Limit theorems for convolutions of probabilities on non Abelian semigroups (Q1114990)

Let S be a locally compact semigroup and P(S) the convolution semigroup of probability measures on S. The paper is concerned with the limiting behaviour of the numbers \(N(\mu,\lambda^ n)=\| \lambda^ n-\mu *\lambda^ n\|\) as n tends to infinity (where \(\mu\),\(\lambda\in P(S))\). Let us state a typical result (Theorem 1): If \(\lambda^ r*\mu =\mu *\lambda^ r\) for some \(r\in {\mathbb{N}}\) and if \(a\cdot (\omega +\mu *\omega)\leq \lambda^ s\) for some \(a\in (0,1]\), \(\omega\in P(S)\) and \(s\in {\mathbb{N}}\), then \(N(\mu,\lambda^ n)=O(1/\sqrt{n}).\) This extends a result of \textit{S. R. Foguel} [Ann. Inst. Henri Poincaré, n. Ser., Sect. B 11, 199-202 (1975; Zbl 0312.60004)]. Moreover, if some power \(\lambda^ r\) commutes with the unit mass \(\epsilon_ x\) in \(x\in S\) then either \(N(\epsilon_ x,\lambda^ n)\equiv 2\) or \(N(\epsilon_ x,\lambda^ n)=O(1/\sqrt{n})\) (zero-two law). Finally the Stam semigroup \(L_ 0(\lambda)=\{x\in S:\) lim N(\(\epsilon_ x,\lambda^ n)=0\}\) of \(\lambda\) is investigated [cf. \textit{H. Heyer}, Probability measures on locally compact groups. (1977; Zbl 0376.60002)]. If \(L_ 0(\lambda)=S\) (i.e. if \((\lambda^ n)_{n\geq 1}\) is left asymptotically equidistributed) the limit behaviour of the sequence \((\lambda^ n)_{n\geq 1}\) is considered.