When is a probability measure determined by infinitely many projections?
type: Item
instance of: scholarly article
title: When is a probability measure determined by infinitely many projections?
zbMATH Open document ID: 0878.60006
author: Claude Bélisle
author: Jean-Claude Massé
author: Thomas J. Ransford
published in: The Annals of Probability
publication date: 1997-12-14T00:00:00Z
review text: The well-known Cramér-Wold theorem states that a Borel probability measure \(\mu\) on \(\mathbb{R}^d\) is uniquely determined by the totality of its one-dimensional projections. Various conditions are examined under which a probability measure is determined by an infinite family \(\mathcal L\) of \((d-1)\)-dimensional subspaces of \(\mathbb{R}^d\). The question can be seen in the context of the work of \textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 3, 131-142 (1952; Zbl 0048.10804)]. The authors formulate conditions on \(\mathcal L\) and the moment generating function of \(\mu\). In particular, a finite moment generating function of \(\mu\) in a neighborhood of the origin implies the theorem of Rényi. The paper concludes with a brief discussion for discrete probability measures.
reviewed by: Claudia Klüppelberg
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 1018408
zbMATH Keywords: Cramér-Wold theorem
zbMATH Keywords: Borel probability measure
zbMATH Keywords: orthogonal projection
zbMATH Keywords: moment generating function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1024404418
OpenAlex ID: W1985828364
cites work: Some Theorems on Distribution Functions
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cites work: Q4340161
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DOI: 10.1214/AOP/1024404418
description: scientific article; zbMATH DE number 1018408