Asymptotics of random processes with immigration I: scaling limits

TL;DR

This paper studies the asymptotic behavior of a class of random processes with immigration, revealing a variety of limiting processes including Gaussian, stable Lévy motions, and their sums under different distributional assumptions.

Contribution

It provides a comprehensive analysis of the weak convergence of scaled processes with immigration, characterizing a rich class of limit processes based on covariance regular variation and stable law domains.

Findings

Includes Gaussian processes with explicit covariance functions

Identifies fractionally integrated stable Lévy motions as limits

Shows convergence to conditionally Gaussian processes under certain conditions

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 $S_k := \xi_1+\ldots+\xi_k$, $k \in \mathbb{N}_0$ and $Y(t) := \sum_{k\geq 0} X_{k+1}(t-S_k) 1_{\{S_k \leq t\}}$, $t\geq 0$. We call the process $(Y(t))_{t \geq 0}$ 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 L\'evy 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.