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...

Out of n i.i.d. random vectors in d let X*n be the one closest to the origin. We show that X*n has a nondegenerate limit distribution if and only if the common probability distribution satisfies a condition of multidimensional regular variation. The result is then applied to a problem of density...

Let X1, X2,..., Xn be n independent, identically distributed, non negative random variables and put and Mn = [logical and operator]ni=1 Xi. Let [varrho](X, Y) denote the uniform distanc distributions of random variables X and Y; i.e. . We consider [varrho](Sn, Mn) when P(X1x) is slowly varying...

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.