Jaap Geluk

Second-order regular variation, convolution and the central limit theorem

Second-order regular variation is a refinement of the concept of regular variation which is useful for studying rates of convergence in extreme value theory and asymptotic normality of tail estimators. For a distribution tail 1 - F which possesses second-order regular variation, we discuss how this property is inherited by 1 - F2 and 1 - F*2. We also discuss the relationship of central limit behavior of tail empirical processes, asymptotic normality of Hills estimator and second-order regular variation.

TAILS OF SUBORDINATED LAWS: THE REGULARLY VARYING CASE

Suppose Xi, I = 1, 2, ... are i.i.d. positive random variables with d.f. F. We assume the tail d.f. to be regularly varying with 0 x) as x --> [infinity] where SN = [Sigma]N1Xi and N,Xi(i >= 1) independent with [Sigma][infinity]n=0P(N = n)xn analytic at x = 1 is studied under an additional smoothness condition on F. As an application we give the asymptotic behaviour of the expected population size of an age-dependent branching process.

An adaptive optimal estimate of the tail index for MA(l) time series

For samples of random variables with a regularly varying tail estimating the tail index has received much attention recently. For the proof of asymptotic normality of the tail index estimator second-order regular variation is needed. In this paper we first supplement earlier results on convolution given by Geluk et al. (Stochastic Process. Appl. 69 (1997) 138-159). Secondly, we propose a simple estimator of the tail index for finite moving average time series. We also give a subsampling procedure in order to estimate the optimal sample fraction in the sense of minimal mean squared error.

On bootstrap sample size in extreme value theory

It has been known for a long time that for bootstrapping the probability distribution of the maximum of a sample consistently, the bootstrap sample size needs to be of smaller order than the original sample size. See Jun Shao and Dongsheng Tu (1995), Ex. 3.9,p. 123. We show that the same is true if we use the bootstrap for estimating an intermediate quantile.

On functions with small differences

Abstract An Abel-Tauber theorem is proved and applied to multiplicative arithmetic functions.

On the domain of attraction of exp(-exp(-x))

A characterization of the domain of attraction of A(x) for maxima is given in terms of conditional moments.