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Limit theorems for random walks conditioned to stay positive. (English) Zbl 0756.60062

This paper is concerned with a random walk \(\{S_ n\}\) on the integers whose drift is negative but which is conditioned to stay positive. Let \(A_ n=\{S_ k\geq 0\), \(1\leq k\leq n\}\) and \(A=\lim_{n\to\infty}A_ n\). Here \(P(A)=0\) but it is shown that for a large class of events \(B\), \(\lim_{n\to\infty}P(B\mid A_ n)\) provides a workable definition of \(P(B\mid A)\). Specific limits are calculated in terms of associated Markov chains in the cases where (i) the random walk is simple or (ii) belongs to the discrete exponential family.

MSC:

60G50 Sums of independent random variables; random walks
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