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

Characterization of a subclass of Tweedie distributions by a property of generalized stability