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
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
