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Consider an asymmetric random walk starting at 0 with probability $p$ moving right and probability $q = 1-p$ moving left, where $p > \frac 12$. Then it is well known that for arbitrary $x > 0$, let $T_x$ be the first hitting time at $x$, then $T_x$ is almost surely finite and its moments can all be calculated using Wald's identity. However, I am wondering if anyone could point me to the references bounding the probability $\mathbb P(T_x > n)$?

Thanks!

shong
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  • here they compute the pdf for $T_{a}$: https://math.stackexchange.com/questions/2794960/distribution-of-stopping-time-for-possibly-biased-random-walk?rq=1 – Thomas Kojar Mar 12 '21 at 17:28

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