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• W2281569708 abstract "Let $(X_1, xi_1), (X_2,xi_2),ldots$ be i.i.d.~copies of a pair $(X,xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,infty)$ and $xi$ is a positive random variable. Define $S_k := xi_1+ldots+xi_k$, $k in mathbb{N}_0$ and $Y(t) := sum_{kgeq 0} X_{k+1}(t-S_k) 1_{{S_k leq t}}$, $tgeq 0$. We call the process $(Y(t))_{t geq 0}$ random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $ttoinfty$. Under the assumptions that the covariance function of $X$ is regularly varying in $(0,infty)times (0,infty)$ in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable L'evy motions and their sums when the law of $xi$ belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of $xi$ belongs to the domain of attraction of a stable law with infinite mean." @default.
• W2281569708 created "2016-06-24" @default.
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• W2281569708 date "2014-05-04" @default.
• W2281569708 modified "2023-09-27" @default.
• W2281569708 title "Asymptotics of random processes with immigration I: scaling limits" @default.
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