Asymptotics of random processes with immigration II: convergence to stationarity

Abstract

Let $X_1,X_2,\dots$ be random elements of the Skorokhod space $D(\mathbb{R})$ and ${\xi }_1,{\xi }_2,\dots$ positive random variables such that the pairs $(X_1,{\xi }_1),(X_2,{\xi }_2),\dots$ are independent and identically distributed. We call the random process $(Y(t){)}_{t\in \mathbb{R}}$ defined by $Y(t):={\sum }_{k\ge 0}{X}_{k+1}(t-{\xi }_1-\dots -{\xi }_k)1_{\{{\xi }_1+\dots +{\xi }_k\le t\}}$, $t\in \mathbb{R}$ 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.

arXiv comment20 pages, accepted for publication in Bernoulli