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# Asymptotics of random processes with immigration II: convergence to stationarity

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### Abstract

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.

### Most cited references10

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### Stability of perpetuities

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### Weak convergence of probability measures and random functions in the function space D[0,∞)

(1973)
This paper extends the theory of weak convergence of probability measures and random functions in the function space D[0,1] to the case D [0,∞), elaborating ideas of C. Stone and W. Whitt. 7)[0,∞) is a suitable space for the analysis of many processes appearing in applied probability.
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### Author and article information

###### Journal
27 November 2013
2015-10-09
###### Article
1311.6923