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Let $X_1, X_2, ...$ be i.i.d. random variables, $\mathbb{E} X_1 > 0$, and let $S_n = \sum\limits _{i = 1} ^n X_i$. Define $\tau = \max \{n \in \mathbb{N}: S_n \leq 0 \}$ with the convention $\tau = 0$ if $S_n > 0$ for all $n \in \mathbb{N}$.

Apart from the asymptotic bounds, is there something known about distribution of $\tau$, at least in some specific cases? I am interested in particular in the case $X_i = \xi _i - a$, where $\xi _1, \xi _2, ...$ are i.i.d. unit exponentials and $a \in (0,1)$. For the Brownian motion with drift the distribution of the last zero crossing is given in Theorem 2.1 in this paper. The settings in this question are a little bit similar to those here and here.

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For a $\pm 1$ random walk, the walk will return to 0 a geometric number of time's and then head off to $\infty$. You can get the distribution of the return time from Spitzer's formula, if not more easily. The same is true, I think, of any integer valued random walk the goes up by 1's.$$$$ The quantity $P(\tau = 0)$ can be related to the distribution of the first increasing ladder height, also available from Spitzer's formula. Feller Vol 2's chapters on the ladder height random variables are a reasonable place to start.

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  • $\begingroup$ Indeed, for a $\pm 1$ simple random walk it is an exercise. $\endgroup$ – Sinusx Dec 24 '20 at 14:31
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There is an exact expression in [1] for $E(\tau_1)$, the mean first ladder epoch, in the Gaussian case. See Theorem 1.1. there, and note that the first ladder height has mean which is the first ladder epoch times the mean of one increment by Wald's identity. As noted in the comment, this is different from the last negative time.

[1] Chang, Joseph T., and Yuval Peres. "Ladder heights, Gaussian random walks and the Riemann zeta function." The Annals of Probability 25, no. 2 (1997): 787-802. https://projecteuclid.org/download/pdf_1/euclid.aop/1024404419

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  • $\begingroup$ In the paper $\tau$ is defined as the first ladder epoch, however in the question it is the last time the walk takes a negative value. Unless I am missing something, $\tau$ as defined in the question can be much larger than $\tau$ in the paper. This happens for instance if $S_n$ goes up a bit, then goes down and stays below $0$ for a while before finally starting to rise. Thus, the relation between two different $\tau$'s is not clear. One could try to condition on the first ladder height, but it is not obvious whether something explicit comes out. $\endgroup$ – Sinusx Dec 25 '20 at 21:43
  • $\begingroup$ Agreed and revised answer accordingly. It now does not directly address your question... $\endgroup$ – Yuval Peres Dec 25 '20 at 22:58

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