A counterexample to the Cantelli conjecture through the Skorokhod embedding problem

• Kleptsyn, Victor ^[1] ; Kurtzmann, Aline ^[2]
  1. [1] University of Rennes 1

     University of Rennes 1

     Arrondissement de Rennes, Francia

     
  2. [2] University of Lorraine

     University of Lorraine

     Arrondissement de Nancy, Francia

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 5, 2015, págs. 2250-2281
• Idioma: inglés
• DOI: 10.1214/14-AOP932
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• Resumen

  • In this paper, we construct a counterexample to a question by Cantelli, asking whether there exists a nonconstant positive measurable function φ such that for i.i.d. r.v. X,Y of law N(0,1), the r.v. X+φ(X)⋅Y is also Gaussian.

    This construction is made by finding an unusual solution to the Skorokhod embedding problem (showing that the corresponding Brownian transport, contrary to the Root barrier, is not unique). To find it, we establish some sufficient conditions for the continuity of the Root barrier function.