## Bernoulli

• Bernoulli
• Volume 23, Number 2 (2017), 1279-1298.

### Asymptotics of random processes with immigration II: Convergence to stationarity

#### Abstract

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.

#### Article information

Source
Bernoulli Volume 23, Number 2 (2017), 1279-1298.

Dates
First available in Project Euclid: 4 February 2017

https://projecteuclid.org/euclid.bj/1486177399

Digital Object Identifier
doi:10.3150/15-BEJ777

Mathematical Reviews number (MathSciNet)
MR3606766

Zentralblatt MATH identifier
1377.60046

#### Citation

Iksanov, Alexander; Marynych, Alexander; Meiners, Matthias. Asymptotics of random processes with immigration II: Convergence to stationarity. Bernoulli 23 (2017), no. 2, 1279--1298. doi:10.3150/15-BEJ777. https://projecteuclid.org/euclid.bj/1486177399

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