Stationary limits of shot noise processes
Authors:
G. K. Verovkin and A. V. Marynych
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 101 (2020), 67-83
MSC (2020):
Primary 60F05; Secondary 60K05
DOI:
https://doi.org/10.1090/tpms/1112
Published electronically:
January 5, 2021
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The weak convergence of centered shot noise processes to a stationary $\mathcal {L}_2$-process is proved under the assumption that the response function is mean square integrable and variance of the step of the underlying random walk is finite. Properties of the limiting stationary process are studied.
References
- Søren Asmussen, Applied probability and queues, 2nd ed., Applications of Mathematics (New York), vol. 51, Springer-Verlag, New York, 2003. Stochastic Modelling and Applied Probability. MR 1978607
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- Harald Cramér and M. R. Leadbetter, Stationary and related stochastic processes, Dover Publications, Inc., Mineola, NY, 2004. Sample function properties and their applications; Reprint of the 1967 original. MR 2108670
- William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- Allan Gut, Stopped random walks, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2009. Limit theorems and applications. MR 2489436
- Alexander Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions, Stochastic Process. Appl. 123 (2013), no. 6, 1987–2010. MR 3038496, DOI https://doi.org/10.1016/j.spa.2013.01.019
- Alexander Iksanov, Renewal theory for perturbed random walks and similar processes, Probability and its Applications, Birkhäuser/Springer, Cham, 2016. MR 3585464
- Alexander Iksanov, Zakhar Kabluchko, and Alexander Marynych, Weak convergence of renewal shot noise processes in the case of slowly varying normalization, Statist. Probab. Lett. 114 (2016), 67–77. MR 3491974, DOI https://doi.org/10.1016/j.spl.2016.03.015
- Alexander Iksanov, Zakhar Kabluchko, Alexander Marynych, and Georgiy Shevchenko, Fractionally integrated inverse stable subordinators, Stochastic Process. Appl. 127 (2017), no. 1, 80–106. MR 3575536, DOI https://doi.org/10.1016/j.spa.2016.06.001
- Alexander Iksanov, Alexander Marynych, and Matthias Meiners, Limit theorems for renewal shot noise processes with eventually decreasing response functions, Stochastic Process. Appl. 124 (2014), no. 6, 2132–2170. MR 3188351, DOI https://doi.org/10.1016/j.spa.2014.02.007
- Alexander Iksanov, Alexander Marynych, and Matthias Meiners, Asymptotics of random processes with immigration II: Convergence to stationarity, Bernoulli 23 (2017), no. 2, 1279–1298. MR 3606766, DOI https://doi.org/10.3150/15-BEJ777
- Zakhar Kabluchko and Alexander Marynych, Renewal shot noise processes in the case of slowly varying tails, Theory Stoch. Process. 21 (2016), no. 2, 14–21. MR 3662592
- C. Klüppelberg and T. Mikosch, Delay in claim settlement and ruin probability approximations, Scand. Actuar. J. 2 (1995), 154–168. MR 1366822, DOI https://doi.org/10.1016/0167-6687%2896%2982365-1
- P. Läuger, Shot noise in ion channels, Biochimica et Biophysica Acta (BBA) — Biomembranes 413 (1975), no. 1, 1–10.
- A. V. Marynych, A note on convergence to stationarity of random processes with immigration, Theory Stoch. Process. 20 (2015), no. 1, 84–100. MR 3502397
- A. Marynych, Limit theorems for random processes with regeneration, Doctorate dissertation, Kyiv, 2017; Available at http://do.unicyb.kiev.ua/marynych/wp-content/uploads/2017/04/final_{w}ith_{h}yperrefs.pdf (In Ukrainian)
- Sidney Resnick, Adventures in stochastic processes, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1181423
- B. A. Rogozin, Asymptotic analysis of the renewal function, Teor. Verojatnost. i Primenen. 21 (1976), no. 4, 689–706 (Russian, with English summary). MR 0420900
- W. Schottky, Spontaneous current fluctuations in electron streams, Ann. Physics 57 (1918), 541–567.
- Walter L. Smith, Asymptotic renewal theorems, Proc. Roy. Soc. Edinburgh Sect. A 64 (1954), 9–48. MR 60755
- Hermann Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications (New York), Springer-Verlag, New York, 2000. MR 1741181
- D. Vere-Jones, Stochastic models for earthquake occurrence, J. Roy. Statist. Soc. Ser. B 32 (1970), 1–62. MR 272087
- Ward Whitt, Stochastic-process limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002. An introduction to stochastic-process limits and their application to queues. MR 1876437
References
- S. Asmussen, Applied Probability and Queues, Springer Verlag, New York, 2003. MR 1978607
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York, 1968. MR 0233396
- H. Cramér and M. R. Leadbetter, Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications, Dover Publications, New York, 2013. MR 2108670
- W. Feller, An Introduction to Probability Theory and its Applications, 2nd edition, vol. 2, John Wiley & Sons, Inc., New York, 1971. MR 0270403
- A. Gut, Stopped Random Walks. Limit Theorems and Applications, 2nd edition, Springer Verlag, New York, 2009. MR 2489436
- A. Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions, Stoch. Process. Appl. 123 (2013), no. 6, 1987–2010. MR 3038496
- A. Iksanov, Renewal Theory for Perturbed Random Walks and Similar Processes, Birkhäuser, Basel, 2016. MR 3585464
- A. Iksanov, Z. Kabluchko, and A. Marynych, Weak convergence of renewal shot noise processes in the case of slowly varying normalization, Statist. Probab. Lett. 114 (2016), 67–77. MR 3491974
- A. Iksanov, Z. Kabluchko, A. Marynych, and G. Shevchenko, Fractionally integrated inverse stable subordinators, Stoch. Process. Appl. 127 (2017), no. 1, 80–106. MR 3575536
- A. Iksanov, A. Marynych, and M. Meiners, Limit theorems for renewal shot noise processes with eventually decreasing response functions, Stoch. Process. Appl. 124 (2014), no. 6, 2132–2170. MR 3188351
- A. Iksanov, A. Marynych, and M. Meiners, Asymptotics of random processes with immigration II: convergence to stationarity, Bernoulli 23 (2017), no. 2, 1279–1298. MR 3606766
- Z. Kabluchko and A. Marynych, Renewal shot noise processes in the case of slowly varying tails, Theory Stoch. Process. 21(37) (2016), no. 2, 14–21. MR 3662592
- C. Klüppelberg and T. Mikosch, Delay in claim settlement and ruin probability approximations, Scand. Actuar. J. 2 (1995), 154–168. MR 1366822
- P. Läuger, Shot noise in ion channels, Biochimica et Biophysica Acta (BBA) — Biomembranes 413 (1975), no. 1, 1–10.
- A. Marynych, A note on convergence to stationarity of random processes with immigration, Theory Stoch. Process. 20(36) (2015), no. 1, 84–100. MR 3502397
- A. Marynych, Limit theorems for random processes with regeneration, Doctorate dissertation, Kyiv, 2017; Available at http://do.unicyb.kiev.ua/marynych/wp-content/uploads/2017/04/final_{w}ith_{h}yperrefs.pdf (In Ukrainian)
- S. Resnick, Adventures in Stochastic Processes, 4th edition, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 1181423
- B. A. Rogozin, Asymptotics of Renewal Functions, Teor. Veroyatnost. Primenen. 21 (1976), no. 4, 689–706; English translation in Theory Probab. Appl. 21 (1977), no. 4, 669–686. MR 0420900
- W. Schottky, Spontaneous current fluctuations in electron streams, Ann. Physics 57 (1918), 541–567.
- W. Smith, Asymptotic renewal theorems. II, Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 64 (1953), no. 1, 9–48. MR 60755
- H. Thorisson, Coupling, Stationarity, and Regeneration, Springer Verlag, New York, 2000. MR 1741181
- D. Vere-Jones, Stochastic models for earthquake occurrence, J. Roy. Statist. Soc. Ser. B 32 (1970), 1–62. MR 272087
- W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and their Application to Queues, Springer Series in Operations Research, Springer Verlag, New York, 2002. MR 1876437
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2020):
60F05,
60K05
Retrieve articles in all journals
with MSC (2020):
60F05,
60K05
Additional Information
G. K. Verovkin
Affiliation:
Faculty for Computer Science and Cybernetics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
glebverov@gmail.com
A. V. Marynych
Affiliation:
Faculty for Computer Science and Cybernetics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
MR Author ID:
848771
Email:
marynych@unicyb.kiev.ua
Keywords:
Stochastic processes with immigration,
$\mathcal {L}_2$-processes,
shot noise processes,
stationary stochastic processes,
stationary renewal process,
renewal theory
Received by editor(s):
August 8, 2019
Published electronically:
January 5, 2021
Article copyright:
© Copyright 2020
American Mathematical Society