Weak convergence of the empirical characteristic function

Abstract:

Let $P$ be a probability measure on ${\mathbf {R}}$ such that the density $f(x)$ for $P$ exists and there exists ${x_0} > 0$ such that $f(x) + f( - x)$ is decreasing for all $\left | x \right | \geqslant {x_0}$. Let $c(t)$ be the characteristic function for $P$, ${c_n}(t)$ the empirical characteristic function, and let ${C_n}(t): = {n^{1/2}}({c_n}(t) - c(t))$. New necessary and sufficient metric entropy conditions are obtained for the weak convergence of ${C_n}(t)$ on the space of continuous complex valued functions on $[ - \tfrac {1}{2},\tfrac {1}{2}]$. The result is used to characterize the weak convergence of ${C_n}(t)$ in terms of the tail behavior of $P$ and it also provides the first step towards a generalization of the Borisov-Dudley-Durst theorem. It also provides a partial response to a challenge raised by Dudley.