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For a symmetric random walk $S_n$, if we have a bound $P(S_n \in A) < \delta$, do we get a bound on the amount of time $S_n$ spends in $A$?

Let $S_n$ be a symmetric random walk on the integers. Let $A$ be a subset of bounded upper density with a bound $P(S_n \in A) < \delta(\epsilon)$. Do we get a bound like $P\left(\frac{\#\{n = 1, \cdots, N \mid S_n \in A\}}{N} > \epsilon \right) < \epsilon$?

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    $\begingroup$ A trivial counter-example is the symmetric $\pm 1$ random walk, $A$ any set of odd integers, and $n$ even. Then $P(S_n \in A) = 0$, but the mean occupation time of $A$ can be arbitrarily close to $\tfrac{1}{2}$. I suppose at least some aperiodicity condition is needed here. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Dec 14, 2019 at 20:36

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Assuming the hypothesis holds for all $n$, and $\delta(\epsilon)=\epsilon^2$, the conclusion holds by Markov’s inequality.

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