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Suppose we have a sequence of independent and identically distributed random variables $(X_n)_{n\ge 1}$ such that $$ P(X_n=1)=p,\quad P(X_n=0)=r,\quad P(X_n=-1)=q $$ with $p,q,r\in[0,1]$, $p+q+r=1$, and $pq>0$ to avoid trivialities.

Choose integers $a,b$, $a<0<b$, and set $$ \tau=\inf\left\{n\ge 1; \sum_{k=1}^nX_k\in\{a,b\}\right\}, $$ i.e. $\tau$ is the time when the random walk defined by $S_0=0$ and $S_n=\sum_{k=1}^nX_k$ leaves for the first time the interval $[a+1,b-1]$. Or it can be interpreted as the time when a gambler has either lost his initial capital $a$ or won an amount $b$.

One can use Doob's optional stopping theorem or the strong Markov property to determine the law of $\tau$, see, e.g., https://mathoverflow.net/questions/125038/stopping-time-expectation-for-gamblers-ruin (where only the symmetric case $p=q=\frac{1}{2}$, $r=0$ is treated).

Does anybody know a reference (article or textbook) where the general case is treated? Many thanks in advance!

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  • $\begingroup$ mathoverflow.net/questions/133399/… related $\endgroup$ – Alexander Chervov Jun 30 '13 at 19:54
  • $\begingroup$ It is not (research level). $\endgroup$ – Did Jul 4 '13 at 9:12
  • $\begingroup$ To go to the obvious, did you try Feller? The chances are high that the relevant martingale is in there. $\endgroup$ – Did Jul 4 '13 at 13:51
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Does anybody know a reference (article or textbook) where the general case is treated?

Feller Vol 1, XIV.4-5.

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