Asymptotics of random processes with immigration II: convergence to stationarity

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. We call the random process \((Y(t))_{t \in \mathbb{R}}\) defined by \(Y(t):=\sum_{k \geq 0}X_{k+1}(t-\xi_1-\ldots-\xi_k)1_{\{\xi_1+\ldots+\xi_k\leq t\}}\), \(t\in\mathbb{R}\) 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.