On the Moment-Generating Functions of Extrema and Their Complements for Almost Semicontinuous Integer-Valued Poisson Processes on Markov Chains
For an integer-valued compound Poisson process with geometrically distributed jumps of a certain sign [these processes are called almost upper (lower) semicontinuous] defined on a finite regular Markov chain, we establish relations (without projections) for the moment-generating functions of extrema and their complements. Unlike the relations obtained earlier in terms of projections, the proposed new relations for the moment-generating functions are determined by the inversion of the perturbed matrix cumulant function. These matrix relations are expressed via the moment-generating functions for the distributions of the corresponding jumps.
English version (Springer): Ukrainian Mathematical Journal 67 (2015), no. 8, pp 1164-1182.
Citation Example: Gusak D. V., Herych M. S. On the Moment-Generating Functions of Extrema and Their Complements for Almost Semicontinuous Integer-Valued Poisson Processes on Markov Chains // Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1034-1049.