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

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

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:

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

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:

$⟨z^n⟩=\underset{\mathrm{\Gamma }}{\int }{e}^{in\theta }{f}_{WL}(\theta ;\mu ,c)d\theta =e^{in\mu -\sqrt{c|n|}(1-i\mathrm{sgn}(n))}.$

where $\mathrm{\Gamma }$ 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:

$⟨z⟩=e^{i\mu -\sqrt{c}(1-i)}$

The mean angle is

${\theta }_{\mu }=\mathrm{A}\mathrm{r}\mathrm{g}⟨z⟩=\mu +\sqrt{c}$

and the length of the mean resultant is

$R=|⟨z⟩|=e^{-\sqrt{c}}$

See also

• Wrapped distribution
• Directional statistics

References

• Fisher, N. I. (1996). Statistical Analysis of Circular Data. Cambridge University Press. ISBN 978-0-521-56890-6. https://books.google.com/books?id=IIpeevaNH88C&q=%22circular+variance%22+fisher. Retrieved 2010-02-09.