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Convergence to stable laws in the space \(\mathcal D\). (English) Zbl 1326.60039

Summary: We study the convergence of centered and normalized sums of independent and identically distributed random elements of the space \(\mathcal D\) of càdlàg functions, endowed with Skorokhod’s \(J_{1}\) topology, to stable distributions in \(\mathcal D\). Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

MSC:

60F17 Functional limit theorems; invariance principles
60G52 Stable stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K05 Renewal theory
60G70 Extreme value theory; extremal stochastic processes
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References:

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