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May 2017 Asymptotics of random processes with immigration I: Scaling limits
Alexander Iksanov, Alexander Marynych, Matthias Meiners
Bernoulli 23(2): 1233-1278 (May 2017). DOI: 10.3150/15-BEJ776

Abstract

Let $(X_{1},\xi_{1}),(X_{2},\xi_{2}),\ldots$ be i.i.d. copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. Define $S_{k}:=\xi_{1}+\cdots+\xi_{k}$, $k\in\mathbb{N}_{0}$ and $Y(t):=\sum_{k\geq0}X_{k+1}(t-S_{k})\mathbf{1}_{\{S_{k}\leq t\}}$, $t\geq0$. We call the process $(Y(t))_{t\geq0}$ random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $t\to\infty$. Under the assumptions that the covariance function of $X$ is regularly varying in $(0,\infty)\times(0,\infty)$ in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable Lévy motions and their sums when the law of $\xi$ belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of $\xi$ belongs to the domain of attraction of a stable law with infinite mean.

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Alexander Iksanov. Alexander Marynych. Matthias Meiners. "Asymptotics of random processes with immigration I: Scaling limits." Bernoulli 23 (2) 1233 - 1278, May 2017. https://doi.org/10.3150/15-BEJ776

Information

Received: 1 June 2015; Published: May 2017
First available in Project Euclid: 4 February 2017

zbMATH: 06701625
MathSciNet: MR3606765
Digital Object Identifier: 10.3150/15-BEJ776

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

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Vol.23 • No. 2 • May 2017
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