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On a gambler’s ruin problem. (English) Zbl 0965.60043
A particle moves in independent steps on the real line. Its initial position is 0. The $$u$$th step is described by a random variable $$X_u$$ such that $$X_u=1$$ w.p. $$p$$, $$X_u=-1$$ w.p. $$q$$, and $$X_u=0$$ w.p. $$r$$, where $$p+q+r=1$$. Let $$a$$ and $$b$$ be positive integers. Consider barriers at points $$-b$$ and $$a$$. If the particle reaches $$-b$$, it is immediately returned to the given point $$j$$, $$-b<j<a$$. The barrier $$a$$ is absorbing. The authors give an explicit result for the absorption probabilities. The derivation is based on a generating function.

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 60G50 Sums of independent random variables; random walks 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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