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On factorization components of the sojourn times for semicontinuous random walks in a strip. (Russian, English) Zbl 1032.60042

Sib. Mat. Zh. 44, No. 4, 800-809 (2003); translation in Sib. Math. J. 44, No. 4, 629-637 (2003).
Let \(\xi_1,\xi_2,\dots\) be independent identically distributed random variables and let \(S_n=\xi_1+\cdots+\xi_n\). Given an interval \((\gamma_1,\gamma_2]\) of the real axis, let \[ u((\gamma_1,\gamma_2],n) =\text{Card}\{k\in[1,n]:S_k\in(\gamma_1,\gamma_2]\} \] be the sojourn time of the random walk \(S_n\) in the interval \((\gamma_1,\gamma_2]\). The special case of so-called semicontinuous random walk is considered when \(\xi\)’s take their values in integers not greater than \(1\). For a random walk of this type, the author obtains explicit representations for factorization components for the distribution of the functional \(u((\gamma_1,\gamma_2],n)\).

MSC:

60G50 Sums of independent random variables; random walks
62E10 Characterization and structure theory of statistical distributions
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