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On overshoots and hitting times for random walks. (English) Zbl 0942.60032

Consider an oscillating integer valued random walk on \(\mathbb{Z}\) up to the first hitting time of some fixed integer \(x>0\). Suppose there is a fee to be paid each time the random walk crosses the level \(x\), and that the amount corresponds to the so-alled overshoot. The author derives the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables one to state the problem in terms of the intersection of certain regenerative sets.

MSC:

60G50 Sums of independent random variables; random walks
60K05 Renewal theory
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