Probabilities on simply connected nilpotent Lie groups

Abstract

This Chapter is devoted to investigations into (semi-) stability phenomena on simply connected nilpotent Lie groups, showing that probabilities on this considerably small class of groups have similar behavior as on vector spaces. It will turn out (in Chapter III) that such investigations on general locally compact groups can be reduced to investigations on simply connected nilpotent Lie groups (for semistable laws under the assumption that G is a Lie group). This Chapter contains the main part of the investigations; one of the key results will provide an identification of continuous convolution semigroups on a group G with continuous convolution semigroups on the tangent space ℒ (G) =: V, a finite-dimensional vector space. Hence the majority of objects studied in Chapter I has a counterpart in the group case; in particular, limits of operator-normalized sums of i.i.d. random variables correspond to automorphismnormalized products and vice versa. (In the sequel we shall call this correspondence the translation procedure, cf. 2.1.3–2.1.7.)