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A REMARK ON POSITIVE ISOTROPIC RANDOM VECTORS Alvaro Arias and Alexander Koldobsky
 

Summary: A REMARK ON POSITIVE ISOTROPIC RANDOM VECTORS
Alvaro Arias and Alexander Koldobsky
Abstract. A random vector X = (X1 : : : Xn) is positive isotropic if P (X1 > 0) >
0, its coordinates are non-negative and identically distributed random variables, and
there exists a function c : R
n
+ ! R+ so that, for every a 2 R
n
+ n f0g, the random
variables
Pai Xi and c(a)X1 are identically distributed. We study the properties of
the function c( ) and prove that c( ) cannot be a norm unless the coordinates of X
are equal with probability 1.
1. Introduction
It is well-known that for every q 2 (0 1) there exist q-stable random vectors
with non-negative coordinates. The classical example is given by the measure q
on R
n
+ = fx 2 R
n : xi 0 i = 1 ::: ng whose Laplace transform is equal to

  

Source: Arias, Alvaro - Department of Mathematics, University of Denver

 

Collections: Mathematics