The Annals of Probability

Recurrence of Stationary Sequences

Donald Geman, Joseph Horowitz, and Joel Zinn

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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.

Article information

Ann. Probab., Volume 4, Number 3 (1976), 372-381.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G10: Stationary processes
Secondary: 28A65

Stationary sequence recurrence flow $\phi$-mixing invariant set


Geman, Donald; Horowitz, Joseph; Zinn, Joel. Recurrence of Stationary Sequences. Ann. Probab. 4 (1976), no. 3, 372--381. doi:10.1214/aop/1176996086.

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