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Functional laws for trimmed Lévy processes. (English) Zbl 1400.60067

Summary: Two different ways of trimming the sample path of a stochastic process in \(\mathbb D[0, 1]\): global (‘trim as you go’) trimming and record time (‘lookback’) trimming are analysed to find conditions for the corresponding operators to be continuous with respect to the (strong) \(J_{1}\)-topology. A key condition is that there should be no ties among the largest ordered jumps of the limit process. As an application of the theory, via the continuous mapping theorem, we prove limit theorems for trimmed Lévy processes, using the functional convergence of the underlying process to a stable process. The results are applied to a reinsurance ruin time problem.

MSC:

60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G70 Extreme value theory; extremal stochastic processes
60F17 Functional limit theorems; invariance principles
91B30 Risk theory, insurance (MSC2010)
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References:

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