
∑n
i=1(Xi − E Xi )/bn = 0 almost surely. The hypotheses stipulate that two series converge, where
the convergence of the first series involves the growth rates of {Var Xn, n ≥ 1} and {bn, n ≥ 1} and the convergence of the second
series involves the growth rate of {supn≥1 |Cov (Xn, Xn+k)|, k ≥ 1}.
c© 2008 Elsevier B.V. All rights reserved.
MSC: 60F15
1. Introduction
Let {Xn, n ≥ 1} be a sequence of square-integrable random variables defined on a probability space (Ω ,F, P)
and let {bn, n ≥ 1} be a sequence of positive constants. The random variables {Xn, n ≥ 1} are not assumed to be
independent. We assume that there exists a sequence of constants {ρk, k ≥ 1} such that
sup
n≥1
|Cov (Xn, Xn+k)| ≤ ρk, k ≥ 1 (1.1)
and we provide conditions on the growth rates of {Var Xn, n ≥ 1}, {bn, n ≥ 1} and {ρk, k ≥ 1} under which (i) the
series
n∑
i=1
(X i − E X i )/bi converges almost surely (a.s.) as n → ∞ (1.2)
∗ Corresponding author. Tel.: +886 3 5742642; fax: +886 3 5723888.
E-mail address: tchu@math.nthu.edu.tw (T.-C. Hu).
0167-7152/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.spl.2008.01.073
2000 T.-C. Hu et al. / Statistics and Probability Letters 78 (2008) 1999–2005
and (ii) {Xn, n ≥ 1} obeys the strong law of large numbers (SLLN)
lim
n→∞
n∑
i=1
(X i − E X i )/bn = 0 a.s. (1.3)
The history and literature on the SLLN problem for dependent summands is not nearly as extensive and complete
as it is for the case of independent summands. It appears that the best result establishing a SLLN for a sequence
of correlated random variables satisfying (1.1) and Var Xn = O(1) is that of Lyons (1988) wherein bn ≡ n. We
point out that Lyons (1988) did not provide conditions for a series of correlated random variables to converge a.s.; he
only treated the SLLN problem. The only SLLN result for correlated summands that we are aware of satisfying (1.1)
without assuming that Var Xn = O(1) is due to Hu et al. (2005) where again bn ≡ n. This result established a link
between the Golden Ratio ϕ and the SLLN. Other results on the SLLN problem for a sequence of correlated random
variables are those of Chandra (1991), Gapos̆kin (1975), Móricz (1977, 1985), Serfling (1970b, 1980).
In the current work, the main result, Theorem 1, provides conditions under which (1.2) and (1.3) hold with
{bn, n ≥ 1} being more general than only bn ≡ n in that n = O(bn) where again it is not assumed that Var Xn = O(1).
The proof of Theorem 1 is classical in nature and is based on the general “method of subsequences”. This method was
apparently developed initially by Rajchman (1932) (see Chung (1974), p. 103) and has since been used by numerous
other authors. However, the key inequality used in our proof is a much more recent result due to Serfling (1970a).
The plan of the paper is as follows. In Section 2, a lemma which is used in the proof of Theorem 1 is presented.
Theorem 1 is stated and proved in Section 3. In Section 4, Theorem 1 is compared with a well-known result.
2. Preliminaries
Throughout this paper, the symbol C denotes a generic constant (0 < C < ∞) which is not necessarily the same
one in each appearance. For x ≥ 1, the natural (base e) logarithm of x and the logarithm of x to the base 2 will be
denoted, respectively, by log x and Log x . We note that for all x ≥ 1, Log x = C log x where C = 1/ log 2.
The following lemma is used in the proof of Theorem 1.
Lemma 1. Let {Xn, n ≥ 1} be a sequence of square-integrable random variables and suppose that there exists a
sequence of constants {ρk, k ≥ 1} such that
sup
n≥1
|Cov (Xn, Xn+k)| ≤ ρk, k ≥ 1. (2.1)
Let {bn, n ≥ 1} be a sequence of positive constants such that
n = O(bn). (2.2)
Then for all n ≥ 0, m ≥ n + 2, and 0 ≤ q < 1,
E
(
m∑
i=n+1
X i − E X i
bi
)2
≤
m∑
i=n+1
Var X i
b2i
+
C
n1−q
m−n−1∑
k=1
ρk
kq
where C is a constant independent of n and m.
Proof. For all n ≥ 0, m ≥ n + 2, and 0 ≤ q < 1,
E
(
m∑
i=n+1
X i − E X i
bi
)2
=
m∑
i=n+1
Var (X i )
b2i
+ 2
m−1∑
i=n+1
m∑
j=i+1
Cov (X i , X j )
bi b j
≤
m∑
i=n+1
Var X i
b2i
+ C
m−1∑
i=n+1
m∑
j=i+1
ρ j−i
i j
(by (2.1) and (2.2))
=
m∑
i=n+1
Var X i
b2i
+ C
m−1∑
i=n+1
m∑
j=i+1
ρ j−i
j − i
(
1
i
−
1
j
)
T.-C. Hu et al. / Statistics and Probability Letters 78 (2008) 1999–2005 2001
=
m∑
i=n+1
Var X i
b2i
+ C
m−n−1∑
k=1
ρk
k
(
m−k∑
l=n+1
1
l
−
m∑
l=n+k+1
1
l
)
≤
m∑
i=n+1
Var X i
b2i
+ C
m−n−1∑
k=1
ρk
k
(
n+k∑
l=n+1
1
l
)
≤
m∑
i=n+1
Var X i
b2i
+ C
m−n−1∑
k=1
ρk
k
log
(
n + k
n
)
≤
m∑
i=n+1
Var X i
b2i
+
C
n1−q
m−n−1∑
k=1
ρk
kq
(2.3)
since log(1 + x) ≤ x1−q/(1 − q) for all x ≥ 0.
3. The main result
With the preliminaries accounted for, the main result may be stated and proved. We note that the condition (3.2)
is indeed stronger than that the condition
∑
∞
k=1 ρk/k < ∞ of Lyons (1988). However, as we remarked in Section 1,
Lyons (1988) treated the case Var Xn = O(1) and bn ≡ n to prove a SLLN and he did not establish a result along the
lines of (3.3). We also note that the condition (3.1) is automatic if
Var Xn
b2n
= O
(
1
n(log n)3(log log n)1+ε
)
for some ε > 0.
For example, suppose that Var Xn ∼ n/(log n)2. If bn ∼ n log n, then (2.2) and (3.1) hold whereas if bn ∼ n, then
(2.2) holds but (3.1) fails.
Theorem 1. Let {Xn, n ≥ 1} be a sequence of square-integrable random variables and suppose that there exists a
sequence of constants {ρk, k ≥ 1} such that (2.1) holds. Let {bn, n ≥ 1} be a sequence of positive constants satisfying
(2.2). Suppose that
∞∑
n=1
(Var Xn)(log n)2
b2n
< ∞ (3.1)
and
∞∑
k=1
ρk
kq
< ∞ for some 0 ≤ q < 1. (3.2)
Then
n∑
i=1
X i − E X i
bi
converges a.s. as n → ∞ (3.3)
and if bn ↑, the SLLN
lim
n→∞
n∑
i=1
(X i − E X i )
bn
= 0 a.s. (3.4)
obtains.
Proof. Note at the outset that (2.2) ensures that limn→∞ bn = ∞. Thus if bn ↑, then the conclusion (3.4) follows
immediately from (3.3) and the Kronecker lemma. To prove (3.3), note that for all n ≥ 1,
sup
m>n
E
(
m∑
i=1
X i − E X i
bi
−
n∑
i=1
X i − E X i
bi
)2
= sup
m>n
E
(
m∑
i=n+1
X i − E X i
bi
)2
2002 T.-C. Hu et al. / Statistics and Probability Letters 78 (2008) 1999–2005
≤ sup
m>n
(
m∑
i=n+1
Var X i
b2i
+
C
n1−q
m−n−1∑
k=1
ρk
kq
)
(by Lemma 1)
=
∞∑
i=n+1
Var X i
b2i
+
C
n1−q
∞∑
k=1
ρk
kq
= o(1) as n → ∞ (by (3.1) and (3.2), and q < 1).
Then by the Cauchy Convergence Criterion (see, e.g., Chow and Teicher (1997, p. 99)), there exists a random variable
S on (Ω ,F, P) with E S2 < ∞ such that
n∑
i=1
X i − E X i
bi
L2
→ S.
Thus for all n ≥ 1,
m∑
i=1
X i − E X i
bi
−
2n∑
i=1
X i − E X i
bi
L2
→ S −
2n∑
i=1
X i − E X i
bi
as m → ∞
whence (see, e.g., Chow and Teicher (1997, p. 101))
E
(
S −
2n∑
i=1
X i − E X i
bi
)2
= lim
m→∞
E
(
m∑
i=1
X i − E X i
bi
−
2n∑
i=1
X i − E X i
bi
)2
= lim
m→∞
E
(
m∑
i=2n+1
X i − E X i
bi
)2
≤ lim
m→∞
(
m∑
i=2n+1
Var X i
b2i
+
C
2n(1−q)
m−2n−1∑
k=1
ρk
kq
)
(by Lemma 1)
≤
∞∑
i=2n
Var X i
b2i
+
C
2n(1−q)
∞∑
k=1
ρk
kq
. (3.5)
Next, it will be shown that
2n∑
i=1
X i − E X i
bi
→ S a.s. (3.6)
For arbitrary ε > 0,
∞∑
n=1
P
{∣∣∣∣∣ 2
n∑
i=1
X i − E X i
bi
− S
∣∣∣∣∣ > ε
}
≤
1
ε2
∞∑
n=1
E
(
2n∑
i=1
X i − E X i
bi
− S
)2
(by the Markov inequality)
≤
1
ε2
∞∑
n=1
(
∞∑
i=2n
Var X i
b2i
+
C
2n(1−q)
∞∑
k=1
ρk
kq
)
(by (3.5))
= C
∞∑
i=2
[Log i]∑
n=1
Var X i
b2i
+ C
(
∞∑
n=1
1
2n(1−q)
)(
∞∑
k=1
ρk
kq
)
≤ C
∞∑
i=2
(Var X i ) log i
b2i
+ C
∞∑
k=1
ρk
kq
(since q < 1)
< ∞ (by (3.1) and (3.2)).
Then by Borel–Cantelli lemma and the arbitrariness of ε > 0, (3.6) follows.
T.-C. Hu et al. / Statistics and Probability Letters 78 (2008) 1999–2005 2003
We will now verify that
max
2n−1<k≤2n
∣∣∣∣∣∣
k∑
i=1
X i − E X i
bi
−
2n−1∑
i=1
X i − E X i
bi
∣∣∣∣∣∣ → 0 a.s. as n → ∞. (3.7)
For a ≥ 0 and n ≥ 1, let the joint distribution function of Xa+1, . . . , Xa+n be denoted by Fa,n . Define a functional g
on {Fa,n : a ≥ 0, n ≥ 1} by
g(Fa,n) =
a+n∑
i=a+1
Var X i
b2i
+ 2
a+n−1∑
i=a+1
a+n∑
j=i+1
|Cov (X i , X j )|
bi b j
, a ≥ 0, n ≥ 1
where the second term is interpreted as 0 if n = 1. Then for a ≥ 0, k ≥ 1, and m ≥ 1,
g(Fa,k) + g(Fa+k,m) =
a+k∑
i=a+1
Var X i
b2i
+ 2
a+k−1∑
i=a+1
a+k∑
j=i+1
|Cov (X i , X j )|
bi b j
+
a+k+m∑
i=a+k+1
VarX i
b2i
+ 2
a+k+m−1∑
i=a+k+1
a+k+m∑
j=i+1
|Cov (X i , X j )|
bi b j
≤
a+k+m∑
i=a+1
Var X i
b2i
+ 2
a+k+m−1∑
i=a+1
a+k+m∑
j=i+1
|Cov (X i , X j )|
bi b j
= g(Fa,k+m).
Moreover, it follows from (2.3) that for all a ≥ 0 and n ≥ 1,
E
(
a+n∑
i=a+1
X i − E X i
bi
)2
≤ g(Fa,n).
Then by Serfling’s (1970a) generalization of the Rademacher–Menchoff fundamental maximal inequality for the
partial sums of orthogonal random variables (see also Stout (1974), Sections 2.3 and 2.4), for all a ≥ 0 and n ≥ 1,
E
(
max
1≤k≤n
∣∣∣∣∣ a+k∑
i=a+1
X i − E X i
bi
∣∣∣∣∣
)2
≤ (Log2n)2g(Fa,n). (3.8)
Thus for arbitrary ε > 0,
∞∑
n=1
P
 max2n−1<k≤2n
∣∣∣∣∣∣
k∑
i=1
X i − E X i
bi
−
2n−1∑
i=1
X i − E X i
bi
∣∣∣∣∣∣ > ε

≤ 1 +
1
ε2
∞∑
n=2
E
 max
2n−1<k≤2n
∣∣∣∣∣∣
k∑
i=1
X i − E X i
bi
−
2n−1∑
i=1
X i − E X i
bi
∣∣∣∣∣∣
2 (by the Markov inequality)
= 1 +
1
ε2
∞∑
n=2
E
 max
2n−1<k≤2n
∣∣∣∣∣∣
k∑
i=2n−1+1
X i − E X i
bi
∣∣∣∣∣∣
2
= 1 +
1
ε2
∞∑
n=2
E
 max
1≤k≤2n−1
∣∣∣∣∣∣
2n−1+k∑
i=2n−1+1
X i − E X i
bi
∣∣∣∣∣∣
2
≤ 1 +
1
ε2
∞∑
n=2
(Log(2 · 2n−1))2g(F2n−1,2n−1) (by (3.8))
2004 T.-C. Hu et al. / Statistics and Probability Letters 78 (2008) 1999–2005
≤ 1 + C
∞∑
n=2
(Log2n−1)2
 2n∑
i=2n−1+1
Var X i
b2i
+ 2
2n−1∑
i=2n−1+1
2n∑
j=i+1
|Cov (X i , X j )|
bi b j

≤ 1 + C
∞∑
n=2
2n∑
i=2n−1+1
(Var X i )(Log i)2
b2i
+ C
∞∑
n=2
n2
2n−1∑
i=2n−1+1
2n∑
j=i+1
ρ j−i
i j
(by (2.1) and (2.2))
≤ 1 + C
∞∑
i=1
(Var X i )(log i)2
b2i
+ C
∞∑
n=2
n2
(2n−1)1−q
2n−1−1∑
k=1
ρk
kq
(by arguing as in the proof of Lemma 1)
≤ 1 + C
∞∑
i=1
(Var X i )(log i)2
b2i
+ C
(
∞∑
n=2
n2
(2n−1)1−q
)(
∞∑
k=1
ρk
kq
)
< ∞
by (3.1) and (3.2), q < 1, and
n2
(2n−1)1−q
= O
((
1
2
1−q
2
)n)
.
Then by the Borel–Cantelli lemma and the arbitrariness of ε > 0, (3.7) follows.
Next, for k ≥ 2, let n ≥ 1 be such that 2n−1 < k ≤ 2n . Then∣∣∣∣∣ k∑
i=1
X i − E X i
bi
− S
∣∣∣∣∣ ≤
∣∣∣∣∣∣
k∑
i=1
X i − E X i
bi
−
2n−1∑
i=1
X i − E X i
bi
∣∣∣∣∣∣+
∣∣∣∣∣∣
2n−1∑
i=1
X i − E X i
bi
− S
∣∣∣∣∣∣
≤ max
2n−1< j≤2n
∣∣∣∣∣∣
j∑
i=1
X i − E X i
bi
−
2n−1∑
i=1
X i − E X i
bi
∣∣∣∣∣∣+
∣∣∣∣∣∣
2n−1∑
i=1
X i − E X i
bi
− S
∣∣∣∣∣∣
→ 0 a.s. as k → ∞
by (3.6) and (3.7) thereby proving that
lim
k→∞
k∑
i=1
X i − E X i
bi
= S a.s.
and hence (3.3) is established.
4. Concluding comments
To conclude, we compare Theorem 1 with a well-known result. The following proposition is essentially Corollary
2.4.1 of Stout (1974, p. 28).
Proposition 1. Let {Xn, n ≥ 1} be a sequence of square-integrable random variables and suppose that there exists a
sequence of constants {rk, k ≥ 1} such that
sup
n≥1
|Cov (Xn, Xn+k)|
(Var Xn · Var Xn+k)1/2
≤ rk, k ≥ 1.
Let {bn, n ≥ 1} be a sequence of positive constants. If
∞∑
n=1
(Var Xn)(log n)2
b2n
< ∞
and
∞∑
k=1
rk < ∞, (4.1)
T.-C. Hu et al. / Statistics and Probability Letters 78 (2008) 1999–2005 2005
then
n∑
i=1
X i − E X i
bi
converges a.s. as n → ∞.
To compare Theorem 1 with Proposition 1, observe that condition (2.2) is not needed in Proposition 1. In general,
the conditions (3.2) and (4.1) are not comparable; however, if Var Xn = O(1), then (3.2) is weaker than (4.1).
References
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On convergence properties of sums of dependent random variables under second moment and covariance restrictions

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On convergence properties of sums of dependent random variables under second moment and covariance restrictions

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