Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Maximal -inequalities for demimartingales | 3 Journal of Inequalities and Applications a SpringerOpen Journal Gong Journal of Inequalities and Applications 2011 2011 59 http content 2011 1 59 RESEARCH Open Access Maximal -inequalities for demimartingales Xiaobing Gong1 2 Correspondence xbgong@ 1Key Laboratory of Numerical Simulation of Sichuan Province Neijiang Sichuan 641112 China Full list of author information is available at the end of the article Springer Abstract In this paper we establish some maximal -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 Keywords demimartingale maximal -inequalities Doob s inequality 1. Introduction Definition Let S1 S2 . be an L 1 sequence of random variables. Assume that for j 1 2 . B Sj i - Sj f 0 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 is equivalent to the condition that Sj j 1 is a martingale with the natural choice of s-algebras. If the function f is assumed to be nonnegative and not necessarily nondecreasing then the condition is equivalent to the condition that Sj j 1 is a submartingale with the natural choice of s -algebras. A martingale with the natural choice of s-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 s-algebras cf. 1 example A 2 p. 10 . Definition is due to Newman and Wright 3 . Relevant to the .
