How do conditional moments of stable vectors depend on the spectral measure? (Q1343595)

A random vector \((X_ 1, X_ 2)\) is said to be symmetric \(\alpha\)- stable, \(0 < \alpha < 2\), if its characteristic function has the form \[ \varphi_{X_ 1, X_ 2} (t,r) = \exp \Bigl(-\int | ts_ 1 + rs_ 2 |^ \alpha \Gamma (ds) \Bigr), \] where \(s=(s_ 1,s_ 2)\) denotes an element of the unit circle \(S_ 2\) in \(\mathbb{R}^ 2\), where \(\Gamma\) is a finite symmetric measure on the Borel subsets of \(S_ 2\) (the spectral measure), and where all integrals are taken over \(S_ 2\). It is known that, for \(p>0\), we have \(E | X_ 2 |^ p < \infty\) if and only if \(p < \alpha\); however, it is possible to have the conditional moments \(C_ p(x) = E(| X_ 2 |^ p \mid X_ 1 = x) < \infty\) a.e. when \(p \geq \alpha\). The main result of this paper -- with a lengthy, technical proof -- gives the following sufficiency condition: Assume that \(\int | s_ 1 |^{-\nu} \Gamma (ds) < \infty\) for some number \(\nu>0\). Then \(C_ p(x)<\infty\) a.e. whenever \(p \leq \alpha + \nu\) and \(p<2 \alpha + 1\). The authors also remark that the result cannot be true for \(p \geq 2 \alpha + 1\), and that, with additional arguments, the symmetry assumption on \(\Gamma\) could be dropped.