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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.

##### MSC:
 60F05 Central limit and other weak theorems 60K05 Renewal theory