Let \(\mathbb{G}\) be a second countable locally compact Abelian topological group. A probability measure \(\mu\) is called weakly infinitely divisible (infinitely divisible in the sense of Parthasarathy [K. R. Parthasarathy, Probability Measures on Metric Spaces. Probability and Mathematical Statistics. A Series of Monographs and Textbooks. New York-London: Academic Press. (1967; Zbl 0153.19101)]), if \(\forall n\in\mathbb{N}\) there exist ’roots’ \(\mu_{(n)}\in M^1(\mathbb{G})\) and \(x_n\in\mathbb{G}\) such that \(\mu = \mu_{(n)}^n \star \delta_{x_n}\). If all elements \(x\in\mathbb{G}\) are divisible the set of ’weakly infinitely divisible’ laws \(I_w\) is just the set of infinitely divisible laws \(I :=\left\{ \mu\in I_w: x_n = 0 \;\forall n\right\}\). In general, \(I_w \supseteq I\). \(I_w\) is the set of limits of row products of independent uniformly infinitesimal arrays. According to Parthasarathy’s representation, any \(\mu\in I_w\) splits as convolution product of an idempotent \(\omega_K\), a shift \(\delta_x\), a Gaussian law \(\gamma\) and a generalized Poissonian law \(e(\eta)\).
The authors investigate three concrete examples of compact Abelian groups \(K\): the torus \(\mathbb{T}\), the \(p\)-adic integers \(\Delta_p\) and the \(p\)-adic solenoid \(S_p\). They characterize \(I_w\) for each of these groups in the following way: There exist subgroups \(\Gamma\) of \(\mathbb{R}^\infty\) (in the examples \(\Delta_p\) and \(S_p\) the group \(\Gamma\) is not locally compact), homomorphisms \(\varphi: \Gamma \to K\) (\(K=\mathbb{T}, \Delta_p, S_p\)), and they construct explicitly random variables taking values in \(\Gamma\), such that their \(\varphi\)-images are distributed according to the afore mentioned idempotents, shifts, Gaussian and Poissonian parts respectively.