## Bernoulli

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

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

Alexander Iksanov, Alexander Marynych, and Matthias Meiners

#### 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**

Received: June 2015

First available in Project Euclid: 4 February 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.3150/15-BEJ777

**Mathematical Reviews number (MathSciNet)**

MR3606766

**Zentralblatt MATH identifier**

1377.60046

**Keywords**

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

#### 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