On the lack of memory for distributions of overshoot functionals in the case of upper almost semicontinuous processes defined on a Markov chain
Authors:
D. V. Gusak and E. V. Karnaukh
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 99 (2019), 77-89
MSC (2010):
Primary 60K37; Secondary 60G51
DOI:
https://doi.org/10.1090/tpms/1081
Published electronically:
February 27, 2020
MathSciNet review:
3908657
Full-text PDF
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Additional Information
Abstract: We study the question on whether or not the property of lack of memory that is valid for the geometrical and exponential distributions remains valid for hitting functionals in the case of upper almost semicontinuous processes defined on a Markov chain. If a boundary is attainable and the state of the environment is known at the moment when this boundary is attained, then the lack of memory holds only for an overshoot over a boundary $x\ge 0$ and the distribution of the overshoot does not depend on the overshoot moment as well as on $x$. The distribution of an undershoot for a boundary $x$ is determined via the distribution of the undershoot for a zero boundary. A similar property is proved for a jump crossing a boundary $x$.
References
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Additional Information
D. V. Gusak
Affiliation:
Department of Probability Theory and Mathematical Analysis, Faculty of Mathematics, Uzhgorod National University, Universytets’ka Street, 14, Uzhgorod, 88000 Ukraine
Email:
husakdv@ukr.net
E. V. Karnaukh
Affiliation:
Department of Statistics and Probability Theory, Faculty of Mechanics and Mathematics, Oles Honchar Dnipro National University, Gagarin Avenue, 72, Dnipro, 49000 Ukraine
Email:
ievgen.karnaukh@gmail.com
Keywords:
Almost semicontinuous processes defined on a Markov chain,
lack of memory,
passage functionals for a positive boundary,
main factorization identity
Received by editor(s):
May 6, 2018
Published electronically:
February 27, 2020
Article copyright:
© Copyright 2020
American Mathematical Society
