The central limit problem on locally compact groups

It is shown that the limit μ of a commutative infinitesimal triangular system Δ on a totally disconnected locally compact groupG is embeddable in a continuous one-parameter convolution semigroup if either (1)G is a compact extension of a closed solvable normal subgroup or (2)G is discrete and Δ is normal or (3)G is a discrete linear group over a field of characteristic zero. For a special triangular system of convolution powers\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {\mu _\nu ^{\kappa _\nu } \to \mu ,\mu _n \to \delta _3 } \right)$$ \end{document}, the above is shown to hold without any of the conditions (1)–(3). For a general locally compact groupG necessary conditions are obtained for the embeddability of a shift of limit μ of Δ; in particular, the conditions are trivially satisfied whenG is abelian. Also, the embedding of a limit of a symmetric system onG is shown to hold under condition (1) as above.