On the class of exponential statistical structures of type B

The article is devoted to the study of exponential statistical structures of type B, which constitute an important subclass of exponential families of probability distributions. This class is characterized by a number of analytical and probabilistic properties that make it a convenient tool for solving both theoretical and applied problems in mathematical statistics. The relevance of this research lies in the need to generalize known classes of distributions and to develop a unified framework for their analysis, which is essential for applications in stochastic modeling, machine learning, and financial mathematics. The paper proposes a formal definition of type B distributions based on the Laplace transform of dominating measures and a system of functional-differential equations describing their structure. Necessary and sufficient conditions for a statistical structure to belong to class B are established, and it is proved that such structures can be represented through a dominating measure with an explicit Laplace transform. The obtained results make it possible to describe a wide range of well-known one-dimensional and multivariate distributions, including the Binomial, Poisson, Normal, Gamma, Polynomial, and Logarithmic distributions, as well as specific cases such as the Borel–Tanner and Random Walk distributions. Particular attention is given to the proof of structural theorems that determine the stability of class B under linear transformations and the addition of independent random vectors. It is shown that if a distribution belongs to class B, its linear transformations and sums also belong to this class with the corresponding parameters. Recursive relations for initial and central moments as well as for semi-invariants are obtained, providing an efficient analytical and computational framework for their evaluation. Furthermore, the tails of type B distributions are investigated using the properties of the Laplace transform. As a result, new exponential inequalities for estimating the probabilities of large deviations are derived, which extend classical approaches to the analysis of statistical distributions. The obtained results can be applied in theoretical studies and in practical problems of stochastic modeling.