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Large deviations of generalized renewal process

  • Gavriil A. Bakay EMAIL logo und Aleksandr V. Shklyaev
Veröffentlicht/Copyright: 14. August 2020
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Abstract

Let (ξ(i), η(i)) ∈ ℝd+1, 1 ≤ i < ∞, be independent identically distributed random vectors, η(i) be nonnegative random variables, the vector (ξ(1), η(1)) satisfy the Cramer condition. On the base of renewal process, NT = max{k : η(1) + … + η(k) ≤ T} we define the generalized renewal process ZT = i=1NTξ(i). Put IΔT(x) = {y ∈ ℝd : xjyj < xj + ΔT, j = 1, …, d}. We find asymptotic formulas for the probabilities P(ZTIΔT(x)) as ΔT → 0 and P(ZT = x) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of (ξ(1), η(1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.


Originally published in Diskretnaya Matematika (2019) 31, №2, 21–55 (in Russian).


References

[1] Embrechts P., Klüppelberg, C., “Some aspects of insurance mathematics”, Theory Probab. Appl., 38:2 (1993), 262–295.10.1137/1138025Suche in Google Scholar

[2] Anscombe, F, “Large-sample theory of sequential estimation”, Biometrika, 36:3-4 (1949), 455–458.10.1093/biomet/36.3-4.455Suche in Google Scholar

[3] Borovkov A.A., “Integro-local limit theorems for compound renewal processes”, Theory Probab. Appl., 62:2, 175–195.10.1137/S0040585X97T988551Suche in Google Scholar

[4] Borovkov A. A., Mogulskii A. A., “Large deviation principles for trajectories of compound renewal processes. I”, Theory Probab. Appl., 60:2 (2016), 207–221.10.1137/S0040585X97T987582Suche in Google Scholar

[5] Frolov A.N., Martikainen A. I., Steinebach J., “On probabilities of small deviations for compound renewal processes”, Theory Probab. Appl., 52:2, 328–337.10.1137/S0040585X97983043Suche in Google Scholar

[6] Mogulskii, A., “Integro-local limit theorems for multidimensional compound renewal processes”, Analytical and Computational Methods in Probability Theory and its Applications (ACMPT-2017), M.: izd-vo RUDN, 2017, 495–499.Suche in Google Scholar

[7] Fenchel W., “On conjugate convex functions”, Canad. J. Math., 1 (1949), 73–77.10.1007/978-3-0348-0439-4_7Suche in Google Scholar

[8] Borovkov A.A., Mogulskii A.A., “On large and superlarge deviations of sums of independent random vectors under Cramér’s condition. I”, Theory Probab. Appl., 51:2 (2007), 227–255.10.1137/S0040585X9798230XSuche in Google Scholar

[9] Dembo A., Zeitouni O., Large Deviations Techniques and Applications, XVI, Springer, Berlin-Heidelberg, 2010, 396 pp.10.1007/978-3-642-03311-7Suche in Google Scholar

[10] Kirkland S., Neumann M., Group Inverses of M-matrices and their Applications, New York: Chapman and Hall/CRC Press, 2012, 311 pp.10.1201/b13054Suche in Google Scholar

[11] Kolmogorov A.N., “Local limit theorem for classical Markov chains”, Izv. Akad. Nauk SSSR, 13:4 (1949), 281–300 (in Russian).Suche in Google Scholar

[12] Iscoe I., Ney P., Nummelin E., “Large deviations of uniformly recurrent Markov additive processes”, Adv. Appl. Math., 1:4 (121985), 373–412.10.1016/0196-8858(85)90017-XSuche in Google Scholar

[13] Miller H., “A convexity property in the theory of random variables defined on a finite Markov chain”, Ann. Math. Statist., 32:4 (1961), 1260–1272.10.1214/aoms/1177704865Suche in Google Scholar

[14] Meyer D., Matrix Analysis and Applied Linear Algebra, Philadelphia: SIAM, 2000, xii + 718 pp.10.1137/1.9780898719512Suche in Google Scholar

[15] Borovkov A.A., Mogulskii A. A., “Integro-local limit theorems for compound renewal processes under Cramér’s condition. I”, Siberian Math. J., 59:3 (2018), 383–402.10.1134/S0037446618030023Suche in Google Scholar

[16] Borovkov A.A., Mogulskii A. A., “Integro-local limit theorems for compound renewal processes under Cramér’s condition. II”, Siberian Math. J., 59:4 (2018), 578–597.10.1134/S003744661804002XSuche in Google Scholar

[17] Mogulskii A. A., Prokopenko E. I., “Integro-local theorems for multidimensional compound renewal processes, when Cramér’s condition holds. I”, Sibirskie Élektronnye Matematicheskie Izvestiya, 2018, No15, 475–502 (in Russian).Suche in Google Scholar

[18] Mogulskii A. A., Prokopenko E. I., “Integro-local theorems for multidimensional compound renewal processes, when Cramér’s condition holds. II”, Sibirskie №lektronnye Matematicheskie Izvestiya, 2018, No15, 503–527 (in Russian).Suche in Google Scholar

[19] Mogulskii A. A., Prokopenko E. I., “Integro-local theorems for multidimensional compound renewal processes, when Cramér’s condition holds. III”, Sibirskie Elektronnye Matematicheskie Izvestiya, 15 (2018), 528–553 (in Russian).Suche in Google Scholar

Received: 2017-12-01
Revised: 2018-07-24
Published Online: 2020-08-14
Published in Print: 2020-08-26

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 30.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/dma-2020-0020/pdf?lang=de
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