# Directional statistics - Entropy of wrapped normal (Jacobi theta) distribution

1. Jul 2, 2011

### Rap

If p(x) is a probability distribution on the real number line, the "wrapped" distribution around the unit circle is:
$$p_w(\theta)=\sum_{n=-\infty}^\infty p(\theta+2\pi n)$$
which is periodic with period 2π. The (differential) entropy is:
$$H=-\int_\Gamma p_w(\theta)\ln[p_w(\theta)]\,d\theta$$
where $\Gamma$ is any interval of length 2π. I am having trouble finding the entropy for the wrapped normal distribution. For the wrapped normal distribution with zero mean,
$$p_w(\theta)=\frac{\vartheta _3\left(\frac{\theta }{2},e^{-\frac{\sigma ^2}{2}}\right)}{2 \pi }$$
which is in Mathematica notation, $\vartheta _3(\cdot)$ is the Jacobi theta function:
$$\vartheta _3(u,q)=1+2\sum_{n=1}^\infty q^{n^2}\cos(2nu)$$
Is anyone familiar enough with Jacobi theta functions to give some guidance in calculating the entropy? Mathematica is not giving much help.

Last edited: Jul 2, 2011
2. Jul 3, 2011

### bpet

You'd be lucky if a closed form expression exists. Perhaps try numerical integration or an asymptotic expansion?