Asymptotic tail behaviour of phase-type scale mixture distributions

We consider phase-type scale mixture distributions which correspond to distributions of a product of two independent random variables: a phase-type random variable Y and a non-negative but otherwise arbitrary random variable S called the scaling random variable. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction. Particular focus is given to phase-type scale mixture distributions where the scaling random variable S has discrete support - such a class of distributions has been recently used in risk applications to approximate heavy-tailed distributions. Our results are complemented with several examples.