Weak Convergence of the Empirical Characteristic Function

Let X be a real valued random variable with probability distribution function $F(x)$ and characteristic function $c(t)$. Let Fₙ(x) be the nth empirical distribution function associated with X and cₙ(t) the characteristic function of Fₙ(x). Necessary and sufficient conditions are obtained for the weak convergence of √n cₙ(t) - c(t) on the space of continuous complex valued functions on -1/2, 1/2.