Moment generating funtions of extremums and their complemets for upper semi-continuous lattice Poison process on Markov chain

Abstract

On the ¯nite regular Markov chain it is con- sidered latticed Poisson process, jumps of which take arbitrary integer negative values, and posi- tive jumps are equal 1. Such processes are called upper semi-continuous. For these processes the relations for moment generating funtions (m. g. f.) of minimum and complements to maximum of the process are established without proective operation. Obtained relations for considered m. g. f. were de¯ned in [6] in terms of projection of coresponding component of factorization. New relations for these m. g. f. are established by the invertions of the cumulant function, which is represented in terms of generating transforma- tion for distribution function of negative jumps.