Does asymptotic linearity of the regression extend to stable domains of attraction?

C. D. Hardin, Jr., G. Samorodnitsky, and M. S. Taqqu (1991,Ann. Appl. Probab. 1 582–612) have shown that the regression E[Y | X = x] is typically asymptotically linear when (X, Y) is an α-stable random vector with α < 2. We provide necessary and sufficient conditions for asymptotic linearity of E[ Y | X + δ = z], where (X,Y) is an α-stable random vector and δ is a random variable, independent of (X, Y), such that X + δ is in the domain of normal attraction of X. Asymptotic linearity does not always hold even when E[Y| X = x] is linear. For some distributions of δ, the asymptotic rate of E[Y| X + δ = z] fluctuates.