On asymptotic approximation of ratio models for weakly dependent sequences

Abstract

Let {Zn, n≥1} be a sequence of nonnegative, weakly dependent random variables, and Xn=∑i=1nωniZi, where true{ωni,1≤i≤n,n≥1true} is an array of nonnegative weights. We show that E[]ftrue(Xntrue)prefix−1 can be asymptotically approximated by []ftrue(EXntrue)prefix−1 for a class of functions f(·) satisfying some mild conditions. Under some general conditions, we also prove that the expectation Etrue[Xnfalse/true(a+Yntrue) αtrue] approximates to EXnfalse/true(a+EYntrue) α with a certain convergence rate for any a > 0 and α>0, where Yn=∑i=1nZi. The results obtained in the article improve and extend some corresponding ones in the literature. Some numerical simulations and a real data example are also provided to support the theoretical results.