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...

A concept of divisibility is introduced for stochastic difference equations. Infinite divisibility then leads to a continuous time process in which a nested sequence of divisible stochastic difference equations can be embedded.

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...