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We consider a certain convolution semigroup $\Theta$ of probabilitydistributions on the group $\mathbb{R}\times \mathbb{Z}(2)$, where $\mathbb{R}$is the group of real numbers and $\mathbb{Z}(2)$ is the additive group of theintegers modulo 2. This semigroup appeared in connection with the study of acharacterization problem of mathematical statistics on $a$-adic solenoidscontaining an element of order 2. We answer the questions that arise in thestudy of arithmetic of the semigroup $\Theta$. Namely, we describe the class ofinfinitely divisible distributions, the class of indecomposable distributions,and the class of distributions which have no indecomposable factors.

Arithmetic of a certain semigroup of probability distributions on the group $\mathbb{R}\times \mathbb{Z}(2)$