Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps
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 (∞).
Similar content being viewed by others
References
D. V. Gusak, “Distributions of the sojourn times of a homogeneous process with independent increments over any level,” Dokl. Akad. Nauk Ukr. SSSR, Ser. A, No. 1, 14–17 (1981).
D. V. Gusak, “How often is the sum of independent random variables larger than a given number?,” Ukr. Mat. Zh., 34, No. 3, 289–295 (1982); English translation: Ukr. Math. J., 34, No. 3, 234–239 (1982).
D. V. Gusak and B. I. Kaplan, “On the distribution of sojourn times in a fixed state for one scheme of random walks with discretely distributed jumps,” in: Analytic Methods in the Problems of Probability Theory [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, 1984, pp. 27–40.
D. V. Gusak and A. M. Rozumenko, “Sojourn times in a fixed state for the processes given by the sums of random numbers of discretely distributed terms,” in: Asymptotic Analysis of Random Evolutions [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, 1994, pp. 74–93.
D. V. Gusak, Limiting Problems for the Processes with Independent Increments in the Theory of Risk [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2007).
B. I. Kaplan, “Asymptotic behavior of the sojourn time in a fixed state for semicontinuous random walks on the Markov chain,” in: Asymptotic Behavior of the Sums of Random Variables on Markov Processes and Periodic Markov Chains [in Russian], Preprint 85.22, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1965), pp. 50–59.
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1021–1029, August, 2011.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0571-y