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A functional limit theorem for random processes with immigration in the case of heavy tails. (English) Zbl 1368.60036

Summary: 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 \(Y(u)=\sum_{k\geq 0}X_{k+1}(u-\xi_1-\cdots -\xi_k)1_{\{ \xi_1+\cdots +\xi_k\leq u\}}\). We obtain a functional limit theorem for the process \((Y(ut))_{u\geq 0}\), as \(t\rightarrow \infty \), 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 \((Y(ut))_{u\geq 0}\), when scaled appropriately, converges weakly in the Skorokhod space \(D(0,\infty)\) to a fractionally integrated inverse stable subordinator.

MSC:

60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60K05 Renewal theory
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