Recurrence of Stationary Sequences

Abstract

Let $\{X_n\}^{+\infty}_{-\infty}$ be a stationary sequence of random variables, with common distribution $\pi(dx)$. If the initial value $X_0$ is repeated with probability one (e.g. when $\pi(dx)$ is discrete), then the "shifted" sequence $\{X_{n+N}\}^\infty_{-\infty}$ is also stationary where $N = N(\omega)$ is the first $n > 0$ for which $X_n(\omega) = X_0(\omega)$. Surprisingly, this may even occur when $\pi(dx)$ is continuous and $\{X_n\}$ is ergodic (although not when $\{X_n\}$ is $\phi$-mixing). For Markov sequences, we also give other conditions which prohibit the a.s. recurrence of $X_0$. For recurrent sequences, we show that when $X_0$ is "conditionally discrete," the invariant $\sigma$-field for the $\{X_{n+N}\}$ process coincides (up to null sets) with $X_0 \vee \mathscr{A}$, the $\sigma$-field generated by $X_0$ and the invariant sets for $\{X_n\}$. Finally, we find an expression for $E(N\mid X_0 \vee \mathscr{A})$ which reduces to Kac's recurrence formula when $X_0$ is an indicator function.