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

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 $Y(u)=\sum_{k\geq0}X_{k+1}(u- \xi_1-\cdots-\xi_k)1\mkern-4.5mu\mathrm{l}_{\{\xi_1+\cdots+\xi_k\leq u\}}$. 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.