Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 598319, 10 pages
doi:10.1155/2008/598319
Research Article
Maximal Inequalities for Dependent Random
Variables and Applications
Soo Hak Sung
Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea
Correspondence should be addressed to Soo Hak Sung, sungsh@mail.pcu.ac.kr
Received 16 April 2008; Revised 3 June 2008; Accepted 7 July 2008
Recommended by Ondrej Dosly
For a sequence {Xn , n ≥ 1} of dependent square integrable random variables and a sequence
{bn , n ≥ 1} of positive numbers, we establish a maximal inequality for weighted sums of dependent
n
random
n variables. Applying this inequality, we obtain the almost sure convergence of i1 Xi /bi
and i1 Xi /bn .
Copyright q 2008 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.
1. Introduction
Throughout this paper let {Xn , n ≥ 1} be a sequence of random variables defined on a
probability space Ω, F, P and let {bn , n ≥ 1} be a sequence of positive numbers. We assume
that there exists a sequence {ρn , n ≥ 1} of nonnegative constants such that
supEXk Xkn ≤ ρn ,
for n ≥ 1.
1.1
k≥1
In this paper, we establish a maximal inequality for weighted sums of the dependent random
variables satisfying 1.1. Applying this inequality, we obtain under some suitable conditions
on the sequence {ρn } that
n
Xi
i1
bi
converges a.s. as n −→ ∞
and the strong law of large numbers SLLN
n
i1 Xi
−→ 0 a.s.
bn
Note that if 0 < bn ↑ ∞, then 1.2 implies 1.3 by the Kronecker lemma.
1.2
1.3
2
Journal of Inequalities and Applications
For a sequence of dependent random variables satisfying 1.1, the SLLNs were
established by Hu et al. 1, 2 and Lyons 3. Lyons 3 obtained an SLLN under the conditions
that VarXn O1 and bn n. Without condition VarXn O1, Hu et al. 1 obtained
an SLLN, where bn n. Hu et al. 2 also obtained an SLLN for more general sequence {bn }
bn n is replaced by n Obn .
For other results on the SLLN for a sequence of correlated random variables, see Chandra
4, Móricz 5, 6, and Serfling 7, 8.
In this paper, we give a sufficient condition under which 1.2 and 1.3 hold. Our results
partially improve those of Hu et al. 1, 2. The technique used in our proof is the wellknown method of subsequences. Note that the maximal inequality is used in the method of
subsequences. Our maximal inequality for weighted sums of the dependent random variables
satisfying 1.1 is sharper than that of Hu et al. 2.
Throughout this paper, log x denotes the natural logarithm.
2. Maximal inequalities for dependent random variables
To prove the maximal inequality for weighted sums of dependent random variables satisfying
1.1, the following lemma is needed.
Lemma 2.1. Let {Xn , n ≥ 1} be a sequence of square integrable random variables satisfying 1.1. Let
{bn , n ≥ 1} be a sequence of positive numbers such that
n ≤ Dbn
∀n ≥ 1 and some constant D > 0.
2.1
Then for all n ≥ 1, m > n, and δ > 0,
m−1
in
m EX X m−n
ρk
D 2 Cδ
i j
≤
1 log k1δ ,
bi bj
max{log 2δ , log nδ } k1 k
ji1
2.2
where Cδ 2δ1 max{1, δ δ e−δ }.
Proof. For simplicity of notation, let In,m and 2.1 that for 1 ≤ n < m,
In,m ≤
m−1
in
≤ D2
m−1 m
in
ji1 EXi Xj /bi bj .
m ρ
j−i
b
bj
i
ji1
m−1
in
m ρ
j−i
ij
ji1
m−n
m−k
ρk
ii k
k1 in
m−n
ρk m−k
1
1
D2
−
k in i i k
k1
D2
≤ D2
Then we get by 1.1
m−n
1
ρk nk−1
.
k in i
k1
2.3
Soo Hak Sung
3
We next estimate
nk−1
in
nk−1
in
1/i. If n 1, then
k
1 1
≤1
i i1 i
k
1
dx ≤ 1 log k ≤ 1 log k1δ .
x
2.4
k
1
k
≤ 2 log 1 .
dx log 1 x
n−1
n
2.5
1
If n ≥ 2, then
nk−1
in
1
≤
i
nk−1
n−1
The log1 k/n is estimated as follows:
k
log 1 n
⎧
1δ
⎪
log nδ
2δδ e−δ
1
⎪
δ −δ 1 log k
⎪
≤
≤
≤
2δ
e
,
log
1
⎪
√
√
⎪
⎪
n
⎨
log nδ n
log nδ
log nδ
≤
⎪
δ
⎪
⎪
log k1δ
1 log k1δ
k 2 log k
⎪
δ
δ
⎪
≤
2
≤
2
,
log
1
⎪
⎩
n log nδ
log nδ
log nδ
if 1 ≤ k ≤
if k >
Thus, we have the desired estimate for In,m :
⎧
m−n
ρk
⎪
⎪
⎪
1 log k1δ ,
if n 1,
⎪ D2
⎪
⎪
k
⎨
k1
In,m ≤
⎪
m−n
⎪
ρk 2 max {2δ , 2δδ e−δ }
⎪
⎪
2
⎪
1 log k1δ , if n ≥ 2,
D
⎪
⎩
δ
k
log n
k1
≤
√
√
n,
n.
2.6
2.7
ρk
D2 2δ1 max{1, δ δ e−δ } m−n
1 log k1δ .
δ
δ
k
max{log 2 , log n } k1
The following lemma is a maximal inequality for general dependent random variables.
Lemma 2.2. Let {Xn , n ≥ 1} be a sequence of square integrable random variables. Then for all a ≥ 0
and n ≥ 1,
2 an
an
ak
an−1
log 2n 2 2
E max Xi EXi 2
EXi Xj .
≤
log 2
1≤k≤n
ia1
ia1
ia1 ji1
2.8
Proof. Let Fa,n be the joint distribution function of Xa1 , . . . , Xan . Define a function g on {Fa,n :
a ≥ 0, n ≥ 1} by
gFa,n an
EXi2 2
ia1
an
an−1
EXi Xj .
ia1 ji1
2.9
4
Journal of Inequalities and Applications
Then we can easily obtain that for a ≥ 0, k ≥ 1, and m ≥ 1,
gFa,k gFak,m ≤ gFa,km .
2.10
Moreover, we have that for all a ≥ 0 and n ≥ 1,
an
E
2
Xi
≤ gFa,n .
2.11
ia1
By Serfling’s 9 generalization of the Rademacher-Menchoff maximal inequality for orthogonal random variables,
2 ak
log 2n 2
E max Xi gFa,n .
≤
log 2
1≤k≤n
ia1
2.12
Thus, the result is proved.
Combining Lemmas 2.1 and 2.2 gives the following maximal inequality for weighted
sums of dependent random variables satisfying 1.1.
Lemma 2.3. Let {Xn , n ≥ 1} be a sequence of square integrable random variables satisfying 1.1. Let
{bn , n ≥ 1} be a sequence of positive numbers satisfying 2.1. Then for all n ≥ 1, m > n, and δ > 0,
2 i
Xj E max n≤i≤m
b jn j
≤
log2m − n 1
log 2
2 m
in
EXi2
bi2
ρk
1δ
,
1 log k
max{log 2δ , log nδ } k1 k
2D2 Cδ
m−n
2.13
where Cδ 2δ1 max{1, δ δ e−δ }.
3. Almost surely convergent series and strong laws of large numbers
In this section, we will assume that {Xn , n ≥ 1} is a sequence of square integrable random
variables satisfying 1.1. A sufficient condition will be given under which 1.2 and 1.3 hold.
We first state and prove one of our main results. The proof is based on the well known
method of subsequences. Our proof is similar to that of Hu et al. 2. However, the maximal
inequality Lemma 2.3 used in the proof is sharper than that of Hu et al. 2.
Theorem 3.1. Let {Xn , n ≥ 1} be a sequence of square integrable random variables satisfying 1.1. Let
{bn , n ≥ 1} be a sequence of positive numbers satisfying 2.1. Suppose that the following conditions
hold:
2
2
2
i ∞
n1 log n EXn /bn < ∞,
∞
ii n1 log n4δ ρn /n < ∞ for some δ > 0.
Then 1.2 holds. Furthermore, if 0 < bn ↑ ∞, then 1.3 holds.
Soo Hak Sung
5
Proof. As noted in the introduction, if 0 < bn ↑ ∞, then 1.2 implies 1.3. To prove 1.2, let
Sn ni1 Xi /bi . By Lemma 2.1 with δ replaced by 3 δ, we have that for m > n,
ESm − Sn 2 ≤
≤
m EX 2
i
2
2
in1 bi
m EX 2
i
2
in1 bi
∞ EX 2
i
in1
bi2
m−1
m EX X
i j
b
b
i
j
in1 ji1
m−n−1
2D2 C3δ
max{log 2
3δ
3δ
, log n
}
k1
∞
ρk
2D2 C3δ
max{log 23δ , log n3δ } k1 k
ρk
1 log k4δ
k
3.1
1 log k4δ −→ 0
as n → ∞ by i and ii. Here C3δ 2δ4 max{1, 3 δ3δ e−3δ }. By the Cauchy convergence
criterion, there exists a random variable S such that ESn − S2 → 0 as n → ∞. It is easy to see
that S2n → S a.s. by the standard method. It remains to show that
max |Sk − S2n | −→ 0 a.s. as n −→ ∞.
3.2
2n <k≤2n1
Using Lemma 2.3, i, and ii, we get that
∞ P max |Sk − S2n | > n1
2n <k≤2n1
≤
≤
≤
∞ 1
2
n|
E
max
|S
−
S
k
2
2 n1 2n <k≤2n1
2 ∞ 2n1 EX 2
log 2n1
1
i
2 n1
1
log 2
i2n 1
bi2
∞ log2i2 EX 2
i
2 log 22 i3
bi2
n
2
−1
2D2 C3δ
log 2n 1
3δ
ρk
1 log k4δ
k
k1
3.3
∞
∞
ρk
n 12 2D2 C3δ 1 log k4δ < ∞.
3δ
3δ
k
2 log 2 n1 n
k1
Then 3.2 follows by the Borel-Cantelli lemma.
Remark 3.2. Hu et al. 2 proved Theorem 3.1 under i and ii .
q
ii ∞
n1 ρn /n < ∞ for some 0 ≤ q < 1.
Since condition ii of Theorem 3.1 is weaker than ii , Theorem 3.1 improves the result
of Hu et al. 2.
We can now establish the following SLLN if condition 2.1 on {bn } is replaced by the
condition 0 < bn ↑ ∞.
Theorem 3.3. Let {Xn , n ≥ 1} be a sequence of square integrable random variables satisfying 1.1. Let
{bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers. Suppose that the following
conditions hold:
2
2
2
i ∞
n1 log n EXn /bn < ∞,
6
Journal of Inequalities and Applications
ii
iii
∞
n1 ρn
∞
∞
in1 log i
2 ∞
n1 EXn
in1
2
/bi2 < ∞,
log i/ibi2 < ∞.
Then 1.3 holds.
To prove Theorem 3.3, we need the following lemma which is due to Fazekas and Klesov
10.
Lemma 3.4. Let {Xn , n ≥ 1} be a sequence of random variables and {bn , n ≥ 1} be a nondecreasing
unbounded sequence of positive numbers. Let {αn , n ≥ 1} be a sequence of nonnegative numbers.
Assume that for each n ≥ 1,
r i
n
E max Xj αi ,
for some constant r > 0.
3.4
≤
1≤i≤n j1
i1
r
If ∞
n1 αn /bn < ∞, then 1.3 holds.
Proof of Theorem 3.3. From Lemma 2.2,
2 n
i
n−1 n
log 2n 2 2
EXi 2
EXi Xj .
≤
E max Xj 1≤i≤n log 2
j1
i1
i1 ji1
3.5
Define αn log 2n/ log 22 An − log 2n − 1/ log 22 An−1 for n ≥ 1, where A0 0 and An n−1 n
i
n
n
2
2
i1 EXi 2 i1
ji1 EXi Xj for n ≥ 1. Then Emax1≤i≤n | j1 Xj | ≤
i1 αi and
log 2n 2
log 2n 2
log 2n − 1 2
αn An − An−1 An−1
−
log 2
log 2
log 2
2 n−1
log 2n
EXn2 2 EXi Xn 3.6
log 2
i1
n−1
n−1
n−2 log 2n 2
log 2n − 1 2
2
−
EXi 2
EXi Xj .
log 2
log 2
i1
i1 ji1
By Lemma 3.4, it is enough to show that
∞
log 2n2 EXn2
bn2
n1
< ∞,
3.7
n−1
∞
log 2n2 EXi Xn < ∞,
2
bn
n1
i1
n−1
∞
log 2n2 − log 2n − 12 bn2
n2
n3
3.9
i1
n−2 ∞
n−1
log 2n2 − log 2n − 12 bn2
EXi2 < ∞,
3.8
EXi Xj < ∞.
3.10
i1 ji1
Clearly 3.7 holds by i. It is easy to see that 3.8–3.10 hold, and the detailed proofs are
omitted.
Soo Hak Sung
7
The following corollary shows that condition ii of Theorem 3.3 can be simplified under
the additional condition 2.1 on {bn }.
Corollary 3.5. Let {Xn , n ≥ 1} be a sequence of square integrable random variables satisfying 1.1.
Let {bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers satisfying 2.1. Suppose
that the following conditions hold:
2
2
2
i ∞
n1 log n EXn /bn < ∞,
∞
ii n1 log n2 ρn /n < ∞,
2
2 ∞
iii ∞
n1 EXn in1 log i/ibi < ∞.
Then 1.3 holds.
Proof. By 2.1, we have that
∞
ρn
n1
∞
log i2
in1
bi2
≤ D2
∞
n1
ρn
∞
∞
log i2
log n2
≤
C
ρ
n
n
i2
in1
n1
3.11
for some constant C > 0. Thus the result follows by Theorem 3.3.
Remark 3.6. Condition ii of Corollary 3.5 is weaker than condition ii of Theorem 3.1. On the
other hand, an additional condition is needed in Corollary 3.5 namely condition iii above.
Using the following lemma, we can omit condition iii of Theorem 3.3 if conditions
2.1 and 3.12 on {bn } are satisfied. If C1 n ≤ bn ≤ C2 nα for all n ≥ 1 and some constants
C1 > 0, C2 > 0, and α > 0, then 2.1 and 3.12 hold.
Lemma 3.7. Let {bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers satisfying
2.1. If
log bn
lim sup
< ∞,
3.12
n → ∞ log n
then
∞
log i
bn2 log2 n in ibi2
O1.
3.13
Proof. Without loss of generality, we may assume that i ≤ bi for all i ≥ 1.
Let fix n. For each k ≥ 1, define mk by mk min{i ≥ n : bi ≥ kbn }. Then bmk ≥ kbn and
n m1 ≤ m2 ≤ · · · . It follows that
∞
∞ m
∞
k1 −1
k1 −1
log i
log i
log i
bn2 bn2 bn2 1 m
≤
by 0 < bn ↑
2
2
2
2
2
2
log n in ibi
log n k1 imk ibi
log n k1 bmk imk i
≤
∞
bn2 2
m
k1 −1
1
2
log i
i
log n k1 kbn imk
mk1 −1
∞
3
log i
log x
1 1 ≤
dx
x
log2 n k1 k 2 i1 i
3
≤
2
∞
1 log 2 log 3 logmk1 − 1
,
2
3
2
log2 n k1 k 2
1
3.14
8
Journal of Inequalities and Applications
mk1 −1
where we assume in the case mk1 mk , the sum imk 0. Since bi ≥ i for all i ≥ 1, bkbn 1 ≥
kbn 1 ≥ kbn and mk ≤ kbn 1 ≤ kbn 1. So we have that
logmk1 − 12 ≤ logk 1bn 2 ≤ 2 logk 12 log bn 2 .
3.15
Substituting this into 3.14, 3.13 holds by 3.12.
The following example shows that Lemma 3.7 fails if 3.12 does not hold.
Example 3.8. Let φ0 1 and φn 2φn−1 for n ≥ 1. Define a sequence {bn , n ≥ 1} by bn φk 1 if φk ≤ n < φk 1. Then 0 < bn ↑ ∞ and bn ≥ n for all n ≥ 1. Since bφn φn 1,
we obtain that
log bφn log φn 1 φn log 2
−→ ∞
log φn
log φn
log φn
3.16
as n → ∞. Hence 3.12 does not hold. We also obtain that for n ≥ 2,
2
bφn
log
2
∞
log i
φn iφn ibi2
≥
≥
2
bφn
φn1−1
log i
log φn
iφn
ibi2
1
φn1−1
2
2
log φn
1
iφn
φn1
log i
i
3.17
log x
dx
x
log2 φn φn
2
1 φn log 2
1
− −→ ∞
2 log φn
2
as n → ∞. So 3.13 does not hold.
If {bn } satisfies 2.1 and 3.12, then we can obtain the following SLLN.
Theorem 3.9. Let {Xn , n ≥ 1} be a sequence of square integrable random variables satisfying 1.1. Let
{bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers satisfying 2.1 and 3.12.
Suppose that the following conditions hold:
2
2
2
i ∞
n1 log n EXn /bn < ∞.
∞
ii n1 log n2 ρn /n < ∞.
Then 1.3 holds.
Proof. By Lemma 3.7 and i, we get
∞
n1
EXn2
∞
log i
∞
log2 n 1EXn2
≤
O1
< ∞,
2
2
bn1
in1 ibi n1
3.18
since logn 1/bn1 ≤ logn 1/bn ≤ 2 log n/bn if n ≥ 2. The result follows by Corollary 3.5.
Soo Hak Sung
9
Remark 3.10. Under condition 3.12, Theorem 3.9 improves Theorem 3.1, since condition ii
of Theorem 3.9 is weaker than condition ii of Theorem 3.1.
If bn n for all n ≥ 1, then {bn } satisfies 2.1 and 3.12. Hence we can obtain the
following.
Corollary 3.11. Let {Xn , n ≥ 1} be a sequence of square integrable random variables satisfying 1.1.
Suppose that the following conditions hold.
2
2
2
i ∞
n1 log n EXn /n < ∞.
∞
ii n1 log n2 ρn /n < ∞.
Then the SLLN holds. Namely,
n
i1 Xi
n
−→ 0 a.s.
3.19
Remark 3.12. Lyons 3 proved an SLLN 3.19 under the conditions that EXn2 O1 and
∞
ρn
n1
n
< ∞.
3.20
When EXn2 O1, condition i of Corollary 3.11 is obviously satisfied. Hu et al. 1 proved
an SLLN 3.19 under conditions 3.21 and 3.22:
∞
Hnϕ1 n1
n2
<∞,
∞
ρn
< ∞,
ϕ−1
n1 n
3.21
3.22
√
where ϕ 1 5/2 1.618 · · · is the golden ratio, and Hx > 0 is a nondecreasing function
on 0, ∞ such that EXn2 ≤ Hn for all n ≥ 1. Condition ii of Corollary 3.11 is weaker than
3.22. In general, condition i of Corollary 3.11 is not comparable with 3.21.
Acknowledgments
The author would like to thank the referees for helpful comments and suggestions that
considerably improved the presentation of this paper. This work was supported by Korea
Science and Engineering Foundation KOSEF grant funded by Korea government MOST
no. R01-2007-000-20053-0.
References
1 T.-C. Hu, A. Rosalsky, and A. I. Volodin, “On the golden ratio, strong law, and first passage problem,”
The Mathematical Scientist, vol. 30, no. 2, pp. 77–86, 2005.
2 T.-C. Hu, A. Rosalsky, and A. Volodin, “On convergence properties of sums of dependent random
variables under second moment and covariance restrictions,” Statistics & Probability Letters. In press.
3 R. Lyons, “Strong laws of large numbers for weakly correlated random variables,” Michigan
Mathematical Journal, vol. 35, no. 3, pp. 353–359, 1988.
10
Journal of Inequalities and Applications
4 T. K. Chandra, “Extensions of Rajchman’s strong law of large numbers,” Sankhyā. Series A, vol. 53, no.
1, pp. 118–121, 1991.
5 F. Móricz, “The strong laws of large numbers for quasi-stationary sequences,” Zeitschrift für
Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 38, no. 3, pp. 223–236, 1977.
6 F. Móricz, “SLLN and convergence rates for nearly orthogonal sequences of random variables,”
Proceedings of the American Mathematical Society, vol. 95, no. 2, pp. 287–294, 1985.
7 R. J. Serfling, “Convergence properties of Sn under moment restrictions,” The Annals of Mathematical
Statistics, vol. 41, no. 4, pp. 1235–1248, 1970.
8 R. J. Serfling, “On the strong law of large numbers and related results for quasistationary sequences,”
Theory of Probability and Its Applications, vol. 25, no. 1, pp. 187–191, 1980.
9 R. J. Serfling, “Moment inequalities for the maximum cumulative sum,” The Annals of Mathematical
Statistics, vol. 41, no. 4, pp. 1227–1234, 1970.
10 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, 2001.