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

Abstract

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.