Local limit theorems for circular/spherical distributions

Question

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)?

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$\textbf{Example:}$ 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?

Answer 1

$\textbf{Partial answer:}$ 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.

Answer 2

You may have a look in the book of Gnedenko and Kolmogorov: Limit distribution for sums of independent random variables.

If I remember correctly, they treat the circular case $S^1$.