Buhrii N. Properties of solutions of Markov renewal equations

The Laplace transform of a distribution, which has regularly varying tail at infinity with exponent -1 is found. The asymptotical behaviour of this Laplace transform at the origin is investigated. The asymptotic of the Laplace transform of the renewal function associated with this distribution at the origin is got. Renewal equation for the semi-Markov process is considered on condition that the distribution tail of the stay time of this process in every fixed state is regularly varying function at infinity with exponent -1. Existence conditions of limit distributions for a functionals family of the semi-Markov process and for time means of random evolutions constructed on the basis of the semi-Markov process and a solution of ordinary differential equations system are received.