A theorem on regularly varying functions in 2 is proved and applied to domains of attraction of stable laws with index 1 [less-than-or-equals, slant] [alpha] [less-than-or-equals, slant] 2. We also present a theory of [Pi]-variation in 2. Unlike the situation in 1 the latter is not connected...

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...

Regular variation of the tail of a multivariate probability distribution is implied by regular variation of the density f provided f satisfies a regularity condition. We give a uniformity condition which controls variation of the function f across rays. Our condition is somewhat more flexible...

A general form for regular variation in IR2 is introduced and applied to domains of attraction of stable distribution in IR2 where the components have different indices. The situation in IRd with d > 2 is more complicated but not essentially different. For simplicity this paper is limited to IR2.

Under a second order regular variation condition, rates of convergence of the distribution of bivariate extreme order statistics to its limit distribution are given both in the total variation metric and in the uniform metric.

Tail index estimation depends for its accuracy on a precise choice of the sample fraction, i.e., the number of extreme order statistics on which the estimation is based. A complete solution to the sample fraction selection is given by means of a two-step subsample bootstrap method. This method...

An estimator is proposed for the index a of a stable distribution (0 # [alpha] [less-than-or-equals, slant] 2) and the asymmetry parameter p (0 [less-than-or-equals, slant] p [less-than-or-equals, slant] 1) based on a sample from a probability distribution in the domain of attraction of the...

In R2 the integral of a regularly varying (RV) function f is regularly varying only if f is monotone. Generalization to R2 of the one-dimensional result on regular variation of the derivative of an RV-function however is straightforward. Applications are given to limit theory for partial sums of...

Let X1, X2,... be independent random variables with distribution functions F1, F2,... respectively, Mn = max {X1,..., Xn} and Ln = min {k [less-than-or-equals, slant] n: Xk = Mn}. Assume that there exist constants an 0 and bn such that (Mn - bn)/an converges in distribution to a non-degenerate...