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

${\displaystyle f_{WL}(\theta$ ;\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 ${\displaystyle \theta +2\pi n-\mu \leq 0}$, ${\displaystyle c}$ is the scale factor and ${\displaystyle \mu }$ is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:

${\displaystyle f_{WL}(\theta$ ;\mu ,c)={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{-in(\theta -\mu )-{\sqrt {c|n|}}\,(1-i\operatorname {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 ${\displaystyle z=e^{i\theta }}$ the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:

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

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

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

The mean angle is

${\displaystyle \theta _{\mu }=\mathrm {Arg} \langle z\rangle =\mu +{\sqrt {c}}}$

and the length of the mean resultant is

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

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. Retrieved 2010-02-09.