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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.
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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

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