On shifted convolution powers of a probability measure
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 \(\lim_{n\to\infty}\mu^ n\tilde\mu^ n\).
weakly relatively shift compact sequence
scientific article; zbMATH DE number 126413