Rényi-mixing of occupation times

Abstract

It is shown that the occupation times of a bounded interval by sums of independent and identically distributed random variables are Renyi-mixing under the classical necessary and sufficient condition of Darling and Kac for the original limit theorem. Some consequences are derived for the occupation times of a random walk in a random number of steps, along with an extension of the Darling–Kac theorem for Revesz-dependent sequences of random variables. As Darling and Kac originally established their limit theorem for occupation times of rather general Markov chains (and continuous-time Markov processes), it is not surprising in itself that condition 2 can be extended for sums of dependent variables. However, despite the fact that Revesz's almost independence is a rather weak form of dependence, the extension is perhaps of some interest in that that it represents a departure from independence, which is not Markovian in character.