Wrapped Lévy distribution

In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle.

Description
The pdf of the wrapped Lévy distribution is



f_{WL}(\theta;\mu,c)=\sum_{n=-\infty}^\infty \sqrt{\frac{c}{2\pi}}\,\frac{e^{-c/2(\theta+2\pi n-\mu)}}{(\theta+2\pi n-\mu)^{3/2}} $$

where the value of the summand is taken to be zero when $$\theta+2\pi n-\mu \le 0$$, $$c$$ is the scale factor and $$\mu$$ is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:



f_{WL}(\theta;\mu,c)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-in(\theta-\mu)-\sqrt{c|n|}\,(1-i\sgn{n})}=\frac{1}{2\pi}\left(1 + 2\sum_{n=1}^\infty e^{-\sqrt{cn}}\cos\left(n(\theta-\mu) - \sqrt{cn}\,\right)\right) $$

In terms of the circular variable $$z=e^{i\theta}$$ the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:


 * $$\langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_{WL}(\theta;\mu,c)\,d\theta = e^{i n \mu-\sqrt{c|n|}\,(1-i\sgn(n))}.$$

where $$\Gamma\,$$ is some interval of length $$2\pi$$. The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:



\langle z \rangle=e^{i\mu-\sqrt{c}(1-i)} $$

The mean angle is



\theta_\mu=\mathrm{Arg}\langle z \rangle = \mu+\sqrt{c} $$

and the length of the mean resultant is

R=|\langle z \rangle| = e^{-\sqrt{c}} $$