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A functional limit theorem for general shot noise processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  04 May 2020

Alexander Iksanov Open the ORCID record for Alexander Iksanov [Opens in a new window]  and
Bohdan Rashytov
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Bohdan Rashytov*
Affiliation:
Taras Shevchenko National University of Kyiv
*
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
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Abstract

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.

Keywords

Hölder continuityshot noise processweak convergence in the Skorokhod space

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 280 - 294
Copyright
© Applied Probability Trust 2020

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References

Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Beachwood, OH: Institute of Mathematical Statistics.Google Scholar
Alsmeyer, G., Iksanov, A. and Marynych, A. (2017). Functional limit theorems for the number of occupied boxes in the Bernoulli sieve. Stoch. Process. Appl. 127, 995–1017.10.1016/j.spa.2016.07.007CrossRefGoogle Scholar
Beran, J., Feng, Y., Ghosh, S. and Kulik, R. (2013). Long-Memory Processes. Probabilistic Properties and Statistical Methods. New York: Springer.CrossRefGoogle Scholar
Billingsley, P. (1968). Convergence of Probability Measures. Chichester: Wiley.Google Scholar
Bingham, N. H. (1973). Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitsth. 26, 273–296.10.1007/BF00534892CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Butkovsky, O. A. (2012). Limit behavior of a critical branching process with immigration. Math. Notes 92, 612–618.CrossRefGoogle Scholar
Chernavskaya, E. A. (2015). Limit theorems for an infinite-server queuing system. Math. Notes 98, 653–666.CrossRefGoogle Scholar
Daley, D. J. (1972). Asymptotic properties of stationary point processes with generalized clusters. Z. Wahrscheinlichkeitsth. 21, 65–76.CrossRefGoogle Scholar
Daw, A. and Pender, J. (2018). Queues driven by Hawkes processes. Stoch. Syst. 8, 192–229.CrossRefGoogle Scholar
Dong, C. and Iksanov, A. (2020). Weak convergence of random processes with immigration at random times. To appear in J. Appl. Prob. 57.Google Scholar
Gnedin, A., Iksanov, A. and Marynych, A. (2012). A generalization of the Erdös–Turán law for the order of random permutation. Combinatorics Prob. Comput. 21, 715–733.CrossRefGoogle Scholar
Gnedin, A., Iksanov, A., Marynych, A. and Möhle, M. (2018). The collision spectrum of $\Lambda$-coalescents. Ann. Appl. Prob. 28, 3857–3883.10.1214/18-AAP1409CrossRefGoogle Scholar
Gut, A. (2009). Stopped Random Walks. Limit Theorems and Applications, 2nd edn. New York: Springer.CrossRefGoogle Scholar
Iglehart, D. L. and Whitt, W. (1971). The equivalence of functional limit theorems for counting processes and associated partial sums. Ann. Math. Statist. 42, 1372–1378.CrossRefGoogle Scholar
Iksanov, A. (2013). Functional limit theorems for renewal shot noise processes with increasing response functions. Stoch. Process. Appl. 123, 1987–2010.CrossRefGoogle Scholar
Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Basel: Birkhäuser.CrossRefGoogle Scholar
Iksanov, A., Jedidi, W. and Bouzzefour, F. (2018). Functional limit theorems for the number of busy servers in a G/G/$\infty$ queue. J. Appl. Prob. 55, 15–29.CrossRefGoogle Scholar
Iksanov, A. and Kabluchko, Z. (2018). A functional limit theorem for the profile of random recursive trees. Electron. Commun. Prob. 23, 87.CrossRefGoogle Scholar
Iksanov, A., Kabluchko, Z. and Marynych, A. (2016). Weak convergence of renewal shot noise processes in the case of slowly varying normalization. Statist. Prob. Lett. 114, 67–77.CrossRefGoogle Scholar
Iksanov, A., Kabluchko, Z., Marynych, A. and Shevchenko, G. (2016). Fractionally integrated inverse stable subordinators. Stoch. Process. Appl. 127, 80–106.CrossRefGoogle Scholar
Iksanov, A., Marynych, A. and Meiners, M. (2014). Limit theorems for renewal shot noise processes with eventually decreasing response functions. Stoch. Process. Appl. 124, 2132–2170.CrossRefGoogle Scholar
Iksanov, A., Marynych, A. and Meiners, M. (2017). Asymptotics of random processes with immigration I: scaling limits. Bernoulli 23, 1233–1278.CrossRefGoogle Scholar
Iksanov, A., Marynych, A. and Meiners, M. (2017). Asymptotics of random processes with immigration II: convergence to stationarity. Bernoulli 23, 1279–1298.10.3150/15-BEJ777CrossRefGoogle Scholar
Iksanov, A., Marynych, A. and Vatutin, V. (2015). Weak convergence of finite-dimensional distributions of the number of empty boxes in the Bernoulli sieve. Theory Prob. Appl. 59, 87–113.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. New York: Springer.10.1007/978-3-662-05265-5CrossRefGoogle Scholar
Kabluchko, Z. and Marynych, A. (2016). Renewal shot noise processes in the case of slowly varying tails. Theor. Stoch. Proc. 21, 14–21.Google Scholar
Koops, D. T., Saxena, M., Boxma, O. J. and Mandjes, M. (2018). Infinite-server queues with Hawkes input. J. Appl. Prob. 55, 920–943.10.1017/jpr.2018.58CrossRefGoogle Scholar
Pang, G. and Zhou, Y. (2018). Functional limit theorems for a new class of non-stationary shot noise processes. Stoch. Process. Appl. 128, 505–544.10.1016/j.spa.2017.05.008CrossRefGoogle Scholar
Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.10.2307/1426114CrossRefGoogle Scholar
Schmidt, V. (1985). On finiteness and continuity of shot noise processes. Optimization 16, 921–933.CrossRefGoogle Scholar
Westcott, M. (1976). On the existence of a generalized shot-noise process. In Studies in Probability and Statistics: Papers in Honour of Edwin J. G. Pitman, ed. E. J. Williams. Amsterdam: North-Holland, pp. 73–88.Google Scholar
Yakovlev, A. and Yanev, N. (2007). Age and residual lifetime distributions for branching processes. Statist. Prob. Lett. 77, 503–513.CrossRefGoogle Scholar