Representation-free Densities of Ratios and Powers of Stable Variables

Source abstract

We derive representation-free densities of ratios and powers of the positive stable random variable given only its simple Laplace transform. This suffices to obtain Laplace/Stieltjes forward transforms of various {\it r.v.} combinations. We state representation-free densities associated with occupation times (Lamperti~\cite{Lamperti}) and Bessel process excursions (Bertoin~{\it et al.}~\cite{Bertoin}) and their forward transforms. We also readily obtain simple representations of these and other densities using a known infinite series representation of the stable density.