Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

Distributions of overshoots for almost continuous stochastic processes defined on a Markov chain


Author: M. S. Gerych
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 94 (2016).
Journal: Theor. Probability and Math. Statist. 94 (2017), 37-52
MSC (2010): Primary 42C40; Secondary 60G12
DOI: https://doi.org/10.1090/tpms/1007
Published electronically: August 25, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the distributions of overshoots for the almost semi-continuous processes defined on a Markov chain. For these processes, we obtain the limit distributions of overshoots for the infinite and zero horizons.


References [Enhancements On Off] (What's this?)

  • [1] Ī. Ī. Ēžov and A. V. Skorohod, Markov processes which are homogeneous in the second component. I, Teor. Verojatnost. i Primenen. 14 (1969), 3–14 (Russian, with English summary). MR 0247666
  • [2] Ī. Ī. Ēžov and A. V. Skorohod, Markov processes which are homogeneous in the second component. II., Teor. Verojatnost. i Primenen. 14 (1969), 679–692 (Russian, with English summary). MR 0267640
  • [3] E. Arjas, On a fundamental identity in the theory of semi-Markov processes, Advances in Appl. Probability 4 (1972), 258–270. MR 0339355, https://doi.org/10.2307/1425998
  • [4] A. A. Mogul'skĭ, Factorization identities for processes with independent increments, given on a finite Markov chain, Teor. Veroyatnost. Mat. Statist. 11 (1974), 86-96; English transl. in Theory Probab. Math. Statist. 11 (1974), 87-98.
  • [5] Søren Asmussen and Hansjörg Albrecher, Ruin probabilities, 2nd ed., Advanced Series on Statistical Science & Applied Probability, vol. 14, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. MR 2766220
  • [6] D. V. Gusak, \cyr Granichnī zadachī dlya protsesīv z nezalezhnimi prirostami na skīnchennikh lantsyugakh Markova ta dlya napīvmarkovs′kikh protsesīv, \cyr Trudi Īnstitutu Matematiki Natsīonal′noï Akademīï Nauk Ukraïni [Proceedings of the Institute of Mathematics of the National Academy of Sciences of the Ukraine], vol. 18, Natsīonal′na Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 1998 (Ukrainian, with English and Ukrainian summaries). MR 1710395
  • [7] E. V. Karnaukh, Boundary problems for a class of processes defined on a Markov chain, Abstract of thesis kand. fiz.-mat. nauk, Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 2007. (Ukrainian)
  • [8] V. I. Lotov and N. G. Orlova, Factorization representations in boundary value problems for random walks defined on a Markov chain, Sibirsk. Mat. Zh. 46 (2005), no. 4, 833–840 (Russian, with Russian summary); English transl., Siberian Math. J. 46 (2005), no. 4, 661–667. MR 2169400, https://doi.org/10.1007/s11202-005-0066-2
  • [9] M. S. Herych and D. V. Husak, On the moment-generating functions of extrema and their complements for almost semicontinuous integer-valued Poisson processes on Markov chains, Ukrainian Math. J. 67 (2016), no. 8, 1164–1182. Translation of Ukraïn. Mat. Zh. 67 (2015), no. 8, 1034–1049. MR 3473711, https://doi.org/10.1007/s11253-016-1144-2
  • [10] D. V. Gusak and A. I. Tureniyazova, Distribution of some functionals for lattice Poisson processes on a Markov chain, Asymptotic methods in the investigation of stochastic models (Russian), Acad. Sci. Ukrain. SSR, Inst. Math., Kiev, 1987, pp. 21–27, 143 (Russian). MR 943353
  • [11] D. V. Gusak, \cyr Granichnī zadachī dlya protsesīv z nezalezhnimi prirostami v teorīï riziku, \cyr Pratsī Īnstitutu Matematiki Natsīonal′noï Akademīï Nauk Ukraïni. Matematika ta ïï Zastosuvannya [Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications], vol. 65, Natsīonal′na Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 2007 (Ukrainian, with English and Ukrainian summaries). MR 2382816
  • [12] M. S. Gerych, A ramification of the main factorization identity for almost semi-continuous lattice Poisson processes defined on a Markov chain, Carpathian Math. Publ. 4 (2012), no. 2, 229-240. (Ukrainian)
  • [13] M. S. Gerych, Generating functions of extremums and their complements for semi-continuous from above lattice Poisson processes defined on a Markov chain, Visn. Kyiv Shevchenko Nat. Univ. Ser. Fiz.-Mat. 1 (2013), 21-27. (Ukrainian)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 42C40, 60G12

Retrieve articles in all journals with MSC (2010): 42C40, 60G12


Additional Information

M. S. Gerych
Affiliation: Department of Probability Theory and Mathematical Analysis, Faculty for Mathematics, Uzhgorod National University, Universytets’ka Street, 14, Uzhgorod 88000, Ukraine
Email: miroslava.gerich@yandex.ua

DOI: https://doi.org/10.1090/tpms/1007
Keywords: Almost semi-continuous process defined on a Markov chain, overshoot functionals, distributions of overshoots for the infinite and zero horizons
Received by editor(s): February 12, 2016
Published electronically: August 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society