Bull. Korean Math. Soc. 43 (2006), No. 1, pp. 213–223
MARCINKIEWICZ-ZYGMUND LAW OF LARGE
NUMBERS FOR BLOCKWISE ADAPTED SEQUENCES
Nguyen Van Quang and Le Van Thanh
Abstract. In this paper we establish the Marcinkiewicz-Zygmund
strong law of large numbers for blockwise adapted sequences. Some
related results are considered.
1. Introduction and notations
In [5] and [8] it was shown that some properties of independent sequences of random variables can be applied to the sequences consisting
of independent blocks. Particularly, it was proved in [8] that if (Xi )∞
i=1 ,
k
k+1
EXi = 0 is a sequence independent in blocks
then it satisfies
P [2 , 2 2 ),−2
(EX
)i
< ∞ implies
the Kolmogorov’s theorem: the condition ∞
i
i=1
the strong law large numbers (s.l.l.n.), i.e.,
n
1X
lim
Xi = 0 a.s.
n→∞ n
i=1
Strong law of large numbers for blockwise independent random variables was studied by V. F. Gaposhkin [4].
Marcinkiewicz-Zygmund type strong law of large numbers was studied by many authors. In 1981, N. Etemadi [3] proved that if {Xn , n ≥ 1}
is a sequence of pairwise i.i.d. random variables with E|X1 | < ∞, then
1 Pn
lim
i=1 (Xi − EX1 ) = 0 a.s.
n→∞ n
Later, in 1985, B. D. Choi and S. H. Sung [2] have shown that if
{Xn , n ≥ 1} are pairwise independent and are dominated in distribution
Received January 21, 2005.
2000 Mathematics Subject Classification: 60F05, 60F15.
Key words and phrases: Blockwise independent, blockwise adapted sequence,
block martingale difference, Marcinkiewicz-Zygmund law of large numbers.
This work was supported by the National Science Council of Vietnam.
214
Nguyen Van Quang and Le Van Thanh
by a random variable X with E|X|p (log+ |X|)2 < ∞, 1 < p < 2, then
n
1 X
(Xi − EXi ) = 0 a.s.
n→∞ p1
n i=1
lim
Recently, D. H. Hong and S. Y. Hwang [6], D. H. Hong and A. I. Volodin
[7] studied Marcinkiewicz-Zygmund strong law of large numbers for double sequence of random variables.
In this paper we establish the Marcinkiewicz-Zygmund strong law of
large numbers for blockwise adapted sequences. Some related results are
considered.
Let {ω(n), n ≥ 1} be a strictly increasing sequence
£ of positive integers
¢
with ω(1) = 1. For each k ≥ 1, we set ∆k = ω(k), ω(k + 1) . We
recall that the sequence {Xi , i ≥ 1} of random variables is blockwise
independent with respect to blocks ∆k , if for any fixed k, the sequences
{Xi }i∈∆k are independent. Let {Fi , i ≥ 1} be a sequence of σ-fields
such that for any fixed k, the sequences {Fi , i ∈ ∆k } are increasing.
The sequence {Xi , i ≥ 1} of random variables is said to be blockwise
adapted to {Fi , i ≥ 1}, if each Xi is measurable with respect to Fi . The
sequence {Xi , Fi , i ≥ 1} is said to be a block martingale difference with
respect to blocks ∆k , if for any fixed k, the sequences {Xi , Fi }i∈∆k are
martingale differences. Denote
Nm = min{n|ω(n) ≥ 2m },
sm = Nm+1 − Nm + 1,
ϕ(i) = max sk if i ∈ [2m , 2m+1 ),
k≤m
(m)
= [2m , 2m+1 ), m ≥ 0,
(m)
= ∆k ∩ ∆(m) , m ≥ 0, k ≥ 1,
∆
∆k
(m)
6= ∅},
(m)
∆k
6= ∅}.
pm = min{k : ∆k
qm = max{k :
Since ω(Nm − 1) < 2m , ω(Nm ) ≥ 2m , ω(Nm+1 ) ≥ 2m+1 for each m ≥ 1,
(m)
the number of nonempty blocks [∆k ] is not large than sm = Nm+1 −
(m)
(m)
(m)
Nm + 1. Assume ∆k 6= ∅, let rk = min{r : r ∈ ∆k }.
The sequence {Xn , n ≥ 1} is said to be stochastically dominated
by a random variable X if there exists a constant C > 0 such that
P {|Xn | > t} ≤ CP {|X| > t} for all nonnegative real numbers t and for
all n ≥ 1.
Marcinkiewicz-Zygmund law of large numbers
215
Finally, the symbol C denotes throughout a generic constant (0 <
C < ∞) which is not necessarily the same one in each appearance.
2. Lemmas
In the sequel we will need the following lemmas.
Lemma 2.1. (Doob’s Inequality) If {Xi , Fi }N
i=1 is a martingale difference, E|Xi |p < ∞ (1 < p < ∞), then
¯p
¯
¯N
¯p
k
¯
¯
X
p p ¯¯X ¯¯
¯
¯
Xi ¯ ≤ (
E ¯max
) E¯
Xi ¯ .
¯
¯ k≤N
¯
¯
p−1
i=1
i=1
The next lemma is due to von Bahr and Esseen [1].
Lemma 2.2. (von Bahr and Esseen [1]) Let {Xi , }N
i=1 be random variables such
P that E{Xm+1 |Sm } = 0 for 0 ≤ m ≤ N − 1, where S0 = 0 and
Sm = m
i=1 Xi for 1 ≤ m ≤ N , then
E|SN |p ≤ C
N
X
E|Xi |p for all 1 ≤ p ≤ 2,
i=1
where C is a constant independent of N .
By lemmas 2.1 and 2.2, we get the following lemma.
p
Lemma 2.3. If {Xi , Fi }N
i=1 is a martingale difference, E|Xi | ≤ ∞ (1
≤ p ≤ 2), then
¯
¯p
k
N
¯
¯
X
X
¯
¯
E ¯max
Xi ¯ ≤ C
E|Xi |p ,
¯ k≤N
¯
i=1
i=1
where C is a constant independent of N .
Proof. In the case p = 1, we have
¯
¯
k
N
N
¯
¯
X
X
X
¯
¯
E ¯max
Xi ¯ ≤ E(
|Xi |) =
E|Xi |.
¯ k≤N
¯
i=1
i=1
i=1
In the case 1 < p ≤ 2,
¯p
¯p
¯
¯
k
N
¯
¯
¯
¯X
X
p
¯
¯
¯
¯
Xi ¯ ≤ (
Xi ¯ (By Lemma 2.1)
E ¯max
)p E ¯
¯
¯
¯ k≤N
¯
p−1
i=1
i=1
≤C
N
X
i=1
E|Xi |p (By Lemma 2.2).
216
Nguyen Van Quang and Le Van Thanh
The proof of the lemma is completed.
Lemma 2.4. If q > 1 and {xn , n ≥ 0} is a sequence of constants such
that lim xn = 0, then
n→∞
lim q
−n
n→∞
n
X
q k+1 xk = 0.
k=0
P∞
Proof. Let s = q + i=0 q −i . For any ² > 0, there exists k0 such that
²
|xk | <
for all k ≥ k0 . Since lim q −n = 0, so, there exists n0 ≥ k0
n→∞
2s ¯
P 0 k+1 ¯ ²
such that ¯q −n kk=0
q xk ¯ < . It follows that, for all n ≥ n0 ,
2
k0
n
n
¯ ¯ −n X
¯
¯ −n X k+1 ¯ ¯ −n X
k+1 ¯
¯
¯q
¯
¯
q xk ≤ q
q xk + q
q k+1 xk ¯
k=0
k=0
k0 +1
²
²
1
≤ + (q + 1 + + · · · )
2 2s
q
²
²
= + = ²,
2 2
which completes the proof.
3. Main result
With the notations and lemmas accounted for, main results may now
be established. Theorem 3.1 establishes the strong law of large numbers
for block martingale differences.
Theorem 3.1. Let {Xi , Fi }∞
i=1 be a block martingale difference with
respect to blocks ∆k , (1 ≤ p ≤ 2). If
∞
X
E|Xi |2
2
i=1
then
ip
< ∞,
Pn
i=1 Xi
1
p
1
n ϕ 2 (n)
→ 0 a.s. as n → ∞.
Proof. Let
(m)
γk
n
¯ X
¯
¯
¯
= max ¯
Xi ¯,
(m)
n∈∆k
(m)
i=rk
m ≥ 0, k ≥ 1;
Marcinkiewicz-Zygmund law of large numbers
γm = 2
−m−1
p
X
1
ϕ− 2 (2m )
(m)
γk ,
217
m ≥ 0.
pm ≤k≤qm
(m)
Using Lemma 2.3 for martingale differences {Xi , Fi , i ∈ ∆k }, we have
X
(m)
E|γk |2 ≤ C
EXi2 , for all m ≥ 0, k ≥ 1.
(m)
i∈∆k
It implies
2
E|γm | ≤ 2
−2m−2
p
ϕ
−1
qm
X
m
(2 )sm
(m)
E|γk |2
k=pm
≤2
qm
X
−2m−2
p
(m)
E|γk |2
k=pm
≤ C2
−2m−2
p
2m+1
X−1
EXi2
i=2m
2m+1 −1
≤C
X X2
i
2
i=2m
ip
.
Thus
∞
X
2
E|γm | ≤ C
m=0
∞
X
X2
i
i=1
i
< ∞.
2
p
By the Markov inequality and the Borel-Cantelli lemma, we get
(3.1)
lim γm = 0 a.s.
m→∞
On the other hand
(3.2)
−m
p
0≤2
1
ϕ− 2 (2m )
qk
m X
X
(k)
γi
≤2
−m
p
k=0 i=pk
m
X
2
k+1
p
k=0
By (3.1), (3.2) and Lemma 2.4, we get
(3.3)
−m
p
lim 2
m→∞
1
ϕ− 2 (2m )
qk
m X
X
k=0 i=pk
(k)
γi
= 0 a.s.
γk .
218
Nguyen Van Quang and Le Van Thanh
(m)
Assume n ∈ ∆k , we have
n
¯
¯ 1
X
¯
¯ − p − 21
Xi ¯
0 ≤ ¯n ϕ (n)
≤2
(3.4)
−m
p
i=1
qk
m X
X
1
ϕ− 2 (2m )
(k)
γi .
k=0 i=pk
By (3.3) and (3.4), we get
n
− p1
1
ϕ− 2 (n)
n
X
Xi → 0 a.s. (as n → ∞).
i=1
The proof is completed.
In the next theorem, we set up the Marcinkiewicz-Zygmund law of
large numbers for blockwise adapted sequences which are stochastically
dominated by a random variable X.
Theorem 3.2. Let {Fi , i ≥ 1} be a sequence of σ-fields such that for
any fixed k, the sequences {Fi , i ∈ ∆k } are increasing and {Xi , i ≥ 1}
is blockwise adapted to {Fi , i ≥ 1}. If {Xi , i ≥ 1} is stochastically
dominated by a random variable X such that either
E|X| log+ |X| < ∞ if p = 1,
or
E|X|p < ∞ if 1 < p < 2,
then
lim
n→∞
n
X
1
1
1
n p ϕ 2 (n)
where ai = EXi if i =
and m ≥ 0.
(m)
rk
(Xi − ai ) = 0 a.s.,
i=1
1
0
(m)
for k ≥ 1
(m)
and bi =
and ai = E(Xi |Fi−1 ) if i 6= rk
0
Proof. Let Xi = Xi I{|Xi | ≤ i p }, bi = EXi if i = rk
0
(m)
E(Xi |Fi−1 ) if i 6= rk for k ≥ 1 and m ≥ 0. We have
0
0
E(Xi − bi )2 ≤ E|Xi |2
Z i p2
P (|Xi |2 > t)dt
=
0
Marcinkiewicz-Zygmund law of large numbers
Z
219
2
ip
≤C
P (|X|2 > t)dt
0
Z
2
ip
=C
¡
¢
2
2
P (t < |X|2 < i p ) + P (i p ≤ |X|2 ) dt
0
=C
¡
Z
1
ip
¢
2
2
x2 dF (x) + i p P (i p ≤ |X|2 ) ,
0
where F (x) is the distribution function of X.
Z
∞
X
1
i=1
i
2
p
1
ip
x2 dF (x) ≤ C
0
i Z
∞
X
1 X
i=1
≤C
2
p
i
1
k=1
k=1 i=k
≤C
i
p−2
p
k
k=1
≤C
∞ Z
X
k=1
2
p
1
kp
1
(k−1) p
Z
x2 dF (x)
(k−1) p
Z
∞
∞ X
X
1
∞
X
1
kp
x2 dF (x)
1
kp
1
(k−1) p
x2 dF (x)
1
kp
1
(k−1) p
xp dF (x)
≤ CE|X|p < ∞,
and
∞
X
2
p
2
P (i ≤ |X| ) =
i=1
∞
X
P (i ≤ |X|p ) ≤ CE|X|p < ∞.
i=1
Hence
0
∞
X
E(Xi − bi )2
2
i=1
ip
0
< ∞.
(m)
For each k > 1 and m ≥ 0, sequence {Xi − bi , Fi , i ∈ ∆k } is a martingale difference. By using the proof of Theorem 3.1, we get
(3.5)
− p1
n
ϕ
− 12
n
X
0
(n)
(Xi − bi ) → 0 a.s. (as n → ∞).
i=1
220
Nguyen Van Quang and Le Van Thanh
Next,
∞
X
0
P (Xi 6= Xi ) =
i=1
∞
X
1
P (|Xi | > i p )
i=1
∞
X
≤C
1
P (|X| > i p )
i=1
(3.6)
≤C
∞
X
P (|X|p > i) ≤ CE|X|p < ∞.
i=1
Finally, we prove that
n
1 X
lim
(ai − bi ) = 0 a.s.
n→∞ p1
n i=1
(3.7)
In the case p=1,
∞
X
−1
n
∞
£
¤ X
E |Xn |I(|Xn | > n) =
n−1
n=1
n=1
∞
X
≤C
=C
≤C
n=1
∞
X
n=1
∞
X
i=1
Z
∞
n
Z
n
¡
¢
P |Xn | > x dx
−1
∞
P (|X| > x)dx
n
n
−1
∞ Z
X
i=n
P (|X| > x)dx
i<x≤i+1
P (|X| > i)
i
X
n−1
n=1
∞
X
≤C
(1 + log i)P (|X| > i) < ∞.
i=1
P∞ −1
This implies that
n=1 n (an − bn ) < ∞ a.s. By using Kronecker’s
lemma, we get (3.7).
In the case 1 < p < 2, since
∞
X
n=1
n
− p1
£
1 ¤
E |Xn |I(|Xn | > n p )
Marcinkiewicz-Zygmund law of large numbers
≤C
∞
X
n
− p1
Z
=C
n
− p1
n=1
≤C
∞
1
∞ Z
X
n
− p1
≤C
i
∞ Z
X
∞
X
n
Z
1
(i+1) p
xdF (x)
1
1
(i+1) p
1
xdF (x)
1
(i+1) p
1
i=1
It follows that
Z
ip
i=1
≤C
xdF (x)
ip
ip
i=1 n=1
p−1
p
1
(i+1) p
1
i=n
∞ X
i
X
∞
X
xdF (x)
np
n=1
∞
X
221
xp dF (x) < ∞.
ip
− p1
(an − bn ) < ∞ a.s..
n=1
By Kronecker’s lemma, we get (3.7).
Combining (3.5), (3.6) and (3.7) we obtain
n
X
1
lim 1 1
(Xi − ai ) = 0 a.s.
n→∞ p 2
n ϕ (n) i=1
This completes the proof of theorem.
The following corollaries extend the classical Marcinkiewicz-Zygmund
strong law of large numbers.
Corollary 3.3. Let {Xi , i ≥ 1} be a sequence of blockwise independent random variables with respect to blocks ∆k . If {Xi , i ≥ 1} is
stochastically dominated by a random variable X, E|X|p < ∞ (1 ≤ p <
2), then
n
X
1
(Xi − EXi ) = 0 a.s.
lim 1 1
n→∞ p 2
n ϕ (n) i=1
Proof. Let Fi = σ(Xr(m) , . . . , Xi ) (the σ-field generated by Xr(m) , . . . ,
(m)
k
k
Xi ) if i ∈ ∆k . Then {Xi , i ≥ 1} is blockwise adapted to {Fi , i ≥ 1}.
(m)
From the independence of sequence {Xi , i ∈ ∆k } we get for all k and
m
(m)
E(Xi |Fi−1 ) = EXi if i 6= rk .
222
Nguyen Van Quang and Le Van Thanh
By the proof of Theorem 3.2, we only need prove for the case p = 1.
In the case p = 1, also using the proof of Theorem 3.2, we get
n
X
1
0
(3.8)
(Xi − EXi ) → 0 (as n → ∞),
n−1 ϕ− 2 (n)
i=1
0
where Xi = Xi I(|Xi | ≤ i). On the other hand
Z ∞
E[|Xi |I(|Xi | > i)] =
P (|Xi | > x)dx
i
Z ∞
≤C
P (|X| > x)dx → 0 as i → ∞.
i
Thus
(3.9)
|n−1
n
n
X
X
0
(EXi − EXi )| ≤ n−1
E[|Xi |I(|Xi | > i)] → 0 as n → ∞.
i=1
i=1
Combining (3.8) and (3.9) we obtain
n
X
1
(Xi − EXi ) = 0 a.s.
lim
n→∞ nϕ 12 (n)
i=1
Corollary 3.4. If ω(k) = 2k (or ω(k) = [q k ], q > 1) and {Xi , i ≥ 1}
is ∆k -independent, P {|Xi | ≥ t} ≤ CP {|X| ≥ t} for all nonnegative real
numbers t, E|X|p < ∞, (1 ≤ p < 2), then
n
1 X
lim 1
(Xi − EXi ) = 0 a.s.
n→∞ p
n i=1
Proof. Really, in that case ϕ(i) = O(1), so, from Corollary 3.3, we
obtain
n
1 X
lim 1
(Xi − EXi ) = 0 a.s.
n→∞ p
n i=1
References
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pairwise independent random variables, Bull. Korean Math. Soc. 22 (1985), no.
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Marcinkiewicz-Zygmund law of large numbers
223
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, Series of block-orthogonal and block-independent systems, Izv. Vyssh.
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Nguyen Van Quang, Department of Mathematics, University of Vinh, 182
Le Duan, Vinh, Nghean, Vietnam
E-mail : [email protected]
Le Van Thanh, Department of Mathematics, University of Vinh, 182 Le
Duan, Vinh, Nghean, Vietnam
E-mail : [email protected]