Inequalities Of Demimartingale With Applications

Let {Ω, F, P} be a probability space and {X_n, n≥1} be a sequence of random variables defined on it. A finit sequence {X₁, X₂,…, X_n} is said to be associated if for any two component wise non-decreasing functions f and g on Rⁿ, Cov{f(X₁,X₂,…,X_n),g(X₁,X₂,…,X_n)}≥0. Suppose X₁,X₂,…,X_n∈L¹, EX_i=0.letS=sum from i=1 to n X_iAssume that forj=1,2,…E[(S_j+1-S_j)f(S₁,…, S_n)]≥0,for all component non-decreasing functions f such that the expectation is defined.Then {S_j,j≥1} is called a demimartingale. Inqualities for demimartingale play an impotant role in the study of demimartingale. Dood's maxmal inquality and upcrossing inquality is tent to the case of demi(sub)martingale introduce by Newman and Right. In this paper, we extent some inqualities for martingale to the case of demi(sub)martingale by useing resuit of referce [6]. And we derive a new kind of inquality. At last, we obtained some theories including convergence theorem, the law of large number and large deviation theorem.