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 $\mathbb{ R}^{ 1}$.
Categories: math.ST math.PR stat.TH