Laws of the Iterated Logarithm for the Empirical Characteristic Function

Abstract

Let $X$ be a real-valued random variable with distribution function $F(x)$ and characteristic function $c(t)$. Let $F_n(x)$ be the $n$th empirical distribution function associated with $X$ and let $c_n(t)$ be the characteristic function $F_n(x)$. Necessary and sufficient conditions in terms of $c(t)$ are obtained for $c_n(t) - c(t)$ to obey bounded and compact laws of the iterated logarithm in the Banach space of continuous complex-valued functions on $\lbrack -1, 1 \rbrack$.