Weak Convergence of the Empirical Characteristic Function

Abstract

Let $X$ be a real valued random variable with probability distribution function $F(x)$ and characteristic function $c(t)$. Let $F_n(x)$ be the $n$th empirical distribution function associated with $X$ and $c_n(t)$ the characteristic function of $F_n(x)$. Necessary and sufficient conditions are obtained for the weak convergence of $\sqrt{n}\lbrack c_n(t) - c(t)\rbrack$ on the space of continuous complex valued functions on $\lbrack -1/2, 1/2\rbrack$.