Generalized exponentially tilted alpha-stable distributions

Tilted stable distributions play a central role in sampling methods and in simulation of rare events. Specifically, this work is referred to a special class of stable distributions: the Exponentially and the Erlang tilted alpha-stable distribution. Relying on this type of distributions, we are able to show, through the use of the Laplace transform, that the Erlang tilted alpha-stable distribution is equal in distribution to the convolution of two independent random variables: one distributed according to an Exponentially tilted alpha-stable distribution and the other according to a mixture of compound Gamma distribution. A matlab code is provided in order to show a proposed sampler for the Gamma tilted alpha-stable distribution, a more general class containing the Erlang tilted one.