  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 $\mathbb{R}^{1}$.
