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September 2015 A counterexample to the Cantelli conjecture through the Skorokhod embedding problem
Victor Kleptsyn, Aline Kurtzmann
Ann. Probab. 43(5): 2250-2281 (September 2015). DOI: 10.1214/14-AOP932


In this paper, we construct a counterexample to a question by Cantelli, asking whether there exists a nonconstant positive measurable function $\varphi$ such that for i.i.d. r.v. $X,Y$ of law $\mathcal{N} (0,1)$, the r.v. $X+\varphi(X)\cdot Y$ 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.


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Victor Kleptsyn. Aline Kurtzmann. "A counterexample to the Cantelli conjecture through the Skorokhod embedding problem." Ann. Probab. 43 (5) 2250 - 2281, September 2015.


Received: 1 June 2013; Revised: 1 March 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1372.60057
MathSciNet: MR3395461
Digital Object Identifier: 10.1214/14-AOP932

Primary: 60G40
Secondary: 60J65

Keywords: Brownian motion , Cantelli conjecture , Gaussian variable , mass transport , Root barrier , Skorokhod embedding , Stefan problem

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 5 • September 2015
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