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I am looking at a random walk that starts at 0 and at every step, either increases or decreases by 1, or doesn't move.

More specifically, $\mathbb{P} (X_{t+1} = X_t) = 1-p,\mathbb{P} (X_{t+1} = X_t +1) = p/2 + q/2 $ and $\mathbb{P} (X_{t+1} = X_t +1) = p/2 - q/2$.

Do we know something about the hitting time of a target $T>0$?

I know some results exist if $p=1$, namely something like $\min(T^2, \frac T q)$, do these results generalise for smaller $p$?

Many thanks!

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  • $\begingroup$ What you call a random walk with self loops is called a lazy random walk in literature. Also, what do you call a target ? Is T a single element ? Is T a finite subset of the integers ? Is T any subset of the integers ? And could you please be more precise about the formula you are looking for and the one you think holds in the case $p=1$ ? $\endgroup$ Commented Jun 12, 2024 at 7:29

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