$$D$$ -Domains of attraction of stable measures on stratified Lie groupsof attraction of stable measures on stratified Lie groups

Abstract

LetG be a stratified Lie group and (μt)t ⩾ 0 be a continuous convolution semigroup of probability measures onG. A probability measurev is said to belong to the\(D\)-domain of attraction of μ1, if there exists a sequence (a n ) of positive real numbers such that\(\delta _{a_n } v^n \to \mu _1 \) weakly, where δ1 denotes the natural dilation onG. We prove convergence criteria for discrete convolution semigroups. These are used to obtain a simple necessary and sufficient condition for the existence of sucha n if (μt)t ⩾ 0 has no Gaussian component. For the proof we introduce the notion of regularly varying measures onG and develop the necessary theory of regular variation.