A counterexample to the Cantelli conjecture through the Skorokhod embedding problem

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.