| Martingale is an important concept in probability theory. Martingale is a special kind of random variable sequence, and its theory has fairly been perfected. Here are some examples:Martingale's maximal inequality (Chow,1960):Let {Xn,Fn,n≥1} be a submartingale, (Ω,Φ)=F0(?)F1(?)F2(?)…,Fi be aσfield. Let {an,n≥1}be a nonincreasing sequence of random variables and a1＞0, an be measurable with respect to Fn-1, i.e. an∈Fn-1, n=1, 2,…Then for(?)ε＞0, Where S+=max{0,S}.Doob's maximal inequalities: Let {Xn,Fn,n≥1} be a submartingale, C is a positive number. Then for (?)n∈N, CP((?) Xk≥C)≤∫((?)Xk≥C) Xn dP.Doob's inequalities for martingale: Let {Tk, k≥1} be a martingale or nonnegative submartingale. For fixed p＞1, suppose E|Tk|p＜∞, k=1, 2,…, n, thenDemisubmartingales were introduced by Newman and Wright in 1982. Naturally, people begin to think about that whether the whole results for martingales are also true for the case of demimartingales.Chow's maximal inequality for submartingale was extended to the case of demisubmartingales by Tasos C. Christofides in 2000. This result serves as a "source" inequality for other inequalities such as the Hajek-Renyi inequality and Doob's maximal inequality and leads to a strong law of large numbers. The partial sum of mean zero associated random variables is a demimartingale. Therefore, the classical Hajek-Renyi inequalities can be easily extended to the case of associated random variables.This paper is based on the former results. First of all, we propose some useful lemmas. Then we reprove the maximal inequality for demisubmartingale and Doob's maximal inequality. Finally, the main results of this paper are proposed in chapter three. We get Doob's inequality for demimartingale and it has the same form as Doob's inequality for martingale. We also get a strong law of large numbers and a strong growth rate for demimartingale. At last we use demisubmartingale's convergence theorem obtaining a strong law.
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