Take the 2-minute tour ×  I have a $\mathbb{Z}_+$-valued
stochastic process $X$ in discrete time, which has unit jumps up
or down. I know the following statement: there exists a random
variable $\tau$, almost surely finite, s.t. the time between
successive returns of $X$ to 0 is stochastically bounded by
$\tau$. That is, if $T_i$ is the time between $i$th and
$(i+1)$th visits to 0, then $P(T_i>n) \leq P(\tau > n)$ for all
$i$. (The process isn't Markov, so there isn't any reason to
think that $T_i$ are iid.)  I would like to derive some
statement of the form: for any $\epsilon > 0$ there exists $K <
\infty$ such that $\lim\inf_{T \to \infty} \frac1T \sum_{n=0}^T
\delta_{X_n < K} \geq 1-\epsilon$. This seems plausible, because
morally $X$ should be upper bounded by $\tau/2$ (it has unit
jumps, and in time $\tau$ needs to return to 0), but I'm having
trouble formalizing this intuition.  share|improve this question
Does $\tau$ have finite expectation? –  Ori Gurel-Gurevich Sep
25 '11 at 16:06 No, it doesn't. Als