Estimation of convolution tail behaviour

Several classes of distribution functions are originated by considering distributions whose tailfunctions satisfy special asymptotic relations. A large class sharing this property is provided by the subexponential class S, in which case the asymptotic relation involves tails of convolution powers. In this paper we introduce a statistic which estimates the asymptotic behaviour of convolution tails of a given distribution function and we show that this statistic is asymptotically normal under appropriate conditions. An important tool and result of independent interest is an asymptotic representation in probability for intermediate order statistics.