How do conditional moments of stable vectors depend on the spectral measure?

AbstractLet (X1,X2) be an α-stable random vector with 0 < α < 2, not necessarily symmetric. Its distribution is characterized by a finite measure Γ on the unit circle called the spectral measure. It is known that if Γ satisfies some integrability condition then the conditional moment E[∥X2∥p∥X1] can exist for some values of p greater than α. This paper provides a sufficient condition on Γ for the existence of the conditional moment E[∥X2∥p∥X1] involving the maximal range of possible p's, namely p < 2α + 1