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

Definition 1.1.

Let be an sequence of random variables. Assume that for

(1.1)

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

(1.2)

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

(1.3)

for all Borel measurable and increasing (or decreasing) set pairs (A set is said increasing (or decreasing) if implies for any ), where

(1.4)

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 ,

(1.5)

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 ,

(1.6)

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 ,

(1.7)

then

(1.8)

and with the growth rate

(1.9)

where

(1.10)

In addition,

(1.11)

If further assumes one that for infinitely many , then

(1.12)

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

1. (i)

   If is a demisubmartingale (or a demimartingale) and is a nondecreasing convex function such that , then is a demisubmartingale.

    
2. (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 ,

(1.13)

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

1. (i)

   Let be a demisubmartingale. Then for any ,

    

(1.14)

(1.15)

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

(1.16)

(1.17)

We point out that there is a mistake in the proof of (1.17), that is,

(1.18)

should be replaced by

(1.19)

In fact, by Lemma 1.5 and Fubini Theorem, we can see that

(1.20)

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

(1.21)

in Shiryaev [10, page 495] should be revised as

(1.22)

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 ,

(2.1)

(2.2)

Proof.

By Lemma 1.5 and Hölder's inequality, we have

(2.3)

where is a real number and satisfies Since for each , we can obtain

(2.4)

therefore,

(2.5)

Similar to the proof of (2.3) and using Lemma 1.5 again, we can see that

(2.6)

For constants and , it follows that

(2.7)

Combining (2.6) and (2.7), we have

(2.8)

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 ,

(2.9)

(2.10)

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

Let and be a demimartingale. Suppose that for each , then for every ,

(2.11)

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

(2.12)

then , and (1.9)-(1.10) hold ( is replaced by ), where

(2.13)

In addition,

(2.14)

If further one assumes that for infinitely many , then

(2.15)

Proof.

By the condition of the theorem, we can see that for all . Thus,

(2.16)

follows from (2.9) for each . By (2.12), we have

(2.17)

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

(2.18)

then for any ,

(2.19)

Proof.

Taking , and in Lemma 1.8, we have

(2.20)

Thus, by (2.20) and (2.18), we can get

(2.21)

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 ,

(2.22)

Proof.

By Fubini theorem, it is easy to check that

(2.23)

It follows from Lemma 1.7(i) and Lemma 1.9(ii) that

(2.24)

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 ,

(2.25)

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

(2.26)

then a.s., and (1.9)-(1.10) hold ( is replaced by ), where

(2.27)

In addition,

(2.28)

If further one assumes that for infinitely many , then

(2.29)

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 , .