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hal-00666733, version 1

A counter-example to the Cantelli conjecture

Victor Kleptsyn () 1, Aline Kurtzmann () 2

(2012-02-03)

Abstract: In this paper, we construct a counter-example to a question by Cantelli, asking whether there exists a non-constant positive measurable function $\varphi$ such that for i.i.d. r.v. $X,Y$ of law $\mN(0,1)$, the r.v. $X+\varphi(X)\cdot Y$ is also Gaussian. For the construction that we propose, we introduce a new tool, the Brownian mass transport: the mass is transported by Brownian particles that are stopped in a specific way. This transport seems to be interesting by itself, turning out to be related to the Skorokhod and Stefan problems.

  • 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
  • CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
  • 2:  Institut Elie Cartan Nancy (IECN)
  • CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
  • Domain : Mathematics/Probability
    Mathematics/Dynamical Systems
    Mathematics/Analysis of PDEs
  • Keywords : Brownian motion – Stefan problem – mass transport – Skorokhod embedding – Cantelli conjecture
  • Comment : 37 pages
 
  • hal-00666733, version 1
  • http://hal.archives-ouvertes.fr/hal-00666733
  • oai:hal.archives-ouvertes.fr:hal-00666733
  • From: Aline Kurtzmann <>
  • Submitted on: Thursday, 9 February 2012 13:51:12
  • Updated on: Friday, 16 March 2012 09:02:38