A note on convergence to stationarity of random processes with immigration

Abstract

Let $X_1,X_2,\dots$ be random elements of the Skorokhod space $D(\mathbb{R})$ and ${\xi }_1,{\xi }_2,\dots$ positive random variables such that the pairs $(X_1,{\xi }_1),(X_2,{\xi }_2),\dots$ are independent and identically distributed. The random process $Y(t):={\sum }_{k\ge 0}{X}_{k+1}(t-{\xi }_1-\dots -{\xi }_k)1_{\{{\xi }_1+\dots +{\xi }_k\le t\}}$, $t\in \mathbb{R}$, 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 Iksanov, Marynych and Meiners (2015+).

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• A functional limit theorem for random processes with immigration in the case of heavy tails