HOW DO CONDITIONAL MOMENTS OF STABLE VECTORS DEPEND ON THE SPECTRAL MEASURE

Let (X(1),X(2)) be an alpha-stable random vector with 0 < alpha < 2, n ot necessarily symmetric. Its distribution is characterized by a finit e measure Gamma on the unit circle called the spectral measure. It is known that if Gamma satisfies some integrability condition then the co nditional moment E[\X(2)\(p)\X(1)] can exist for some values of p grea ter than alpha. This paper provides a sufficient condition on Gamma fo r the existence of the conditional moment E[\xX(2)\(p)\X(1)] involving the maximal range of possible p's, namely p < 2 alpha + 1.