2018
Том 70
№ 6

# Sojourn time of almost semicontinuous integral-valued processes in a fixed state

Gusak D. V.

Abstract

Let $\xi(t)$ be an almost lower semicontinuous integer-valued process with the moment generating function of the negative part of jumps $\xi_k : \textbf{E}[z^{\xi_k} / \xi_k < 0] = \frac{1 − b}{z − b},\quad 0 ≤ b < 1.$ For the moment generating function of the sojourn time of $\xi(t)$ in a fixed state, we obtain relations in terms of the roots $z_s < 1 < \widehat{z}_s$ of the Lundberg equation. By passing to the limit $(s → 0)$ in the obtained relations, we determine the distributions of $l_r(\infty)$.

English version (Springer): Ukrainian Mathematical Journal 63 (2011), no. 8, pp 1176-1186.

Citation Example: Gusak D. V. Sojourn time of almost semicontinuous integral-valued processes in a fixed state // Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1021-1029.

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