Memorandum 1740

Tail behavior of the empirical distribution function of convolutions

W. Albers & W.C.M. Kallenberg

Abstract:
Control charts based on convolutions require study of the tail behavior of the empirical distribution function of convolutions. It is well-known that this empirical distribution function at a fixed argument x is asymptotically normal. The asymptotic normality is extended here to sequences x_n tending to infinity at a suitable rate. At still larger x_n's Poisson limiting distributions come in for the classical empirical distribution function. Surprisingly, this property does not generalize to its convolution counterpart, since for those x_n's it is degenerate at 0 with probability tending to 1. Exact inequalities for the tail behavior are presented as well.

Keywords:   Control chart, exceedance probability, convolution sample quantiles, empirical distribution function, U-statistics, Berry-Esseen bound, tail dependence, asymptotic degeneration, Markov, Chebyshev and Bernstein inequalities

Mathematics Subject Classification:   62E20, 62G30, 62P30

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