On extremes of PSI-processes and gaussian limits of their normalized independent identical distributed sums

Authors

  • Oleg V. Rusakov St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation
  • Roman A. Ragozin St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2022.208

Abstract

We define PSI-process — Poisson Stochastic Index process, as a continuous time random process which is obtained by a manner of a randomization for the discrete time of a random sequence. We consider the case when a double stochastic Poisson process generates this randomization, i. e. such Poisson process has a random intensity. Under condition of existence of the second moment the stationary PSI-processes possess a covariance which coincides with the Laplace transform of the random intensity. In our paper we derive distributions of extremes for a one PSI-process, and these extremes are expressed in terms of Laplace transform of the random intensity. The second task that we solve is a convergence of the maximum of Gaussian limit for normalized sums of i. i. d. stationary PSI-processes. We obtain necessary and sufficient conditions for the intensity under which, after proper centering and normalization, this Gaussian limit converges in distribution to the double Exponential Law. For solution this task we essentially base on the monograph: M.R.Leadbetter, Georg Lindgren, Holder Rootzen (1986) “Extremes and Relative Properties of Random Sequences and Processes”, end essentially use the Tauberian theorem in W. Feller form.

Keywords:

pseudo-poissonian type processes, random intensity, Laplace transform for distributions, Tauberian theorems

Downloads

Download data is not yet available.
 

References

Литература

1. Феллер В. Введение в теорию вероятностей и ее приложения, пер. с англ. Т. 2. Москва, Мир (1984).

2. Невё Ж. Математические основы теории вероятностей, пер. с франц. Москва, Мир (1969).

3. Русаков О.В. Псевдо-пуассоновские процессы со стохастической интенсивностью и класс процессов, обобщающих процесс Орнштейна—Уленбека. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 4 (62), вып. 2, 247–257 (2017). https://doi.org/10.21638/11701/spbu01.2017.208

4. Лидбеттер М., Линдгрен Г., Ротсен Х. Экстремумы случайных последовательностей и процессов, пер. с англ. Москва, Мир (1989).

5. Булинский А. В., Ширяев А.Н. Теория случайных процессов. Москва, Физматлит (2003).

6. Биллингсли П. Сходимость вероятностных мер, пер. с англ. Москва, Наука (1977).

7. Русаков О.В. Относительная компактность сумм независимых одинаково распределенных псевдо-пуассоновских процессов в пространстве Скорохода. Зап. научн. сем. ПОМИ 442, 122–132 (2015).

8. Samorodnitsky G., Taqqu M. S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. In Ser.: Stochastic Modeling Series, vol. 1. New York, Chapman and Hall/CRC (1994).

9. Bingham N.H., Goldie C. M., Teugels J. L. Regular variation. Cambridge, Cambridge University Press (1987).

10. Русаков О.В., Баев Б.А., Якубович Ю.В. О некоторых локальных асимптотических свойствах последовательностей со случайным индексом. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 7 (65), вып. 3, 453–468 (2020). https://doi.org/10.21638/spbu01.2020.308

References

1. Feller W. An introduction o probability theory and its applications. Vol.2. New York, JohnWiley & Sons (1971). [Rus. ed.: Feller W. Vvedenie v teoriju verojatnostej i ee prilozhenija. Moscow, Mir Publ. (1984)].

2. Neveu J. Bases mat´ematiques du calculdes probabilit´es. Paris, Masson Et Cie (1964). [Rus. ed.: Neveu J. Matematicheskie osnovy teorii verojatnostej. Moscow, Mir Publ. (1969)].

3. Rusakov O.V. Pseudo-Poissonian processes with stochastic intensity and a class of processes which generalize the Ornstein—Uhlenbeck process. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 4 (62), iss. 2, 247–257 (2017). https://doi.org/10.21638/11701 /spbu01.2017.208 (In Russian) [Eng. transl.: Vestnik St Petersburg University, Mathematics 50, 153– 160 (2017). https://doi.org/10.3103/S106345411702011X].

4. Leadbetter M.R., Lindgren G., Rootzen H. Extremes and related properties of random sequences and processes. In Ser.: Springer Series in Statistics. New York, Heidelberg, Berlin, Springer-Verlag (1986). [Rus. ed.: Leadbetter M.R., Lindgren G., Rootzen H. Jekstremumy sluchajnyh posledovatel’nostej i processov. Moscow, Mir Publ. (1989)].

5. Bulinskiy A. V., Shiryaev F.N. Theory of random processes. Moscow, Fizmatlit Publ. (2003). (In Russian)

6. Billingsley P. Convergence of Probability Measures. New York, John Wiley & Sons (1968). [Rus. ed.: Billingsley P. Shodimost’ verojatnostnyh mer. Moscow, Nauka Publ. (1977)].

7. Rusakov O.V. Tightness of Sums of Independent Identically Distributed Pseudo-Poisson Processes in the Skorokhod Space. Zap. Nauchn. Sem. POMI 442, 122–132 (2015). (In Russian) [Eng. transl.: J. Math. Sci. 225, 805–811 (2017). https://doi.org/10.1007/s10958-017-3496-z].

8. Samorodnitsky G., Taqqu M. S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. In Ser.: Stochastic Modeling Series, vol. 1. New York, Chapman and Hall/CRC (1994).

9. Bingham N.H., Goldie C. M., Teugels J. L. Regular variation. Cambridge, Cambridge University Press (1987).

10. Rusakov O. V., Baev B.A., Yakubovich Y. V. On some local asymptotic properties of sequences with a random index. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 7 (65), iss. 3, 453–468 (2020). https://doi.org/10.21638/spbu01.2020.308 (In Russian) [Eng. transl.: Vestnik St Petersburg University, Mathematics 53 (3), 308–319 (2020). https://doi.org/10.1134/S1063454120030115].

Downloads

Published

2022-07-06

How to Cite

Rusakov, O. V., & Ragozin, R. A. (2022). On extremes of PSI-processes and gaussian limits of their normalized independent identical distributed sums. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(2), 269–277. https://doi.org/10.21638/spbu01.2022.208

Issue

Vol. 9 No. 2 (2022)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.