Necessary Conditions for the Existence of Conditional Moments of Stable Random Variables

Renata Cioczek-Georges
Murad S. Taqqu


Let be a symmetric -stable random vector with . Its distribution is characterized by a finite measure on the unit circle called the spectral measure. It is known that if satisfies some integrability condition then the conditional moment can exist for . The paper shows that this sufficient condition is also necessary in the cases if either or if and if . It also provides a sufficient and necessary condition for the existence of (i.e. ) for .