Doob's Type Inequality and Strong Law of Large Numbers for Demimartingales

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).

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Recommended by Andrei Volodin

School of Mathematical Science, Anhui University, Hefei 230039, China

Received 9 October 2009; Revised 14 December 2009; Accepted 28 January 2010

1. Introduction

Definition 1.1.

Let _S1 ,_S2 ,... be an ^L1 sequence of random variables. Assume that for j=1,2,... [figure omitted; refer to PDF] for all coordinatewise nondecreasing functions f such that the expectation is defined. Then {_Sj ,j≥1} is called a demimartingale. If in addition the function f is assumed to be nonnegative, then the sequence {_Sj ,j≥1} is called a demisubmartingale.

Definition 1.2.

A finite collection of random variables _X1 ,_X2 ,...,_Xm is said to be associated if [figure omitted; refer to PDF] for any two coordinatewise nondecreasing functions f,g on ^...m such that the covariance is defined. An infinite sequence {_Xn ,n≥1} is associated if every finite subcollection is associated.

Definition 1.3.

A finite collection of random variables _X1 ,_X2 ,...,_Xn is said to be strongly positive dependent if [figure omitted; refer to PDF] for all Borel measurable and increasing (or decreasing) set pairs (_Λ1 ,_Λ2 )⊂_R1 ×_R2 (A set Λ is said increasing (or decreasing) if x≤(or≥)y implies y∈Λ for any x∈Λ ), where [figure omitted; refer to PDF] An infinite sequence {_Xn ,n≥1} 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 n≥1 ,

[figure omitted; refer to PDF] for all coordinatewise nondecreasing functions f 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 {_Xn ,n≥1} and {_Sn ,n≥1} be sequences of random variables defined on a fixed probability space (Ω,...,P) and I(A) the indicator function of the event A . Denote _S0 =0 , ^X+ =max (0,X) , ^X- =max (0,-X) , log x=_{log e} x=ln x , ^{log +} x=ln (max (x,1)) . The main results of this paper depend on the following lemmas.

Lemma 1.5 (see Wang [4, Theorem 2.1 ]).

Let {_Sn ,n≥1} be a demimartingale and g a nonnegative convex function on ... with g(0)=0 and g(_Si )∈^L1 ,i≥1 . Let {_ck ,k≥1} be a nonincreasing sequence of positive numbers. Then for any [varepsilon]>0 , [figure omitted; refer to PDF]

Lemma 1.6 (see Fazekas and Klesov [8, Theorem 2.1 ] and Hu et al. [5, Lemma 1.5 ]).

Let {_Xn ,n≥1} be a random variable sequence and _Sn =^∑i=1n_Xi for n≥1 . Let _b1 ,_b2 ,... be a nondecreasing unbounded sequence of positive numbers and _α1 ,_α2 ,... nonnegative numbers. Let p and C be fixed positive numbers. Assume that for each n≥1 , [figure omitted; refer to PDF] then [figure omitted; refer to PDF] and with the growth rate [figure omitted; refer to PDF] where [figure omitted; refer to PDF] In addition, [figure omitted; refer to PDF] If further assumes one that _αn >0 for infinitely many n , then [figure omitted; refer to PDF]

Lemma 1.7 (see Christofides [3, Lemma 2.1 , Corollary 2.1 ]).

(i) If {_Sn ,n≥1} is a demisubmartingale (or a demimartingale) and g is a nondecreasing convex function such that g(_Si )∈^L1 ,i≥1 , then {g(_Sn ),n≥1} is a demisubmartingale.

(ii) If {_Sn ,n≥1} is a demimartingale, then {^Sn+ ,n≥1} is a demisubmartingale and {^Sn- ,n≥1} is a demisubmartingale.

Lemma 1.8 (see Hu et al. [9, Theorem 2.1 ]).

Let {_Sn ,n≥1} be a demimartingale and {_ck ,k≥1} be a nonincreasing sequence of positive numbers. Let ν≥1 and E|_Sk^|ν <∞ for each k , then for any [varepsilon]>0 and 1≤n≤N , [figure omitted; refer to PDF]

Lemma 1.9 (see Christofides [3, Corollary 2.4 , Theorem 2.1 ]).

(i) Let {_Sn ,n≥1} be a demisubmartingale. Then for any [varepsilon]>0 ,

[figure omitted; refer to PDF]

(ii) Let {_Sn ,n≥1} be a demisubmartingale and {_ck ,k≥1} a nonincreasing sequence of positive numbers. Then for any [varepsilon]>0 ,

[figure omitted; refer to PDF]

Using Lemma 1.5, Wang [4] obtained the following inequalities for demimartingales.

Theorem 1.10 (see Wang [4, Corollary 2.1 ]).

Let {_Sn ,n≥1} be a demimartingale and {_ck ,k≥1} a nonincreasing sequence of positive numbers. Then [figure omitted; refer to PDF] [figure omitted; refer to PDF] We point out that there is a mistake in the proof of (1.17), that is,

[figure omitted; refer to PDF] should be replaced by

[figure omitted; refer to PDF] In fact, by Lemma 1.5 and Fubini Theorem, we can see that

[figure omitted; refer to PDF] The rest of the proof is similar to Corollary 2.1 in Wang [4].

The same problem exists in Shiryaev [10, page 495, in the proof of Theorem 2 ] and Krishna and Soumendra [11, page 414, in the proof of Theorem 13.2.13 ]. For example, the following inequality

[figure omitted; refer to PDF] in Shiryaev [10, page 495] should be revised as

[figure omitted; refer to PDF]

2. Main Results and Their Proofs

Theorem 2.1.

Let {_Sn ,n≥1} be a demimartingale and g a nonnegative convex function on ... with g(0)=0 . Let {_ck ,k≥1} be a nonincreasing sequence of positive numbers. p>1 . Suppose that E(g(_Sk )^)p <∞ for each k≥1 , then for every n≥1 , [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Proof.

By Lemma 1.5 and Hölder's inequality, we have [figure omitted; refer to PDF] where q is a real number and satisfies 1/p+1/q=1. Since E(g(_Sk )^)p <∞ for each k≥1 , we can obtain [figure omitted; refer to PDF] therefore, [figure omitted; refer to PDF] Similar to the proof of (2.3) and using Lemma 1.5 again, we can see that [figure omitted; refer to PDF] For constants a≥0 and b>0 , it follows that [figure omitted; refer to PDF] Combining (2.6) and (2.7), we have [figure omitted; refer to PDF] Thus, (2.2) follows from (2.8) immediately. The proof is complete.

Remark 2.2.

If we take g(x)=|x| 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 _ck ≡1 for each k≥1 . Then for every n≥1 , [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Corollary 2.4 (Doob's type maximal inequality for demimartingales).

Let p>1 and {_Sn ,n≥1} be a demimartingale. Suppose that E|_Sk^|p <∞ for each k≥1 , then for every n≥1 , [figure omitted; refer to PDF]

Theorem 2.5.

Let {_Sn ,n≥1} be a demimartingale and g a nonnegative convex function on ... with g(0)=0 . Let {_bn ,n≥1} be a nondecreasing unbounded sequence of positive numbers. p>1 . Suppose that E(g(_Sk-1 )^)p ≤E(g(_Sk )^)p <∞ for each k≥1 and [figure omitted; refer to PDF] then _{lim n[arrow right]∞} (g(_Sn )/_bn )=0 a.s. , and (1.9)-(1.10) hold (_Sn is replaced by g(_Sn ) ), where [figure omitted; refer to PDF] In addition, [figure omitted; refer to PDF] If further one assumes that _αk >0 for infinitely many k , then [figure omitted; refer to PDF]

Proof.

By the condition of the theorem, we can see that _αn ≥0 for all n≥1 . Thus, [figure omitted; refer to PDF] follows from (2.9) for each n≥1 . By (2.12), we have [figure omitted; refer to PDF] Therefore, _{lim n[arrow right]∞} (g(_Sn )/_bn )=0 a.s. 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 g(x) is a nonnegative and nondecreasing convex function on ... with g(0)=0 , then the condition "E(g(_Sk-1 )^)p ≤E(g(_Sk )^)p for each k≥1 " is satisfied.

Remark 2.6.

Theorem 2.5 generalizes and improves the results of Theorem 2.2 in Christofides [3] and Theorem 2.7 in Prakasa Rao [6].

Theorem 2.7.

Let p>1 and {_Sn ,n≥1} a demimartingale with E|_Sk^|p <∞ for each k≥1 . Let {_bn ,n≥1} be a nondecreasing sequence of positive numbers. If [figure omitted; refer to PDF] then for any 0<r<p , [figure omitted; refer to PDF]

Proof.

Taking n=1 , ν=p and _ck =1/_bk in Lemma 1.8, we have [figure omitted; refer to PDF] Thus, by (2.20) and (2.18), we can get [figure omitted; refer to PDF]

Theorem 2.8.

Let {_Sn ,n≥1} be a demisubmartingale and g a nondecreasing and nonnegative convex function on ... with g(0)=0 and g(_Si )∈^L1 ,i≥1 . Let {_ck ,k≥1} be a nonincreasing sequence of positive numbers. Then for all 0<p<1 and each n≥1 , [figure omitted; refer to PDF]

Proof.

By Fubini theorem, it is easy to check that [figure omitted; refer to PDF] It follows from Lemma 1.7(i) and Lemma 1.9(ii) that [figure omitted; refer to PDF] Therefore, (2.22) follows from the above statements immediately.

Corollary 2.9.

Let the conditions of Theorem 2.8 be satisfied with _ck ≡1 for each k≥1 . Then for all 0<p<1 and each n≥1 , [figure omitted; refer to PDF]

By Corollary 2.9, we can get the following theorem.

Theorem 2.10.

Let {_Sn ,n≥1} be a demisubmartingale and g a nondecreasing and nonnegative convex function on ... with g(0)=0 and g(_Si )∈^L1 ,i≥1 . Let {_bn ,n≥1} be a nondecreasing unbounded sequence of positive numbers. If there exists some 0<p<1 such that [figure omitted; refer to PDF] then _{lim n[arrow right]∞} (g(_Sn )/_bn )=0 a.s., and (1.9)-(1.10) hold (_Sn is replaced by g(_Sn ) ), where [figure omitted; refer to PDF] In addition, [figure omitted; refer to PDF] If further one assumes that _αk >0 for infinitely many k , then [figure omitted; refer to PDF]

Similar to the proof of Theorem 2.8 and using Lemma 1.9(i), we can get the following.

Theorem 2.11.

Let {_Sn ,n≥1} be a nonnegative demisubmartingale. Then for all 0<p<1 , E[_{max 1≤k≤n Sk}^]p ≤(1/(1-p))(E_Sn^)p .

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.

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