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May 2017 Asymptotics of random processes with immigration II: Convergence to stationarity
Alexander Iksanov, Alexander Marynych, Matthias Meiners
Bernoulli 23(2): 1279-1298 (May 2017). DOI: 10.3150/15-BEJ777


Let $X_{1},X_{2},\ldots$ be random elements of the Skorokhod space $D(\mathbb{R})$ and $\xi_{1},\xi_{2},\ldots$ positive random variables such that the pairs $(X_{1},\xi_{1}),(X_{2},\xi_{2}),\ldots$ are independent and identically distributed. We call the random process $(Y(t))_{t\in\mathbb{R}}$ defined by $Y(t):=\sum_{k\geq0}X_{k+1}(t-\xi_{1}-\cdots-\xi_{k})\mathbf{1}_{\{\xi_{1}+\cdots+\xi_{k}\leq t\}}$, $t\in\mathbb{R}$ random process with immigration at the epochs of a renewal process. Assuming that $X_{k}$ and $\xi_{k}$ are independent and that the distribution of $\xi_{1}$ is nonlattice and has finite mean we investigate weak convergence of $(Y(t))_{t\in\mathbb{R}}$ as $t\to\infty$ in $D(\mathbb{R})$ endowed with the $J_{1}$-topology. The limits are stationary processes with immigration.


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Alexander Iksanov. Alexander Marynych. Matthias Meiners. "Asymptotics of random processes with immigration II: Convergence to stationarity." Bernoulli 23 (2) 1279 - 1298, May 2017.


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

zbMATH: 1377.60046
MathSciNet: MR3606766
Digital Object Identifier: 10.3150/15-BEJ777

Keywords: random point process , renewal shot noise process , stationary renewal process , weak convergence in the Skorokhod space

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


Vol.23 • No. 2 • May 2017
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