On the uniqueness of semistable embedding and domain of semistable attraction for probability measures on \(p\)-adic groups
type: Item
instance of: scholarly article
title: On the uniqueness of semistable embedding and domain of semistable attraction for probability measures on \(p\)-adic groups
zbMATH Open document ID: 1151.43001
author: Riddhi Shah
published in: Tokyo Journal of Mathematics
publication date: 2008-07-21T00:00:00Z
review text: Let \(G\) be a locally compact group, \(\Aut(G)\) be the group of topological automorphisms of \(G\). A probability measure \(\mu\) on \(G\) is said to be semistable if there exist \(\tau\in\Aut(G)\) and \(c\in (0, 1)\) such that \(\mu\) can be embedded in a continuous one-parameter semigroup \(\{\mu_t\}_{t\geq 0}\) of probability measures on \(G\) in such a way that \(\mu=\mu_1\) and \(\tau(\mu_t)=\mu_{ct}\) for all \(t\geq 0\). A measure \(\mu\) is said to be normal if \(\mu*\widetilde\mu= \widetilde\mu*\mu\), where \(\widetilde\mu(B)=\mu(B^{-1})\) for all Borel subsets \(B\) of \(G\). The main result of the article is Theorem 2.1: Any normal semistable measure on a \(p\)-adic Lie group has a unique semistable embedding. This implies the uniqueness of semistable embedding of any operator-semistable measure on a finite dimensional \(p\)-adic vector space. Theorem 2.1 can be considered as an analogue for \(p\)-adic Lie groups of the well known results by W. Hazod and E. Siebert about the uniqueness of semistable embedding for some real nilpotent Lie groups. In Section 3, for a unipotent \(p\)-adic algebraic group, the author compares the class of semistable measures and the class of measures whose domain of semistable attraction is nonempty.
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 5302206
zbMATH Keywords: semistable measure
zbMATH Keywords: embedding
zbMATH Keywords: \(p\)-adic Lie group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1202136683
OpenAlex ID: W2005108128
cites work: Q3999894
cites work: Contraction subgroups and semistable measures on p-adic Lie groups
cites work: Q4523950
cites work: Q4258048
cites work: Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups
cites work: Q2755106
cites work: Q3717277
cites work: Q4089001
cites work: Q4057087
cites work: Moments and projections of semistable probability measures on \(p\)-adic vector spaces
cites work: Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on \(p\)-adic vector spaces
cites work: Semistable measures and limit theorems on real and \(p\)-adic groups
cites work: Q4213327
cites work: Limits of commutative triangular systems on real and p-adic groups
cites work: Limits of commutative triangular systems on locally compact groups
cites work: Contractive automorphisms on locally compact groups
cites work: Limit theorems for probability measures on totally disconnected groups
cites work: Q4527707
cites work: The Mautner phenomenon for p-adic Lie groups
DOI: 10.3836/TJM/1202136683
description: scientific article; zbMATH DE number 5302206