80

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

**Renata Cioczek-Georges
**

Murad S. Taqqu

### Abstract:

*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 .
*