Moments and polynomial expansions in discrete matrix-analytic models

Søren Asmussen; Mogens Bladt

doi:10.1016/j.spa.2021.12.002

Moments and polynomial expansions in discrete matrix-analytic models

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

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Abstract

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 Lévy 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.

Original language	English
Journal	Stochastic Processes and Their Applications
Volume	150
Pages (from-to)	1165-1188
Number of pages	24
ISSN	0304-4149
DOIs	https://doi.org/10.1016/j.spa.2021.12.002
Publication status	Published - Aug 2022

Keywords

BMAP
Erlangization
Factorial moments
Matrix exponentials
Richardson extrapolation
Wiener–Hopf factorization

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10.1016/j.spa.2021.12.002

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title = "Moments and polynomial expansions in discrete matrix-analytic models",

abstract = "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 L{\'e}vy 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.",

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author = "S{\o}ren Asmussen and Mogens Bladt",

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language = "English",

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T1 - Moments and polynomial expansions in discrete matrix-analytic models

AU - Asmussen, Søren

AU - Bladt, Mogens

PY - 2022/8

Y1 - 2022/8

N2 - 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 Lévy 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.

AB - 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 Lévy 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.

KW - BMAP

KW - Erlangization

KW - Factorial moments

KW - Matrix exponentials

KW - Richardson extrapolation

KW - Wiener–Hopf factorization

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U2 - 10.1016/j.spa.2021.12.002

DO - 10.1016/j.spa.2021.12.002

M3 - Journal article

AN - SCOPUS:85121784807

SN - 0304-4149

VL - 150

SP - 1165

EP - 1188

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

ER -

Moments and polynomial expansions in discrete matrix-analytic models

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Keywords

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