  We introduce a class of distributions originating from an exponential family
and having a property related to the strict stability property. A
characteristic function representation for this family is obtained and its
properties are investigated. The proposed class relates to stable distributions
and includes Inverse Gaussian distribution and Levy distribution as special
cases. Due to its origin, the proposed distribution has a sufficient statistic.
Besides, it combines stability property at lower scales with an exponential
decay of the distribution's tail and has an additional flexibility due to the
convenient parametrization. Apart from the basic model, certain generalizations
are considered, including the one related to geometric stable distributions.
