Sojourn time of almost semicontinuous integral-valued processes in a fixed state - Gusak, D. V. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
Let ξ(t) be an almost lower semicontinuous integer-valued process with the moment generating function of the negative part of jumps ξ_k : E[z^ξ_k / ξ_k < 0] = 1 − bz − b,0 ≤ b < 1. For the moment generating function of the sojourn time of ξ(t) in a fixed state, we obtain relations in terms of the roots z_s < 1 < z_s of the Lundberg equation. By passing to the limit (s → 0) in the obtained relations, we determine the distributions of l_r(∞).