Inadmissibility and Transience

We discuss the relation between the statistical question of inadmissibility and the probabilistic question of transience. Brown (1971) proved the mathematical link between the admissibility of the mean of a Gaussian distribution and the recurrence of a Brownian motion, which holds for \(\mathbb{R}^{2}\) but not for \(\mathbb{R}^{3}\) in Euclidean space. We extend this result to symmetric, non-Gaussian distributions, without assuming the existence of moments. As an application, we prove that the relation between the inadmissibility of the predictive density of a Cauchy distribution under a uniform prior and the transience of the Cauchy process differs from dimensions \(\mathbb{R}^{1}\) to \(\mathbb{R}^{2}\). We also show that there exists an extreme model that is inadmissible in \(\mathbb{R}^{1}\).