# Local limit theorems for circular/spherical distributions

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?

• Your question is most interesting, though I cannot claim to comprehend it. Have you considered posting such an advanced topic on Mathoverflow? Commented May 14, 2021 at 6:08
• Ok, I just post it on MathOverflow. Commented May 14, 2021 at 7:26

$$\textbf{Partial answer:}$$ 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.
If I remember correctly, they treat the circular case $$S^1$$.