A class of negative-definite functions

Abstract

The associated probability measures here correspond to the symmetric stable laws of Paul Lev see Theorem 1 below. It does not appear to me, however, that the connection between normed vector spaces and symmetric stable laws is obvious from Le'vy's presentation, and I think the connection is an illuminating one. The central idea of the present article is that the terminology LV-norm is apt. It is well known (cf. [2 ]) that if 0 <,B ? 1 then J/P is a proper negative-definite function whenever V is. A fortiori, an LP-norm is an Lr-norm for 1 < r < p. The simplest example of an L2-norm is a function J/ of the form V (x) = I 'x| where EG'. Since sums and positive multiples of negative-definite functions are negative-definite, the LPnorms on G form a cone which contains, in particular, functions of the type