This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis–Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.
The research of Kouji Yano was supported by JSPS KAKENHI grant no.’s JP19H01791 and JP19K21834 and by JSPS Open Partnership Joint Research Projects grant no. JPJSBP120209921. This research was supported by RIMS and by ISM.
"Infinite convolutions of probability measures on Polish semigroups." Probab. Surveys 19 129 - 159, 2022. https://doi.org/10.1214/22-PS6