I. V. Ostrovskii's work on arithmetic of probability laws (Q6642739)

In the setting of the semigroup of probability measures on the real line with the operation of convolution, denoted by \(*\), this paper surveys the background and contributions of the work of I.~V.~Ostrovskii. Of particular interest is the class \(I_0\) of probability measures which do not have indecomposable components, where a measure \(P\) is called indecomposable if, when writing \(P=P_1*P_2\), it follows that either \(P_1\) or \(P_2\) is a degenerate (i.e., a point mass). This class \(I_0\) is known to be strictly smaller than the set of infinitely divisible distributions. Many results on the structure of \(I_0\) are discussed, including the construction of two classes of distributions from \(I_0\) and its relationship with a class of infinitely divisible distributions with discrete spectral Lévy measure studied by Linnik. Decompositions of multidimensional probability distributions are also considered, as are related results on growth and other properties of characteristic functions.