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

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Ukrainian Mathematical Journal Aims and scope

Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps

$$ {\xi_k}:E\left[ {{{{{z^{{\xi_k}}}}} \left/ {{{\xi_k} < 0}} \right.}} \right] = \frac{{1 - b}}{{z - b}},\,\,\,0 \leqslant b < 1. $$

For the moment generating function of the sojourn time of ξ(t) in a fixed state, we deduce relations in terms of the roots \( {z_s} < 1 < {\hat{z}_s} \) of the Lundberg equation. Passing to the limit as s → 0 in the relations obtained as a result, we determine the distributions of l r (∞).

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References

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1021–1029, August, 2011.

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Gusak, D.V. Sojourn time of almost semicontinuous integral-valued processes in a fixed state. Ukr Math J 63, 1176–1186 (2012). https://doi.org/10.1007/s11253-012-0571-y

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  • DOI: https://doi.org/10.1007/s11253-012-0571-y

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