A note on convergence to stationarity of random processes with immigration

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. 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 $t\to \mathrm{\infty }$ 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+).