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
,
.