NECESSARY CONDITIONS FOR THE EXISTENCE OF CONDITIONAL MOMENTS OF STABLE RANDOM-VARIABLES

Let (X(1),X(2)) be a symmetric alpha-stable random vector with 0 < alp ha < 2. Its distribution is characterized by a finite measure Gamma on the unit circle called the spectral measure. It is known that if Gamm a satisfies some integrability condition then the conditional moment E [\X(2)\(p)\X(1) = x] can exist for alpha less than or equal to p < 2 a lpha + 1. The paper shows that this sufficient condition is also neces sary in the cases alpha less than or equal to p < 2 alpha + 1 if eithe r 0 < alpha less than or equal to 1/2 or 1 < alpha less than or equal to 3/2, alpha < p less than or equal to 2 if 1/2 < alpha less than or equal to 1 and alpha < p less than or equal to 4 if 3/2 < alpha < 2. I t also provides a sufficient and necessary condition for the existence of E[\X(2)\(alpha)\X(1) = x] (i.e. p = alpha) for 0 < alpha < 2.