A note on convergence to stationarity of random processes with immigration

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 $Y(t):=\sum_{k \geq 0}X_{k+1}(t-\xi_1-\ldots-\xi_k){\mathbf{1}}_{\{\xi_1+\ldots+\xi_k\leq t\}}$, $t\in\mathbb{R}$, 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 [2, 3]. Unlike the corresponding result in [3] arbitrary dependence between $X_1$ and $\xi_1$ is allowed.