On asymptotic approximation of ratio models for weakly dependent sequences

Let {Z(n), n >= 1} be a sequence of nonnegative, weakly dependent random variables, and X-n= n-ary sumation i=1n omega(ni)Z(i), where {omega(ni),1 <= i <= n,n >= 1} is an array of nonnegative weights. We show that Ef(Xn)-1 can be asymptotically approximated by f(EXn)(-1) for a class of functions f(center dot) satisfying some mild conditions. Under some general conditions, we also prove that the expectation E[X-n/(a+Y-n) alpha] approximates to EXn/(a+EYn)(alpha) with a certain convergence rate for any a > 0 and alpha>0, where Y-n= n-ary sumation i=1(n)Z(i). 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. (C) 2022 Statistical Society of Canada