Geodesic
A note on convergence to stationarity of random processes with immigration
Teoriâ slučajnyh processov, Tome 20 (2015) no. 1, pp. 84-100.

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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.
Keywords: Random marked point process, renewal shot noise process, stationary renewal process, weak convergence in the Skorokhod space, processes with immigration. 
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A. V. Marynych. A note on convergence to stationarity of random processes with immigration. Teoriâ slučajnyh processov, Tome 20 (2015) no. 1, pp. 84-100. http://geodesic.mathdoc.fr/item/THSP_2015_20_1_a5/

[1] Patrick Billingsley, Convergence of probability measures, John Wiley Sons, Inc., New York-London-Sydney, 1968 | MR

[2] Alexander Iksanov, Alexander Marynych, Matthias Meiners, “Asymptotics of random processes with immigration I: scaling limits”, Bernoulli, 2016 (to appear)

[3] Alexander Iksanov, Alexander Marynych, and Matthias Meiners, “Asymptotics of random processes with immigration {II}: convergence to stationarity”, Bernoulli, 2016 (to appear)

[4] Norman Kaplan, “Limit theorems for a {$GI/G/\infty $} queue”, Ann. Probability, 3:5 (1975), 780–789 | DOI | MR | Zbl

[5] Torgny Lindvall, “Weak convergence of probability measures and random functions in the function space {$D(0,\,\infty )$}”, J. Appl. Probability, 10 (1973), 109–121 | DOI | MR | Zbl

[6] Klaus Matthes, Johannes Kerstan, Joseph Mecke, Infinitely divisible point processes, Translated from the German by B. Simon, Wiley Series in Probability and Mathematical Statistics, John Wiley Sons, Chichester-New York-Brisbane, 1978 | MR | Zbl

[7] Mark M. Meerschaert, Peter Straka, “Semi-Markov approach to continuous time random walk limit processes”, Ann. Probab., 42:4 (2014), 1699–1723 | DOI | MR | Zbl

[8] Sidney Resnick, Adventures in stochastic processes, Birkhäuser Boston, Inc., Boston, MA, 1992 | MR | Zbl

[9] Karl Sigman, Stationary marked point processes, Stochastic Modeling Series, Chapman Hall, New York, 1995 | MR | Zbl

[10] Hermann Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications, Springer-Verlag, New York, 2000 | MR | Zbl

[11] Ward Whitt, “Some useful functions for functional limit theorems”, Math. Oper. Res., 5:1 (1980), 67–85 | DOI | MR | Zbl

[12] Ward Whitt, Stochastic-process limits, Springer Series in Operations Research, Springer-Verlag, New York, 2002 | MR | Zbl