Here, by a statistical manifold I mean a $d$-dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. What are there known examples of statistical manifolds which are complete Riemannian manifolds of dimension $d\geq 1$?

Here, by a statistical manifold I mean a $d$-dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. What are some well-studied/interesting examples of statistical manifolds which are complete Riemannian manifolds of dimension $d\geq 1$?

Here, by a statistical manifold I mean a finite dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. Are there known examples of statistical manifolds which are complete Riemannian manifolds?

Here, by a statistical manifold I mean a $d$-dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. What are there known examples of statistical manifolds which are complete Riemannian manifolds of dimension $d\geq 1$?