Conditional moments and linear regression for stable random variables

Jointly [alpha]-stable random variables with index 0 < [alpha] < 2 have only finite moments of order less than [alpha], but their conditional moments can be higher than [alpha]. We provide conditions for this to happen and use the existence of the conditional moments to study the regression E(X2X1=x). We show that if (X1, X2) is a symmetric [alpha]-stable random vector, then under appropriate conditions, the regression is well-defined even when [alpha] [less-than-or-equals, slant] 1 and is linear in x. The results are applied to different classes of symmetric [alpha]-stable processes.