Moment convergence rates in the law of iterated logarithm for moving average process under dependence
Zhang and Wu Journal of Inequalities and Applications (2016) 2016:253
DOI 10.1186/s13660-016-1190-1
RESEARCH
Open Access
Moment convergence rates in the law of
iterated logarithm for moving average
process under dependence
Yayun Zhang and Qunying Wu*
*
Correspondence:
College of Science, Guilin University
of Technology, Guilin, 541006,
P.R. China
Abstract
We assume that Xk = +∞
i=–∞ ai ξi+k is a moving average process and {ξi , –∞ < i < +∞}
is a doubly infinite sequence of identically distributed and dependent random
variables with zero mean and finite variance and {ai , –∞ < i < +∞} is an absolutely
summable sequence of real numbers. Under suitable conditions of dependence, we
get the precise rates in the law of iterated logarithm for the first moment of the partial
sums of the moving average process.
MSC: 60F15
Keywords: precise asymptotics; the law of iterated logarithm; moment
convergence; moving average; ϕ -mixing
1 Introduction
Suppose that {ξi , –∞ < i < +∞} is a doubly infinite sequence of identically distributed random variables with zero mean and finite variance and {ai , –∞ < i < +∞} is an absolutely
summable sequence of real numbers and
Xk =
+∞
ai ξi+k ,
k ≥ ,
()
i=–∞
is a moving average process based on {ξi , –∞ < i < +∞}. Set Sn = nk= Xk , n ≥ .
Under the assumption that {ξi , –∞ < i < +∞} is a sequence of independent identically
distributed (i.i.d.) random variables, many limiting results have been obtained for the moving average process {Xk , k ≥ }. Burton and Dehling [] got a large deviation principle; Yang
[] established the central limit theorem and the law of the iterated logarithm; Li et al. []
and Zhang [] obtained the complete convergence; and complete moment convergence
was proved by Li [] and Li and Zhang [].
In this article we will discuss the case of ϕ-mixing. Suppose that {ξi , –∞ < i < +∞} is a
sequence of identically distributed and ϕ-mixing random variables with
k
∞
ϕ(m) := sup P(B|A) – P(B), A ∈ F–∞
→ ,
, P(A) = , B ∈ Fk+m
m → ∞,
k≥
© 2016 Zhang and Wu. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Zhang and Wu Journal of Inequalities and Applications (2016) 2016:253
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where Fab = σ (ξi , a ≤ i ≤ b). By definition, we know that {Xk , k ≥ } is ϕ-mixing if {ξi , –∞ <
i < +∞} is a sequence of ϕ-mixing distributed random variables.
Since Hsu and Robbins [] introduced complete convergence, there have been extensions in several directions. Some authors studied the precise asymptotics of complete convergence on moving average. For example, Li and Zhang [] got precise asymptotics in the
law of the iterated logarithm of moving average and Xiao and Yin [] got moment convergence rates in the law of logarithm for moving average process under dependence, etc.
Xiao and Yin [] studied precise asymptotics in the law of iterated logarithm for the
first moment convergence of i.i.d. random variables. They got the following result.
Theorem . Let {Y , Yn , n ≥ } be a sequence of i.i.d.random variables and set Un =
n
i= Yi . N is the standard normal random variable and EY = , EY = σ , then, for d >
and β > ,
lim εβ/d
ε
∞
√
dσ E|N |β/d+
(log log n)β–
d/
E
|U
.
|
–
εσ
n(log
log
n)
=
n
+
n/ log n
β(β + d)
n=
()
Inspired by Xiao and Yin [], we extend this result to moving average processes under
ϕ-mixing random variable. Throughout this article, log x := ln(x ∨ e), and the symbol c
denotes a positive constant which may be different in various places, and [x] denotes the
largest integer which is not greater than x, and N is the standard normal random variable.
Now, we state the main result of this article.
Theorem . Suppose {Xi , i ≥ } is defined as in (), where {ai , –∞ < i < +∞} is a sequence
of real numbers with +∞
i=–∞ |ai | < ∞, and {ξi , –∞ < i < +∞} is a sequence of identically
distributed ϕ-mixing random variables with zero mean and finite variance and < σ :=
∞ / m
+∞
Eξ + ∞
k= Eξ ξk < ∞,
m= ϕ ( ) < ∞, τ := σ |
i=–∞ ai |. Then, for d > and β > ,
lim εβ/d
ε
∞
√
dτ E|N |β/d+
(log log n)β–
d/
|
–
ετ
=
E
|S
.
n(log
log
n)
n
+
n/ log n
β(β + d)
n=
()
Remark . Let ai = for i = and ai = for i = , that is to say, Xk = ξk with Eξ =
n
, Eξ < ∞ and < σ := Eξ + ∞
k= Eξ ξk < ∞, then, for Sn =
k= ξk , () still holds for
τ = σ when {Xk ; k ≥ } is a sequence of identically distributed ϕ-mixing random variables.
Therefore, this result extends Theorem ..
2 Lemmas
To prove our main result, we need the following lemmas.
Lemma . (Burton and Dehling []) Let +∞
i=–∞ ai be an absolutely convergent series of
a
and
k
≥
,
then
real numbers with a = +∞
i=–∞ i
k
+∞ i+n
aj = |a|k .
lim
n→∞ n
i=–∞ j=i+
Zhang and Wu Journal of Inequalities and Applications (2016) 2016:253
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Lemma . (Shao []) Let {Xi , i ≥ } be a sequence of ϕ-mixing random variables with
zero means and finite second moments. Set Sn = ni= Xi . Then
ESn
[log n]
≤ ,n exp
ϕ / i
i=
max EXi .
≤i≤n
()
If there is some Cn such that max ESi ≤ Cn , then, for all q ≥ , there exists a C = C(q, ϕ(·))
≤i≤n
such that
E max |Si |q ≤ C Cnq/ + E max |Xi |q .
≤i≤n
≤i≤n
()
Lemma . (Lin and Zhou []) Let {Xi , i ≥ } be defined as in (), and {ξi , –∞ < i < +∞}
be a sequence of identically distributed ϕ-mixing random variables with zero means and
∞ / m
finite variances and < σ = Eξ + ∞
k= Eξ ξk < ∞,
m= ϕ ( ) < ∞, then
Sn D
√ −→ N (, ),
τ n
+∞
τ = σ
ai ,
()
i=–∞
D
where −→ denotes convergence in distribution.
3 Proof of main result
In this section, for < ε < and M > , we set
b(ε) = exp exp Mε–/d .
()
Without loss of generality, we assume that τ = .
Theorem . will be proved if we show the following propositions.
Proposition . Under the conditions of Theorem ., we have
lim εβ/d
ε
∞
dE|N |β/d+
(log log n)β–
E |N | – ε(log log n)d/ + =
,
n log n
β(β + d)
n=
lim εβ/d
M→∞
(log log n)β–
E |N | – ε(log log n)d/ + = ,
n log n
()
()
n>b(ε)
where () uniformly holds true with respect to < ε < .
Proof See the proof of Proposition and Proposition in Xiao and Yin [].
Proposition . Under the conditions of Theorem ., we have
lim εβ/d
ε
(log log n)β–
E |N | – ε(log log n)d/
+
n log n
n≤b(ε)
√
– E |Sn |/ n – ε(log log n)d/ + = .
()
Zhang and Wu Journal of Inequalities and Applications (2016) 2016:253
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√
Proof Let n = supx∈R |P(|N | ≥ x) – P(|Sn |/ n ≥ x)|. By Lemma ., then we have (...truncated)
