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A counterexample to the Cantelli conjecture through the Skorokhod embedding problem

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.
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Submitted on : Thursday, February 9, 2012 - 1:51:12 PM
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Victor A. Kleptsyn, Aline Kurtzmann. A counterexample to the Cantelli conjecture through the Skorokhod embedding problem. Annals of Probability, Institute of Mathematical Statistics, 2015, 43 (5), pp.2250-2281. ⟨10.1214/14-AOP932⟩. ⟨hal-00666733⟩

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