Related Distributions

Symmetric Laplace distributions can be extended in various ways. As we discussed in Chapter 3, skewness may be introduced, leading to asymmetric Laplace laws. Next, one can consider a more general class of distributions whose ch.f.’s are positive powers of Laplace ch.f.’s. These are marginal distributions of the Lévy process $$ \left\{ {Y(t),t \geqslant 0\left. {} \right\}} \right. $$ with independent increments, for which Y(1) has symmetric or asymmetric Laplace distribution. We term such a process the Laplace motion. Finally, one obtains a wider class of limiting distributions consisting of geometric stable laws, by allowing for infinite variance of the components in the geometric compounds (2.2.1). More generally, if the random number of components in the summation (2.2.1) is distributed according to a discrete law v on positive integers, a wider class of v-stable laws is obtained as the limiting distributions. This chapter is devoted to a discussion of all such related distributions and random variables.