A counterexample to the Cantelli conjecture through the Skorokhod embedding problem

TL;DR

This paper constructs a counterexample to Cantelli's conjecture by solving an unusual case of the Skorokhod embedding problem, revealing non-uniqueness in Brownian transport solutions.

Contribution

It introduces a novel counterexample to Cantelli's conjecture using a unique solution to the Skorokhod embedding problem, challenging previous assumptions.

Findings

Counterexample to Cantelli's conjecture constructed

Non-uniqueness of Brownian transport shown

Conditions for continuity of Root barrier established

Abstract

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.