## Bernoulli

- Bernoulli
- Volume 23, Number 2 (2017), 1233-1278.

### Asymptotics of random processes with immigration I: Scaling limits

Alexander Iksanov, Alexander Marynych, and Matthias Meiners

#### 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.

#### Article information

**Source**

Bernoulli, Volume 23, Number 2 (2017), 1233-1278.

**Dates**

Received: June 2015

First available in Project Euclid: 4 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1486177398

**Digital Object Identifier**

doi:10.3150/15-BEJ776

**Mathematical Reviews number (MathSciNet)**

MR3606765

**Zentralblatt MATH identifier**

06701625

**Keywords**

random process with immigration renewal theory shot noise processes weak convergence of finite-dimensional distributions

#### Citation

Iksanov, Alexander; Marynych, Alexander; Meiners, Matthias. Asymptotics of random processes with immigration I: Scaling limits. Bernoulli 23 (2017), no. 2, 1233--1278. doi:10.3150/15-BEJ776. https://projecteuclid.org/euclid.bj/1486177398