On shifted convolution powers of a probability measure
type: Item
instance of: scholarly article
title: On shifted convolution powers of a probability measure
zbMATH Open document ID: 0760.60009
DOI: 10.1007/BF02571446
author: Peter Eisele
published in: Mathematische Zeitschrift
publication date: 1993-03-07T00:00:00Z
full work available at URL: https://eudml.org/doc/174459
review text: It is shown that for a probability measure \(\mu\) on a locally compact group \(G\) with weakly relatively shift compact sequence \((\mu^ n)_{n\in\mathbb{N}}\) (or equivalently with convergent sequence \((\mu^ n\tilde\mu^ n)_{n\in\mathbb{N}})\) there is a compact subgroup \(H\) of \(G\) such that firstly \(\text{supp}(\mu)\subset Hx=xH\) and secondly \(\mu^ n x^{-n}@>n>>\omega_ H\) for all \(x\in\text{supp}(\mu)\) if \(\mu\) is a normal measure, i.e. if \(\mu\tilde\mu=\tilde\mu\mu\). For the affine group \(\mathbb{A}\) on \(\mathbb{R}\) the measures \(\mu\in{\mathcal P}(\mathbb{A})\) with convergent sequence \((\mu^ n\tilde\mu^ n)_{n\in\mathbb{N}}\) are characterized by a condition on the support and an integrability condition. Here the first point of the above result is false for nonnormal measures while the second point is still true. Under the hypothesis of an idempotent limit \(\omega_ H\) of \((\mu^ n\tilde\mu^ n)_{n\in\mathbb{N}}\), which implies that the subgroup \(H\) contains \(\text{supp}(\mu)\) in a coset, the normality of \(H\) is investigated and characterized. As finally shown, even in the case of a measure \(\mu\) with a support contained in the coset of a compact subgroup, the convergence of \((\mu^ n\tilde\mu^ n)_{n\in\mathbb{N}}\) does neither imply the convergence of \((\mu^ n x^{-n})_{n\in\mathbb{N}}\) for all \(x\in\text{supp}(\mu)\) nor the idempotency of \(\lim_{n\to\infty}\mu^ n\tilde\mu^ n\).
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 126413
zbMATH Keywords: convolution powers
zbMATH Keywords: locally compact group
zbMATH Keywords: weakly relatively shift compact sequence
MaRDI profile type: Publication
cites work: More on limit theorems for iterates of probability measures on semigroups and groups
cites work: On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups
cites work: Q5723435
cites work: Q4153376
cites work: Q5780864
cites work: Bemerkung über Faltungspotenzen eines Wahrscheinlichkeitsmaßes auf kompakten Gruppen
cites work: Limit theorems for probability measures on non-compact groups and semi-groups
cites work: Q4176223
description: scientific article; zbMATH DE number 126413