Let $(X_1,X_2,\ldots)$ be a stationary ergodic process with each $X_n$ a real random variable taking values in $[-1,+1]$. Suppose that $\mathbb{E}[X_n]=0$. Let $S_n = \sum_{k=1}^n X_k$. Is the process $(S_1,S_2,\ldots)$ necessarily recurrent, in the sense that there exists some $M$ such that almost surely $|S_n| \leq M$ infinitely often?
Ergodic Theory – Recurrence of Ergodic Processes
ergodic-theorypr.probability
Best Answer
Yes. I will use some different notation, but the idea is the same. Let $(\Omega,\mu)$ be a probability space, and let $\sigma \colon \Omega \to \Omega$ be an ergodic measure preserving transformation. Let $f$ be a measurable function taking values in $[-1,1]$ such that $\int f\,d\mu=0$. Write $S_nf(\omega)=f(\omega)+\ldots+f(\sigma^{n-1}\omega)$.
I claim that for $\mu$-a.e. $\omega$, $|S_nf(\omega)|\le c$ infinitely often for any $c>\frac 12$. Here is a proof by contradiction. Suppose that $c>\frac 12$ and there is a set of positive measure $A$ such that $|S_nf(\omega)|\le c$ only finitely many times for $\omega\in A$. Then since $c-(-c)>1$, then for $\omega\in A$, either $S_nf(\omega)>c$ for all sufficiently large $n$ or $S_nf(\omega)<-c$ for all sufficiently large $n$.
Negating $f$ if necessary, we may pick an $N>0$, and a subset $B$ of $A$ of positive measure such that $S_nf(\omega)>c$ for all $n\ge N$. Now we consider the induced (first return) dynamical system on $B$, which is also ergodic. The $k$th return of $\omega$ to $B$ satisfies $t_k(\omega)/k\to 1/\mu(B)$. But by definition of $B$, we see (by splitting up the summation) that if $\omega\in B$, $$ S_{t_{kN}}f(\omega)= \sum_{j=0}^{k-1} S_{t_{(j+1)N}(\omega)-t_{jN}(\omega)} f(\sigma^{t_{jN}(\omega)})>ck. $$ This implies $S_{t_{kN}(\omega)}f(\omega)/t_{kN}(\omega)= [S_{t_{kN}(\omega)}f(\omega)/k] / [t_{kN}(\omega)/k ]$ has a positive limit superior. That contradicts the Birkhoff ergodic theorem.