Fluctuations of Sequences which Converge in Distribution

Abstract

A sequence $\{Y_n\}^\infty_{n=1}$ of random variables with values in a metric space is mixing with limiting distribution $G$ if $P(\{Y_n \in A\}\mid B) \rightarrow G(A)$ for all $G$-continuity sets $A$ and all events $B$ that have positive probability. It is shown that if $\{Y_n\}$ is mixing with limiting distribution $G$ and if the support of $G$ is separable, then the range $\{Y_n(\omega); n \geqq 1\}$ is dense in the support of $G$ almost surely. A theorem that, under rather general conditions, establishes mixing for the summation processes based on a martingale is given, and as an application it is shown that, under certain conditions, the range of the periodogram is dense in $R^+$ almost surely.