Non - linear regression of stable randomvariables

Let (X 1 ; X 2) be an-stable random vector, not necessarily symmetric, with 0 < < 2. The paper investigates the regression E(X 2 jX 1 = x) for all values of. We give conditions for the existence of the conditional moment E(jX 2 j p jX 1 = x) when p , and we obtain an explicit form of the regression E(X 2 jX 1 = x) as a function of x. Although this regression is, in general, not linear, it can be linear even when the vector (X 1 ; X 2) is skewed. We give a necessary and suucient condition for linearity and characterize the asymptotic behavior of the regression as x ! 1. The behavior of the regression functions is also illustrated graphically.