- Research Article
- Open access
- Published:
Doob's Type Inequality and Strong Law of Large Numbers for Demimartingales
Journal of Inequalities and Applications volume 2010, Article number: 838301 (2010)
Abstract
We establish some maximal inequalities for demimartingales which generalize the result of Wang (2004). The maximal inequality for demimartingales is used as a key inequality to establish other results including Doob's type maximal inequality, strong law of large numbers, strong growth rate, and integrability of supremum for demimartingales, which generalize and improve partial results of Christofides (2000) and Prakasa Rao (2007).
1. Introduction
Definition 1.1.
Let be an
sequence of random variables. Assume that for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ1_HTML.gif)
for all coordinatewise nondecreasing functions such that the expectation is defined. Then
is called a demimartingale. If in addition the function
is assumed to be nonnegative, then the sequence
is called a demisubmartingale.
Definition 1.2.
A finite collection of random variables is said to be associated if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ2_HTML.gif)
for any two coordinatewise nondecreasing functions on
such that the covariance is defined. An infinite sequence
is associated if every finite subcollection is associated.
Definition 1.3.
A finite collection of random variables is said to be strongly positive dependent if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ3_HTML.gif)
for all Borel measurable and increasing (or decreasing) set pairs (A set
is said increasing (or decreasing) if
implies
for any
), where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ4_HTML.gif)
An infinite sequence is strongly positive dependent if every finite subcollection is strongly positive dependent.
Remark 1.4.
Chow [1] proved a maximal inequality for submartingales. Newman and Wright [2] extended Doob's maximal inequality and upcrossing inequality to the case of demimartingales, and pointed out that the partial sum of a sequence of mean zero associated random variables is a demimartingale. Christofides [3] showed that the Chow's maximal inequality for (sub)martingales can be extended to the case of demi(sub)martingales. Wang [4] obtained Doob's type inequality for more general demimartingales. Hu et al. [5] gave a strong law of large numbers and growth rate for demimartingales. Prakasa Rao [6] established some maximal inequalities for demisubmartingales and N-demisuper-martingales.
It is easily seen that the partial sum of a sequence of mean zero strongly positive dependent random variables is also a demimartingale by the inequality (3) in Zheng [7], that is, for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ5_HTML.gif)
for all coordinatewise nondecreasing functions such that the expectation is defined. Therefore, the main results of this paper hold for the partial sums of sequences of mean zero associated random variables and strongly positive dependent random variables.
Let and
be sequences of random variables defined on a fixed probability space
and
the indicator function of the event
. Denote
,
,
,
,
. The main results of this paper depend on the following lemmas.
Lemma 1.5 (see Wang [4, Theorem ]).
Let be a demimartingale and
a nonnegative convex function on
with
and
. Let
be a nonincreasing sequence of positive numbers. Then for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ6_HTML.gif)
Lemma 1.6 (see Fazekas and Klesov [8, Theorem ] and Hu et al. [5, Lemma
]).
Let be a random variable sequence and
for
. Let
be a nondecreasing unbounded sequence of positive numbers and
nonnegative numbers. Let
and
be fixed positive numbers. Assume that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ7_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ8_HTML.gif)
and with the growth rate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ9_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ10_HTML.gif)
In addition,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ11_HTML.gif)
If further assumes one that for infinitely many
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ12_HTML.gif)
Lemma 1.7 (see Christofides [3, Lemma , Corollary
]).
-
(i)
If
is a demisubmartingale (or a demimartingale) and
is a nondecreasing convex function such that
, then
is a demisubmartingale.
-
(ii)
If
is a demimartingale, then
is a demisubmartingale and
is a demisubmartingale.
Lemma 1.8 (see Hu et al. [9, Theorem ]).
Let be a demimartingale and
be a nonincreasing sequence of positive numbers. Let
and
for each
, then for any
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ13_HTML.gif)
Lemma 1.9 (see Christofides [3, Corollary , Theorem
]).
-
(i)
Let
be a demisubmartingale. Then for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ14_HTML.gif)
-
(ii)
Let
be a demisubmartingale and
a nonincreasing sequence of positive numbers. Then for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ15_HTML.gif)
Using Lemma 1.5, Wang [4] obtained the following inequalities for demimartingales.
Theorem 1.10 (see Wang [4, Corollary ]).
Let be a demimartingale and
a nonincreasing sequence of positive numbers. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ17_HTML.gif)
We point out that there is a mistake in the proof of (1.17), that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ18_HTML.gif)
should be replaced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ19_HTML.gif)
In fact, by Lemma 1.5 and Fubini Theorem, we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ20_HTML.gif)
The rest of the proof is similar to Corollary in Wang [4].
The same problem exists in Shiryaev [10, page 495, in the proof of Theorem ] and Krishna and Soumendra [11, page 414, in the proof of Theorem
]. For example, the following inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ21_HTML.gif)
in Shiryaev [10, page 495] should be revised as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ22_HTML.gif)
2. Main Results and Their Proofs
Theorem 2.1.
Let be a demimartingale and
a nonnegative convex function on
with
. Let
be a nonincreasing sequence of positive numbers.
. Suppose that
for each
, then for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ24_HTML.gif)
Proof.
By Lemma 1.5 and Hölder's inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ25_HTML.gif)
where is a real number and satisfies
Since
for each
, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ26_HTML.gif)
therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ27_HTML.gif)
Similar to the proof of (2.3) and using Lemma 1.5 again, we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ28_HTML.gif)
For constants and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ29_HTML.gif)
Combining (2.6) and (2.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ30_HTML.gif)
Thus, (2.2) follows from (2.8) immediately. The proof is complete.
Remark 2.2.
If we take in Theorem 2.1, then Theorem 2.1 implies Corollary 2.1 in Wang [4].
Corollary 2.3.
Let the conditions of Theorem 2.1 be satisfied with for each
. Then for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ31_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ32_HTML.gif)
Corollary 2.4 (Doob's type maximal inequality for demimartingales).
Let and
be a demimartingale. Suppose that
for each
, then for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ33_HTML.gif)
Theorem 2.5.
Let be a demimartingale and
a nonnegative convex function on
with
. Let
be a nondecreasing unbounded sequence of positive numbers.
. Suppose that
for each
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ34_HTML.gif)
then , and (1.9)-(1.10) hold (
is replaced by
), where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ35_HTML.gif)
In addition,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ36_HTML.gif)
If further one assumes that for infinitely many
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ37_HTML.gif)
Proof.
By the condition of the theorem, we can see that for all
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ38_HTML.gif)
follows from (2.9) for each . By (2.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ39_HTML.gif)
Therefore, follows from Lemma 1.6, (2.16), and (2.17); (1.9), (1.10), (2.14), (2.15) hold. This completes the proof of the theorem.
In Theorem 2.5, if we assume that is a nonnegative and nondecreasing convex function on
with
, then the condition "
for each
" is satisfied.
Remark 2.6.
Theorem 2.5 generalizes and improves the results of Theorem in Christofides [3] and Theorem
in Prakasa Rao [6].
Theorem 2.7.
Let and
a demimartingale with
for each
. Let
be a nondecreasing sequence of positive numbers. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ40_HTML.gif)
then for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ41_HTML.gif)
Proof.
Taking ,
and
in Lemma 1.8, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ42_HTML.gif)
Thus, by (2.20) and (2.18), we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ43_HTML.gif)
Theorem 2.8.
Let be a demisubmartingale and
a nondecreasing and nonnegative convex function on
with
and
. Let
be a nonincreasing sequence of positive numbers. Then for all
and each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ44_HTML.gif)
Proof.
By Fubini theorem, it is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ45_HTML.gif)
It follows from Lemma 1.7(i) and Lemma 1.9(ii) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ46_HTML.gif)
Therefore, (2.22) follows from the above statements immediately.
Corollary 2.9.
Let the conditions of Theorem 2.8 be satisfied with for each
. Then for all
and each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ47_HTML.gif)
By Corollary 2.9, we can get the following theorem.
Theorem 2.10.
Let be a demisubmartingale and
a nondecreasing and nonnegative convex function on
with
and
. Let
be a nondecreasing unbounded sequence of positive numbers. If there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ48_HTML.gif)
then a.s., and (1.9)-(1.10) hold (
is replaced by
), where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ49_HTML.gif)
In addition,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ50_HTML.gif)
If further one assumes that for infinitely many
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838301/MediaObjects/13660_2009_Article_2268_Equ51_HTML.gif)
Similar to the proof of Theorem 2.8 and using Lemma 1.9(i), we can get the following.
Theorem 2.11.
Let be a nonnegative demisubmartingale. Then for all
,
.
References
Chow YS: A martingale inequality and the law of large numbers. Proceedings of the American Mathematical Society 1960, 11: 107–111. 10.1090/S0002-9939-1960-0112190-3
Newman CM, Wright AL: Associated random variables and martingale inequalities. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1982, 59(3):361–371. 10.1007/BF00532227
Christofides TC: Maximal inequalities for demimartingales and a strong law of large numbers. Statistics & Probability Letters 2000, 50(4):357–363. 10.1016/S0167-7152(00)00116-4
Wang JF: Maximal inequalities for associated random variables and demimartingales. Statistics & Probability Letters 2004, 66(3):347–354. 10.1016/j.spl.2003.10.021
Hu SH, Chen GJ, Wang XJ: 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
Prakasa Rao BLS: On some maximal inequalities for demisubmartingales and -demisuper martingales. Journal of Inequalities in Pure and Applied Mathematics 2007, 8(4, article 112):1–17.
Zheng YH: Inequalities of moment and convergence theorem of order statistics of partial sums for a class of strongly positive dependent stochastic sequence. Acta Mathematicae Applicatae Sinica 2001, 24(2):168–176.
Fazekas I, Klesov O: A general approach rate to the strong law of large numbers. Statistics & Probability Letters 2001, 45(3):436–449.
Hu SH, Wang XJ, Yang WZ, Zhao T: The Hà jek-Rènyi-type inequality for associated random variables. Statistics & Probability Letters 2009, 79(7):884–888. 10.1016/j.spl.2008.11.014
Shiryaev AN: Probability, Graduate Texts in Mathematics. Volume 95. 2nd edition. Springer, New York, NY, USA; 1996:xvi+623.
Athreya KB, Lahiri SN: Measure Theory and Probability Theory, Springer Texts in Statistics. Springer, New York, NY, USA; 2006:xviii+618.
Acknowledgments
The authors are most grateful to the Editor Andrei Volodin and an anonymous referee for the careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 10871001, 60803059), Talents Youth Fund of Anhui Province Universities (Grant no. 2010SQRL016ZD), Youth Science Research Fund of Anhui University (Grant no. 2009QN011A), Provincial Natural Science Research Project of Anhui Colleges and the Innovation Group Foundation of Anhui University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Xuejun, W., Shuhe, H., Ting, Z. et al. Doob's Type Inequality and Strong Law of Large Numbers for Demimartingales. J Inequal Appl 2010, 838301 (2010). https://doi.org/10.1155/2010/838301
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/838301