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...