Asymptotic Behavior of Conditional Laws and Moments of $\infty$-Stable Random Vectors, with Application to Upcrossing Intensities

Abstract

We derive upper bounds for the conditional moment $\mathbf{E} \{| X|^{\varrho}|Y=y\}$ of a strictly $\alpha$-stable random vector (X,Y) when $\alpha\neq 1$ and $\varrho\leq 2$ and prove weak convergences for the conditional law $(X/u|Y= u)$ as $u \to \infty$ when $\alpha > 1$. As an example of application, we derive a new result in crossing theory for $\alpha$-stable processes.