A note on convergence to stationarity of random processes with immigration

TL;DR

This paper investigates the long-term behavior of certain random processes with immigration, proving their convergence to a stationary process under specific conditions on the inter-arrival times and the process decay.

Contribution

It extends previous research by establishing weak convergence of processes with immigration to a stationary limit in the Skorokhod space.

Findings

Proves weak convergence of the process as time tends to infinity.

Requires nonlattice distribution with finite mean for inter-arrival times.

Assumes the process decays sufficiently fast for convergence.

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. The random process $Y(t):=\sum_{k \geq 0}X_{k+1}(t-\xi_1-\ldots-\xi_k)1_{\{\xi_1+\ldots+\xi_k\leq t\}}$, $t\in\mathbb{R}$, is called random process with immigration at the epochs of a renewal process. Assuming that the distribution of $\xi_1$ is nonlattice and has finite mean while the process $X_1$ decays sufficiently fast, we prove weak convergence of $(Y(u+t))_{u\in\mathbb{R}}$ as $t\to\infty$ on $D(\mathbb{R})$ endowed with the $J_1$-topology. The present paper continues the line of research initiated in Iksanov, Marynych and Meiners (2015+).