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The influence of sequential extremal processes on the partial sum process. (English) Zbl 1329.62220

Summary: In this paper we derive general upper bounds for the total variation distance between the distributions of a partial sum process in row-wise independent, nonnegative triangular arrays and the sum of a fixed number of corresponding extremal processes. As a special case we receive bounds for the supremum distance between the distribution functions of a partial sum and the sum of corresponding upper extremes which improve upon existing results. The outcome may be interpreted as the influence of large insurance claims on the total loss. Moreover, under an additional infinitesimal condition we also prove explicit bounds for limits of the above quantities. Thereby we give a didactic and elementary proof of the Ferguson-Klass representation of Lévy processes on \(\mathbb{R}_{\geq 0}\) which reflects the influence of extremal processes in insurance.

MSC:

62G30 Order statistics; empirical distribution functions
60G70 Extreme value theory; extremal stochastic processes
60E07 Infinitely divisible distributions; stable distributions
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