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 ,

Proof.

By Lemma 1.5 and Hölder's inequality, we have

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

therefore,

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

For constants and , it follows that

Combining (2.6) and (2.7), we have

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 ,

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

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

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

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

In addition,

If further one assumes that for infinitely many , then

Proof.

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

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

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

then for any ,

Proof.

Taking , and in Lemma 1.8, we have

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

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 ,

Proof.

By Fubini theorem, it is easy to check that

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

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 ,

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

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

In addition,

If further one assumes that for infinitely many , then

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 , .