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A Bernoulli excursion and its various applications. (English) Zbl 0738.60069

Let \(\{\eta_ 0^ +,\eta_ 1^ +,\dots,\eta_{2n}^ +\}\) perform a Bernoulli excursion, i.e. a random walk process in which \(\eta_ 0^ +=\eta_{2n}^ +=0\) while \(\eta_ i^ +\geq 0\). Then define \(2n\omega_ n=\eta_ 0^ ++\dots+\eta_{2n}^ +\). The author derives the distribution, the moments and the asymptotic distribution of \(\omega_ n\). He further beautifully illustrates how the random quantities \(\omega_ n\) naturally appear in random graphs, tournaments, combinatorics, order statistics and inventory control. The limit distribution \(W\) of \(\omega_ n/\sqrt {2n}\) is tied up with the classical Brownian excursion process; a set of interesting properties of \(W\) is obtained as well.

MSC:

60G50 Sums of independent random variables; random walks

Software:

Mathematica
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