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10
votes
Accepted
Random walk to stay in an interval forever
Yes. Indeed, if $s = \sum_{i \geq 1} t_i^2 <1$, then
$$
\mathbb{P}[ \ \ \forall n, \sum_{i=1}^n X_i \in [-1,1] \ \ ] \geq 1-s > 0.
$$
To see this, note that $M_n = |\sum_{i=1}^n X_i|$ is a nonnegativ …