On the Glivenko–Cantelli theorem for generalized empirical processes based on strong mixing sequences

Abstract

Given \\s{Xi, i ⩾ 1\\s} as non-stationary strong mixing (n.s.s.m.) sequence of random variables (r.v.'s) let, for 1 ⩽ i ⩽ n and some γ ϵ [0, 1], F1(x)=γP(Xi<x)+(1-γ)P(Xi⩽x) and Ii(x)=γI(Xi<x)+(1-γ)I(Xi⩽x). For any real sequence \\s{Ci\\s} satisfying certain conditions, let Dn=supx,γmaxN⩽n|∑1NCi[Ii(x)-Fi(x)]|.In this paper an exponential type of bound for P(Dn ε), for any ε >0, and a rate for the almost sure convergence of Dn are obtained under strong mixing. These results generalize those of Singh (1975) for the independent and non-identically distributed sequence of r.v.'s to the case of strong mixing.