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Limit theorems for sums of heavy-tailed variables with random dependent weights. (English) Zbl 1134.60028

The object of study are various limit theorems for scaling limits of \[ V_n(t)=\sum_{j=1}^{\lfloor nt\rfloor}W_j(U_j-\mu), \] where \((U_j)\) are i.i.d. random variables with regularly varying tail of index \(0<\alpha<2\) and \((W_j)\) are random variables fulfilling certain assumptions that allow dependencies among the \((W_j)\). The structure of dependence is shown to be mild by application of Birkhoff’s ergodic theorem or Kingman’s sub-ergodic theorem, e.g., the assumptions are fulfilled if \((W_j)\) is a strictly stationary sequence with a lighter tail than \((U_j)\). The study is motivated by internet traffic models, where \(U_j\) can be considered as the connection time and \(W_j\) as the traffic intensity of the \(j\)-th connection. The authors first show convergence of the finite-dimensional distributions of \(n^{-\alpha}V_n(t)\) in case \(U_1\) belongs to the normal domain of attraction of an \(\alpha\)-stable law. Thus the centering constant is \(\mu=E[U_1]\) if \(1<\alpha<2\) and \(\mu=0\) if \(0<\alpha\leq1\), where \(U_1\) is assumed symmetric if \(\alpha=1\). The limit is shown to be a scale mixture of an \(\alpha\)-stable Lévy process. Additional conditions for the convergence of finite-dimensional distributions are given in case \(U_1\) belongs to the non-normal domain of attraction of an \(\alpha\)-stable law. Essentially, these conditions are further shown to be sufficient for a functional limit theorem in Skorokhod’s \(J_1\)-topology. The authors finally consider functional limit theorems for \(V_n(t)\) with multivariate (dependent) weights \((W_j)\) in the strong \(M_1\)-topology, which is weaker than the \(J_1\)-topology.

MSC:

60F17 Functional limit theorems; invariance principles
60G52 Stable stochastic processes
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
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