On the rate of convergence in a local limit theorem

Abstract

Let Zn = n−½(X1 + X2 + … + Xn), where {Xn} is a sequence of independent and identically distributed random variables with EX1 = 0, and a common distribution function F and characteristic function ω. Suppose |ω|r is integrable for some integer r ≥ 1. For all n ≥ r, then Zn has a probability density function fn obtained by using the inversion formula.