REMARKS ON THE POSNTVTTY OF DENSITIES OF STABLE LAWS
Mark Ashbaugh Balram S. Rajput Kavi Rama-Murthy Carl Sundberg
Abstract: Let and be a non-empty subset of the
-dimensional Euclidean space. It is shown that if satisfies whenever
with then is a convex cone with vertex at 0. This, in particular,
confirms a conjecture of Port and Vitale [4]. Using this result, an elementary, completely
geometric and unified proof is provided for the following known result concerning, the
positivity properties of densities of -stable laws on Let be
a strictly -stable random vector in with truly -dimensional law and let
and be the density of the law and the spectral measure of
respectively. If and the support of is contained in a half-space, then, for any
if and only if belongs to the interior of the convex cone
generated by support of ; and, in all other cases, for all and