The influence of sequential extremal processes on the partial sum process

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, non-negative 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 ?_?≥?0 which reflects the influence of extremal processes in insurance.