Nonlinear regression of stable random variables (Q1182686)

A stable distribution \(\mu\) on \(\mathbb{R}^ 2\) is characterized by the requirement that any linear combination \(aX^{(1)}+bX^{(2)},\;a,b>0\), of independent random vectors with distribution \(\mu\) is equal in distribution to some \(cX+y,\;c>0,\;y\in\mathbb{R}^ 2\), where \(X\) is a random vector with distribution \(\mu\) again. A number of results are obtained on the existence of conditional moments and on the form and nature of the regression function \(x\to E[X_ 2\mid X1=x]\) for stable random vectors \(X=(X_ 1,X_ 2)\); in particular, a necessary and sufficient condition for this function to be linear is given.