Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$):


1. $$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/2-1}}{2 \pi^{d/2} I\_{d/2-1}(\kappa)} \exp(\kappa \mathbf{x}^{\top} \boldsymbol{\mu}), \qquad (\text{called the Fisher-von Mises-Langevin density}),$$
2. $$\mathbf{x}\mapsto \frac{1}{a(\kappa,A)} \exp(\kappa \mathbf{x}^{\top} \boldsymbol{\mu} + \mathbf{x}^{\top} A \mathbf{x}), \qquad (\text{called the Fisher-Bingham density}),$$
3. $$\mathbf{x}\mapsto \frac{\Gamma(d/2)}{2 \pi^{d/2} M(\frac{1}{2},\frac{d}{2},\kappa)} \exp(\kappa (\mathbf{x}^{\top} \boldsymbol{\mu})^2), \qquad (\text{called the Watson density}),$$


where $\kappa\geq 0$ is a concentration parameter, $\boldsymbol{\mu}\in \mathcal{S}^{d-1}$ is a location parameter, $A$ is a symmetric $d\times d$ matrix, and both $a(\kappa,A)$ and $M(\frac{1}{2},\frac{d}{2},\kappa)$ are the appropriate normalizing constants.


I've seen very few central limit theorems in the literature relating to this setting.
In particular, I found absolutely nothing regarding local limit theorems.
If the parameter $\kappa$ approaches some limit ($0$ or $\infty$), do any of these density functions approach a particular limit density (with a properly normalized argument)?




---


$\textbf{Example:}$ As the intensity parameter $\lambda$ of a Poisson$(\lambda)$ distribution tends to $\infty$, the probability mass function tends to the density of a $\text{Normal}(\lambda,\lambda)$ distribution. Is there any analogous results/conjectures in the context of spherical distributions?



Local limit theorems for circular/spherical distributions
$\textbf{Partial answer:}$ Note that
$$\boldsymbol{x}^{\top} \boldsymbol{\mu} = -\frac{1}{2} (\boldsymbol{x} - \boldsymbol{\mu})^{\top}(\boldsymbol{x} - \boldsymbol{\mu}) + 1$$ since $\boldsymbol{x}^{\top}\boldsymbol{x} = \boldsymbol{\mu}^\top\boldsymbol{\mu} = 1$, and it is possible to show (using the integral representation for the modified Bessel function of the first kind) that, as $\kappa\to \infty$,
$$e^{\kappa} \cdot \frac{(\kappa/2)^{d/2-1}}{2 \pi^{d/2} I\_{d/2-1}(\kappa)} = \frac{1}{(2\pi \kappa^{-1})^{(d-1)/2}} \cdot \frac{1}{\sqrt{2\pi} \kappa^{1/2} e^{-\kappa} I\_{d/2 - 1}(\kappa)} \approx \frac{1}{(2\pi \kappa^{-1})^{(d-1)/2}},$$
so that, as $\kappa\to \infty$,
$$\frac{(\kappa/2)^{d/2-1}}{2 \pi^{d/2} I\_{d/2-1}(\kappa)} \exp(\kappa \mathbf{x}^{\top} \boldsymbol{\mu}) \approx \frac{1}{(2\pi \kappa^{-1})^{(d-1)/2}} \exp\Big(-\frac{1}{2 \kappa^{-1}} (\boldsymbol{x} - \boldsymbol{\mu})^{\top}(\boldsymbol{x} - \boldsymbol{\mu})\Big).$$
This heuristic solves the question for 1.


$\textbf{Note:}$ The $d-1$ instead of $d$ comes from the restriction to the $\mathcal{S}^{d-1}$ sphere.


