Laplace probability distributions and related stochastic processes

In English Skew Laplace distributions, which naturally arise in connection with random summation
and quantile regression settings, offer an attractive and flexible alternative to
the normal (Gaussian) distribution in a variety of settings where the assumptions of
symmetry and short tail are too restrictive. The growing popularity of the Laplacebased
models in recent years is due to their fundamental properties, which include a
sharp peak at the mode, heavier than Gaussian tails, existence of all moments, infinite
divisibility, and, most importantly, random stability and approximation of geometric
sums. Since the latter arise quite naturally, these distributions provide useful models
in diverse areas, such as biology, economics, engineering, finance, geosciences,
and physics. We review fundamental properties of these models, which give insight
into their applicability in these areas, and discuss extensions to time series, stochastic
processes, and random fields.