Moments and polynomial expansions in discrete matrix-analytic models

Calculation of factorial moments and point probabilities is considered in integer-valued matrix analytic models at a finite horizon T. Two main settings are considered, maxima of integer-valued downward skipfree Levy processes and Markovian point process with batch arrivals (BMAPs). For the moments of the finite-time maxima, the procedure is to approximate the time horizon T by an Erlang distributed one and solve the corresponding matrix Wiener-Hopf factorization problem. For the BMAP, a structural matrix-exponential representation of the factorial moments of N(T) is derived. Moments are then used as a computational vehicle to provide a converging Gram-Charlier series for the point probabilities. Topics such as change-of-measure techniques and time inhomogeneity are also discussed.(c) 2021 Elsevier B.V. All rights reserved.

Automated summary

This article discusses the calculation of factorial moments and point probabilities in integer-valued matrix analytic models. It focuses on the maxima of integer-valued downward skipfree Levy processes and Markovian point processes with batch arrivals (BMAPs). The finite-time maxima are approximated using an Erlang distributed time horizon T and solved using the matrix Wiener-Hopf factorization problem. The factorial moments of N(T) in BMAP are represented using a structural matrix-exponential representation. Moments are then used to compute converging Gram-Charlier series for point probabilities. Change-of-measure techniques and time inhomogeneity are also discussed.