Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-25T12:59:45.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak convergence of random processes with immigration at random times

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  04 May 2020

Congzao Dong  and
Alexander Iksanov Open the ORCID record for Alexander Iksanov [Opens in a new window]
Congzao Dong*
Affiliation:
Xidian University
Alexander Iksanov*
Affiliation:
Xidian University and Taras Shevchenko National University of Kyiv
*
*Postal address: School of Mathematics and Statistics, Xidian University, 710126 Xi’an, P.R. China.
*Postal address: School of Mathematics and Statistics, Xidian University, 710126 Xi’an, P.R. China.
Get access Rights & Permissions [Opens in a new window]

Abstract

By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

Keywords

Finite-dimensional distributionsrandom process with immigrationweak convergence

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 250 - 265
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. and Hering, H. (1983). Branching Processes. Boston, Birkhäuser.CrossRefGoogle Scholar
Billingsley, P. (1968). Convergence of Probability Measures. New York, Wiley.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Gnedin, A. and Iksanov, A. (2018). On nested infinite occupancy scheme in random environment. Preprint, arXiv:1808.00538.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Applications. New York, Academic Press.Google Scholar
Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Cham, 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. Probab. 23, 87.CrossRefGoogle Scholar
Iksanov, A. and Kabluchko, Z. (2018). Weak convergence of the number of vertices at intermediate levels of random recursive trees. J. Appl. Prob. 55, 1131–1142.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. and Rashytov, B. (2019). A functional limit theorem for general shot noise processes. To appear in J. Appl. Prob. 57.Google Scholar
Marynych, A. and Verovkin, G. (2017). A functional limit theorem for random processes with immigration in the case of heavy tails. Mod. Stoch.: Theory Appl. 4, 93–108.CrossRefGoogle Scholar
Pang, G. and Taqqu, M. S. (2019). Nonstationary self-similar Gaussian processes as scaling limits of power-law shot noise processes and generalizations of fractional Brownian motion. High Frequency 2, 95–112.CrossRefGoogle 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.CrossRefGoogle Scholar