Abstract: Non-Gaussian stable random variables always have infinite variance, but the conditional second moment E[X22|X1] for a jointly α-stable vector (X1,X2) with index 1/2<α<2 may exist when some conditions on the spectral measure are met. Wu and Cambanis (1991) obtained a functional form of the conditional variance Var [X2|X1=x] for symmetric α-stable vectors with 1<α<2. This paper extends their result to the whole range 1/2<α<2 and also provides a formula for the conditional variance in the case where (X1,X2) are skewed and α≠ 1.
Key words and phrases: Stable distributions, bivariate stable distributions, domain of attraction, conditional moments, regression, nonlinear regression.