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

$$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/2-1}}{2 \pi^{d/2} I_{d/2-1}(\kappa)} \exp(\kappa \mathbf{x}^{\top} \boldsymbol{\mu}), \qquad (\text{called the Fisher-von Mises-Langevin density}),$$

$$\mathbf{x}\mapsto \frac{1}{a(\kappa,A)} \exp(\kappa \mathbf{x}^{\top} \boldsymbol{\mu} + \mathbf{x}^{\top} A \mathbf{x}), \qquad (\text{called the Fisher-Bingham density}),$$

$$\mathbf{x}\mapsto \frac{\Gamma(d/2)}{2 \pi^{d/2} M(\frac{1}{2},\frac{d}{2},\kappa)} \exp(\kappa (\mathbf{x}^{\top} \boldsymbol{\mu})^2), \qquad (\text{called the Watson density}),$$

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?

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