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Results tagged with random-walks
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user 21724
0
votes
One dimension random walk. Is hitting time Lipschitz with respect to target?
One should assume that $X$ is not a.s. $0$, in which case $\tau$ is a.s. integer-valued and non-constant, hence a.s. non-continuous, hence a.s. non-Lipschitz.
- 7,249
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 …
- 7,249