I am looking for density distributions over a circular set (think about $[-\pi, \pi[$ or $\Delta+[0, T[$ in general $\forall T \in \mathbb{R}^{+*}, \Delta \in \mathbb{R}$). Is there a density distribution over such sets, which would be as natural as the Gaussian density?
Here is what I am thinking about:
Gaussian density:
$d : \left\{\begin{array}{l}\mathbb{R}\rightarrow\mathbb{R}^+\\t\mapsto\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^2} \end{array}\right.$
Circular Gaussian density:
$\delta : \left\{\begin{array}{l}[0,T[\rightarrow\mathbb{R}^+\\\tau\mapsto\sum_{-\infty}^{+\infty}{d(\tau+Ti)}\end{array}\right.$
Having tried to compute $\delta(\tau)\ \forall \tau \in [0,T[$, I came accross this problem so I'm stuck for now. But does this remind anyone of anything? Can one perform such winding operations over regular density functions?
It turns out that Mathematica knew the answer: this involves elliptic theta functions:
$\delta(\tau)=\frac{1 + 2 \sum _{i=1}^{+\infty } q^{i^2} \cos (2 i u)}{T}$
with $q=e^{-2 \left(\frac{ \pi \sigma }{T}\right)^2}$ and $u=\frac{\pi (\tau -\mu )}{T}$.
This is not a closed form expression though...