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Maximal ϕ-inequalities for demimartingales
Journal of Inequalities and Applications volume 2011, Article number: 59 (2011)
Abstract
In this paper, we establish some maximal ϕ-inequalities for demimartingales that generalize the results of Wang (Stat. Probab. Lett. 66, 347-354, 2004) and Wang et al. (J. Inequal. Appl. 2010(838301), 11, 2010) and improve Doob's type inequality for demimartingales in some cases.
Mathematics Subject Classification (2010): 60E15; 60G48
1. Introduction
Definition 1.1 Let S1, S2, ... be an L1 sequence of random variables. Assume that for j = 1, 2, ...
for all componentwise 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, the sequence {Sj,j ≥ 1} is called a demisubmartingale.
Remark. If the function f is not required to be nondecreasing, then the condition (1.1) is equivalent to the condition that {S j , j ≥ 1} is a martingale with the natural choice of σ-algebras. If the function f is assumed to be nonnegative and not necessarily nondecreasing, then the condition (1.1) is equivalent to the condition that {S j , j ≥ 1} is a submartingale with the natural choice of σ -algebras. A martingale with the natural choice of σ-algebras is a demimartingale. It can be checked that a submartingale is a demisubmartingale (cf. [[1], Proposition 1]). However, there are stochastic processes that are demimartingales but not martingales with the natural choice of σ-algebras (cf. [[1], example A], [[2], p. 10]). Definition 1.1 is due to Newman and Wright [3].
Relevant to the notion of demimartingales is the notion of positive dependence. To that end, we have the following definition.
Definition 1.2 A finite collection of random variables X1, X2, ..., X m is said to be associated if
for any two componentwise nondecreasing functions f, g on Rm such that the covariance is defined. An infinite collection is associated if every finite subcollection is associated.
Remark. Associated random variables were introduced by Esary et al. [4] and have been found many applications especially in reliability theory. Proposition 2 of Newman and Wright [3] shows that the partial sum of a sequence of mean zero associated random variables is a demimartingale.
The connection between demimartingales and martingales pointed out in the previous remark raises the question whether certain results and especially maximal inequalities valid for martingales are also valid for demimartingales. Newman and Wright [3] have extended various results including Doob's maximal inequality and Doob's upcrossing inequality to the case of demimartingales. Christofides [5] showed that Chow's maximal inequality for (sub)martingales can be extended to the case of demi(sub)martingales. Prakasa Rao [6] derived a Whittle-type inequality for demisubmartingales. Wang [7] obtained Doob's type inequality for more general demimartingales. Prakasa Rao [8] established some maximal inequalities for demisubmartingales. Wang et al. [9] established some maximal inequalities for demimartingales that generalize the results of Wang [7]. In this paper, we establish some maximal ϕ-inequalities for demimartingales that generalize the results of Wang [7] and Wang et al. [9], and improve Doob's type inequality for demimartingales in some cases.
2. Demimartingales inequalities
Let denote the class of Orlicz functions, that is, unbounded, nondecreasing convex functions ϕ : [0, + ∞) → [0, +∞) with ϕ(0) = 0. Let denote the set of such that is integrable at 0. Given and a ≥ 0, define
Denote Φ(x) = Φ0(x), x > 0.
We now prove a maximal ϕ-inequality for demimartingales.
Theorem 2.1. Let S1, S2, ... be a demimartingale and g(.) be a nonnegative convex function such that g(0) = 0. Let and {c k , k ≥ 1} be a nonincreasing sequence of positive numbers, define . Then
where , p > 1.
Proof. By Fubini theorem and Theorem 2.1 in [7] we have
The last inequality follows from the Hölder's inequality.
Remark. Let ϕ(x) = xp , p > 1 in Theorem 2.1, then . Hence
Let . We get
which is the inequality (2.1) of Theorem 2.1 in [9].
Let ϕ(x) = (x - 1)+ = max{0, x - 1} in Theorem 2.1. Then . Hence . Therefore
which is the inequality (2.6) in [9]. By the inequality
we have
which is the inequality (2.2) of Theorem 2.1 in [9]. Let c j = 1, j ≥ 1 in inequality (2.2), the inequality (2.10) in [9] is obtained immediately. Let g(x) = |x| in inequality (2.2) we have
which is the inequality (2.10) in [7]. Let c j = 1, j ≥ 1 in inequality (2.3) we have
which is the inequality (2.11) in [9].
Corollary 2.1. Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let . Then
Where , p > 1.
Proof. Let c k = 1, k ≥ 1 in Theorem 2.1 we get (2.5) immediately.
Remark. Let ϕ(x) = xp , p > 1 in Corollary 2.1, then . Hence
Let . We get
which is the inequality (2.9) in [9]. Let g(x) = |x| in the above inequality we get
which is the inequality (2.11) in [9].
Corollary 2.2. Let S1, S2, ... be a demimartingale with S0 = 0 and {c k , k ≥ 1} be a nonincreasing sequence of positive numbers. Let . Then
Where , p > 1.
Proof. Let g(x) = |x| in Theorem 2.1, inequality (2.6) is obtained immediately.
Remark. Let ϕ(x) = xp , p > 1 in Corollary 2.2, then . Hence
Let . We get
which is the inequality (2.9) in [7].
We now prove some other maximal ϕ-inequalities for demimartingales following the techniques in [8].
Theorem 2.2 Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let {c k , k ≥ 1} be a nonincreasing sequence of positive numbers and . Then
for all n ≥ 1, t > 0 and 0 < λ < 1. Furthermore,
for n ≥ 1, a > 0, b > 0 and 0 < λ < 1.
Proof. Let t > 0 and 0 < λ < 1. Theorem 2.1 in [7] implies
Rearranging the last inequality, we get that
for all n ≥ 1, t > 0 and 0 < λ < 1.
Let b > 0. By inequality (2.7), then
for n ≥ 1, a > 0, b > 0, t > 0 and 0 < λ < 1.
Corollary 2.3. Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let . Then
for all n ≥ 1, t > 0 and 0 < λ < 1. Furthermore,
for all n ≥ 1, a > 0, b > 0 and 0 < λ < 1.
Proof. Let c k = 1, k ≥ 1 in Theorem 2.2, Corollary 2.3 follows.
As a special case of Corollary 2.3 is the following corollary.
Corollary 2.4. Let S1, S2, ... be a demimartingale with S0 = 0 and . Then
for all n ≥ 1, t > 0 and 0 < λ < 1. Furthermore,
for all n ≥ 1, a > 0, b > 0 and 0 < λ < 1.
Remark. Theorem 3.1 in [8] is generalized in the case of demimartingales.
As a special case of Theorem 2.2 is the following theorem.
Theorem 2.3 Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let {c k , k ≥ 1} be a nonincreasing sequence of positive numbers and . Then
for all n ≥ 1, a > 0 and 0 < λ < 1. Let in (2.9). Then
for a > 0, n ≥ 1.
Proof. Theorem 2.3 follows from Choosing b = a in (2.8) and observing that .
Let c k = 1, k ≥ 1 in Theorem 2.3 we have the following corollary.
Corollary 2.5. Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let . Then
for all n ≥ 1, a > 0, 0 < λ < 1 and
for a > 0, n ≥ 1.
As a special case of Corollary 2.5 is the following Corollary.
Corollary 2.6. Let S1, S2, ... be a demimartingale with S0 = 0 and . Then
for all n ≥ 1, a > 0, 0 < λ < 1 and
for a > 0, n ≥ 1.
Remark. Theorem 3.2 in [8] is generalized in the case of demimartingales.
Theorem 2.4 Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let {c k , k ≥ 1} be a nonincreasing sequence of positive numbers. Then
Proof. Let ϕ(x) = x in Theorem 2.2. Then Φ1(x) = x ln x - x + 1, . Hence
for all n ≥ 1, b > 0 and 0 < λ < 1. Let b > 1, . Therefore
for all b > 1 and n ≥ 1. Since
the inequality (2.11) can be rewritten in the form
Corollary 2.7. Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let {c k , k ≥ 1} be a nonincreasing sequence of positive numbers. Then
Proof. Let in (2.10). Then we get (2.12).
Corollary 2.8. Let S1, S2, ... be a demimartingale with S0 = 0 and g(.) be a nonnegative convex function such that g(0) = 0. Let {c k , k ≥ 1} be a nonincreasing sequence of positive numbers. Then
Proof. Let b = e in (2.10). Then we get (2.13).
Remark. Inequality (2.13) is a sharper inequality than inequality (2.2) in [9] when
Corollary 2.9. Let S1, S2, ... be a demimartingale with S0 = 0 and {c k , k ≥ 1} be a nonincreasing sequence of positive numbers. Then
Proof. Let g(x) = |x| in (2.13). Then we get (2.14).
Remark. Inequality (2.14) is a sharper inequality than inequality (2.10) in [7] when
Corollary 2.10. Let S1, S2, ... be a demimartingale with S0 = 0. Then
Proof. Let c j = 1, j ≥ 1 and g(x) = |x| in Theorem 2.4. We get inequality (2.15).
Remark. The inequality (3.22) in [8] is generalized in the case of demimartingales.
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Acknowledgements
The author is most grateful to the editor Professor Soo-Hak Sung and anonymous referees for the careful reading of the manuscript and valuable suggestions that helped in significantly improving an earlier version of this paper. This work was supported by the Natural Science Foundation of the Department of Education of Sichuan Province(09ZC071)(China).
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Gong, X. Maximal ϕ-inequalities for demimartingales. J Inequal Appl 2011, 59 (2011). https://doi.org/10.1186/1029-242X-2011-59
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DOI: https://doi.org/10.1186/1029-242X-2011-59