Necessary conditions for the existence of conditional moments of stable random variables

Let (X1, X2) be a symmetric α-stable random vector with 0 < α < 2. Its distribution is characterized by a finite measure Г on the unit circle called the spectral measure. It is known that if Г satisfies some integrability condition then the conditional moment E[|X2|p|X1 = x] can exist for α ⩽ p < 2α + 1. The paper shows that this sufficient condition is also necessary in the cases α < p < 2α + 1 if either 0 < α ⩽ 12 or 1 < α ⩽ 32, α < p ⩽ 2 if 12 < α ⩽ 1 and α < p ⩽ 4 if 32 < α < 2. It also provides a sufficient and necessary condition for the existence of E[|X2|α|X1 = x] (i.e. p = α) for 0 < α < 2.