The Asymptotics For The Tail Probability Of Random Sums With Widely Orthant Dependent

This paper gets Rosenthal inequality for widely orthant dependent, that is, let {X, Xk,k≥1} be a WOD sequence of random variables (r.v.'s) with identically dis-tributed F(x). For any 1≤t≤2,p≥t, exists a constant C(p,t) depending on p, t, we have Especially, it contains END and NOD.Let N be a nonnegative integer-value r.v. which is independent of {X,Xk,k≥1}, SN=∑Xk. when {X, Xk:k≥1} was nonnegative and NOD and N was consistently varying, Under the condition of F∈L∩D, Zong(2010) got the asympotics equivalence P(SN>x)~ENF(x)+P(N>x/EX).Inspired by Zong(2010), this paper takes the result of Zong(2010) promotion from three aspects by using Rosenthal inequality with widely orthant dependent. Firstly, {X,Xk,k≥1} is a r.v.s on R canceling R+; Secondly, we promote NOD to widely orthant dependent; Thirdly, we promote consistently varying of N to L∩D. Finally, applying it to compound renewal risk model, we obtain some finite-time ruin probabil-ity.