Let $X_k$ be $1$ or $-1$ with equal probability $\frac{1}{2}$ if $k > 0$, and $S_n = X_1 + \dots + X_n$. We start at $S_0 = 0$. If $T$ is the time (number of steps) to first return to zero (that is, the first $t>0$ such that $S_t=0$), what is the probability $P(T = t)$, for $t = 2, 4, 6$ and so on? The expectation of $T$ is known to be infinite.
Asked
Active
Viewed 32 times
0
-
@Mike: not really, I think something is not right in that answer (it does not answer my question). I did find two solutions. One from a lecture at Harvard University: https://arxiv.org/pdf/1509.04800.pdf. But they got it wrong. And one on this community, and they got it right: https://math.stackexchange.com/questions/64919/biased-random-walk-and-pdf-of-time-of-first-return – Vincent Granville Aug 06 '22 at 17:13
-
My link does answer your question. You first step is either $+1$ or $-1$. Either way, the probability that $T=t$ is equal the probability that the random walk after the first step reaches $\pm1$ for the first time in $t-1$ steps. This is exactly described by the linked question, with $a=t-1$, and $p=1/2$. – Mike Earnest Aug 06 '22 at 17:17
-
@Mike: In any case, I found the answer, and whether your link solves it or not, there is at least one link on Math.StackExchange that solves it in a way that is clear to me. – Vincent Granville Aug 06 '22 at 17:19