A COUNTEREXAMPLE TO THE CANTELLI CONJECTURE THROUGH THE SKOROKHOD EMBEDDING PROBLEM

• 成果类型: Article
• 署名作者: Kleptsyn, Victor; Kurtzmann, Aline
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Rennes; Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP932
• 发表日期: 2015
• 页码: 2250-2281
• 关键词: Brownian Motion; stopping times; CONSTRUCTION; EXISTENCE

摘要

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