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A note on convergence to stationarity of random processes with immigration. (English) Zbl 1363.60017
Summary: Let \(X_1,X_2,\dots\) be random elements of the Skorokhod space \(D(\mathbb R)\) and \(\xi_1,\xi_2,\dots\) be positive random variables such that the pairs \((X_1,\xi_1), (X_2,\xi_2),\dots\) are independent and identically distributed. The random process \(Y(t) := \sum_{k\geq0} X_{k +1}(t -\xi_1-\cdots-\xi_k){\mathbb I}_{\xi_1 +\cdots+\xi_k\leq t}\), \(t\in\mathbb R\), 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.

60F05 Central limit and other weak theorems
60K05 Renewal theory