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Moment Inequality for
-Mixing Sequences and Its Applications
Journal of Inequalities and Applications volume 2009, Article number: 379743 (2009)
Abstract
Firstly, the maximal inequality for -mixing sequences is given. By using the maximal inequality, we study the convergence properties for
-mixing sequences. The Hájek-Rényi-type inequality, strong law of large numbers, strong growth rate and the integrability of supremum for
-mixing sequences are obtained.
1. Introduction
Let be a random variable sequence defined on a fixed probability space
and
for each
. Let
and
be positive integers. Write
. Given
-algebras
,
in
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ1_HTML.gif)
Define the -mixing coefficients by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ2_HTML.gif)
Definition 1.1.
A random variable sequence is said to be a
-mixing random variable sequence if
as
.
The concept of -mixing random variables was introduced by Dobrushin [1] and many applications have been found. See, for example, Dobrushin [1], Utev [2], and Chen [3] for central limit theorem, Herrndorf [4] and Peligrad [5] for weak invariance principle, Sen [6, 7] for weak convergence of empirical processes, Iosifescu [8] for limit theorem, Peligrad [9] for Ibragimov-Iosifescu conjecture, Shao [10] for almost sure invariance principles, Hu and Wang [11] for large deviations, and so forth. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired. The main purpose of this paper is to study the maximal inequality for
-mixing sequences, by which we can get the Hájek-Rényi-type inequality, strong law of large numbers, strong growth rate, and the integrability of supremum for
-mixing sequences.
Throughout the paper, denotes a positive constant which may be different in various places. The main results of this paper depend on the following lemmas.
Lemma 1.2 (see Lu and Lin [12]).
Let be a sequence of
-mixing random variables. Let
,
,
,
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ3_HTML.gif)
Lemma 1.3 (see Shao [10]).
Let be a
-mixing sequence. Put
. Suppose that there exists an array
of positive numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ4_HTML.gif)
Then for every , there exists a constant
depending only on
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ5_HTML.gif)
for every and
.
Lemma 1.4 (see Hu et al. [13]).
Let be a nondecreasing unbounded sequence of positive numbers and let
be nonnegative numbers. Let
and
be fixed positive numbers. Assume that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ7_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ8_HTML.gif)
and with the growth rate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ9_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ10_HTML.gif)
If further we assume that for infinitely many
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ11_HTML.gif)
Lemma 1.5 (see Fazekas and Klesov [14] and Hu [15]).
Let be a nondecreasing unbounded sequence of positive numbers and let
be nonnegative numbers. Denote
for
. Let
be a fixed positive number satisfying (1.6). If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ13_HTML.gif)
then (1.8)–(1.11) hold.
2. Maximal Inequality for
-Mixing Sequences
Theorem 2.1.
Let be a sequence of
-mixing random variables satisfying
. Assume that
and
for each
. Then there exists a constant
depending only on
such that for any
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ15_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ17_HTML.gif)
where may be different in various places.
Proof.
By Lemma 1.2 for , we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ18_HTML.gif)
which implies (2.1). By (2.1) and Lemma 1.3 (take ), we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ19_HTML.gif)
The proof is completed.
3. Hájek-Rényi-Type Inequality for
-Mixing Sequences
In this section, we will give the Hájek-Rényi-type inequality for -mixing sequences, which can be applied to obtain the strong law of large numbers and the integrability of supremum.
Theorem 3.1.
Let be a sequence of
-mixing random variables satisfying
and let
be a nondecreasing sequence of positive numbers. Then for any
and any integer
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ20_HTML.gif)
where is defined in (2.4) in Theorem 2.1.
Proof.
Without loss of generality, we assume that for all
. Let
. For
, define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ21_HTML.gif)
For , we let
and
be the index of the last nonempty set
. Obviously,
if
and
. It is easily seen that
if
and
is also a sequence of
-mixing random variables. By Markov's inequality and (2.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ22_HTML.gif)
Now we estimate . Let
. Then
follows from the definition of
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ23_HTML.gif)
Thus, (3.1) follows from (3.3) and (3.4) immediately.
Theorem 3.2.
Let be a sequence of
-mixing random variables satisfying
and let
be a nondecreasing sequence of positive numbers. Then for any
and any positive integers
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ24_HTML.gif)
where is defined in (3.1).
Proof.
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ25_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ26_HTML.gif)
For , by Markov's inequality and (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ27_HTML.gif)
For , we will apply Theorem 3.1 to
and
. Noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ28_HTML.gif)
thus, by Theorem 3.1, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ29_HTML.gif)
Therefore, the desired result (3.5) follows from (3.7)–(3.10) immediately.
Theorem 3.3.
Let be a sequence of
-mixing random variables satisfying
and let
be a nondecreasing sequence of positive numbers. Denote
for
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ30_HTML.gif)
then for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ31_HTML.gif)
where is defined in (3.1). Furthermore, if
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ32_HTML.gif)
Proof.
By the continuity of probability and Theorem 3.1, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ33_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ34_HTML.gif)
By Theorem 3.2, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ35_HTML.gif)
Hence, by (3.11) and Kronecker's Lemma, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ36_HTML.gif)
which is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ37_HTML.gif)
So the desired results are proved.
4. Strong Law of Large Numbers and Growth Rate for
-Mixing Sequences
Theorem 4.1.
Let be a sequence of mean zero
-mixing random variables satisfying
and let
be a nondecreasing unbounded sequence of positive numbers. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ38_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ39_HTML.gif)
and with the growth rate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ40_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ41_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ42_HTML.gif)
If further we assume that for infinitely many
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ43_HTML.gif)
Proof.
By (2.4) in Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ44_HTML.gif)
It follows by (4.1) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ45_HTML.gif)
Thus, (4.2)–(4.6) follow from (4.7), (4.8), and Lemma 1.4 immediately. We complete the proof of the theorem.
Theorem 4.2.
Let be a sequence of
-mixing random variables with
.
. Denote
for
and
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ46_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ47_HTML.gif)
and with the growth rate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ48_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ51_HTML.gif)
If further we assume that for infinitely many
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ52_HTML.gif)
In addition, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ53_HTML.gif)
Proof.
Assume that ,
and
,
. By (2.4) in Theorem 2.1, we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ54_HTML.gif)
It is a simple fact that for all
. It follows by (4.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ55_HTML.gif)
That is to say (1.12) holds. By Remark in Fazekas and Klesov [14], (1.12) implies (1.13). By Lemma 1.5, we can obtain (4.10)–(4.15) immetiately. By (4.14), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F379743/MediaObjects/13660_2009_Article_1944_Equ56_HTML.gif)
The proof is completed.
Remark 4.3.
By using the maximal inequality, we get the Hájek-Rényi-type inequality, the strong law of large numbers and the strong growth rate for -mixing sequences. In addition, we get some new bounds of probability inequalities for
-mixing sequences, such as (3.1), (3.5), (3.12), (4.5)–(4.6), and (4.13)–(4.16).
References
Dobrushin RL: The central limit theorem for non-stationary Markov chain. Theory of Probability and Its Applications 1956,1(4):72–88.
Utev SA: On the central limit theorem for -mixing arrays of random variables. Theory of Probability and Its Applications 1990,35(1):131–139. 10.1137/1135013
Chen DC: A uniform central limit theorem for nonuniform -mixing random fields. The Annals of Probability 1991,19(2):636–649. 10.1214/aop/1176990445
Herrndorf N: The invariance principle for -mixing sequences. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1983,63(1):97–108. 10.1007/BF00534180
Peligrad M: An invariance principle for -mixing sequences. The Annals of Probability 1985,13(4):1304–1313. 10.1214/aop/1176992814
Sen PK: A note on weak convergence of empirical processes for sequences of -mixing random variables. Annals of Mathematical Statistics 1971,42(6):2131–2133. 10.1214/aoms/1177693079
Sen PK: Weak convergence of multidimensional empirical processes for stationary -mixing processes. The Annals of Probability 1974,2(1):147–154. 10.1214/aop/1176996760
Iosifescu J: Limit theorem for -mixing sequences. Proceedings of the 5th Conference on Probability Theory, September 1977 1–6.
Peligrad M: On Ibragimov-Iosifescu conjecture for -mixing sequences. Stochastic Processes and Their Applications 1990,35(2):293–308. 10.1016/0304-4149(90)90008-G
Shao QM: Almost sure invariance principles for mixing sequences of random variables. Stochastic Processes and Their Applications 1993,48(2):319–334. 10.1016/0304-4149(93)90051-5
Hu S, Wang X: Large deviations for some dependent sequences. Acta Mathematica Scientia. Series B 2008,28(2):295–300. 10.1016/S0252-9602(08)60030-2
Lu CR, Lin ZY: Limit Theory for Mixing Dependent Sequences. Science Press, Beijing, China; 1997.
Hu S, Chen G, Wang X: On extending the Brunk-Prokhorov strong law of large numbers for martingale differences. Statistics & Probability Letters 2008,78(18):3187–3194. 10.1016/j.spl.2008.06.017
Fazekas I, Klesov O: A general approach to the strong law of large numbers. Theory of Probability and Its Applications 2001,45(3):436–449. 10.1137/S0040585X97978385
Hu S: Some new results for the strong law of large numbers. Acta Mathematica Sinica 2003,46(6):1123–1134.
Acknowledgments
The authors are most grateful to the Editor Sever Silvestru Dragomir, the referee Professor Mihaly Bencze and an anonymous referee for careful reading of the manuscript and valuable suggestions and comments which helped to significantly improve an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 10871001, 60803059) and the Innovation Group Foundation of Anhui University.
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Xuejun, W., Shuhe, H., Yan, S. et al. Moment Inequality for -Mixing Sequences and Its Applications.
J Inequal Appl 2009, 379743 (2009). https://doi.org/10.1155/2009/379743
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DOI: https://doi.org/10.1155/2009/379743