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

Alexander Marynych, Glib Verovkin

Abstract


Let $(X_k,\xi_k)_{k\in\mathbb{N}}$ be a sequence of independent 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. The random process with immigration $(Y(u))_{u\in\mathbb{R}}$ is defined as the a.s. finite sum $Y(u)=\sum_{k\geq0}X_{k+1}(u-\xi_1-\cdots-\xi_k) {1\!\!1}_{\{\xi_1+\cdots+\xi_k\leq u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\geq 0}$, as $t\to\infty$, when the law of $\xi$ belongs to the domain of attraction of an $\alpha$-stable law with $\alpha\in(0,1)$, and the process $X$ oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\geq0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty)$ to a fractionally integrated inverse stable subordinator.

Keywords


Fractionally integrated inverse stable subordinators; random process with immigration; shot noise process

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References


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DOI: http://dx.doi.org/10.15559/17-VMSTA76

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