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

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 $Y(u)={\sum }_{k\ge 0}{X}_{k+1}(u-{\xi }_1-\cdots -{\xi }_k)1{\mathrm{l}}_{\{{\xi }_1+\cdots +{\xi }_k\le u\}}$. We obtain a functional limit theorem for the process $(Y(ut){)}_{u\ge 0}$, 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 $(Y(ut){)}_{u\ge 0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty )$ to a fractionally integrated inverse stable subordinator.

Published at http://dx.doi.org/10.15559/17-VMSTA76 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

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• Functional limit theorems for the number of busy servers in a $G/G/\infty$ queue