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

:$f_(\theta;\mu,c)=\sum_^\infty \sqrt\,\frac$

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:

:$f_(\theta;\mu,c)=\frac\sum_^\infty e^=\frac\left(1 + 2\sum_^\infty e^\cos\left(n(\theta-\mu) - \sqrt\,\right)\right)$

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

:$\langle z^n\rangle=\int_\Gamma e^\,f_(\theta;\mu,c)\,d\theta = e^.$

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:

:$\langle z \rangle=e^$

The mean angle is

:$\theta_\mu=\mathrm\langle z \rangle = \mu+\sqrt$

and the length of the mean resultant is

:$R=, \langle z \rangle, = e^$

See also

* Wrapped distribution
* Directional statistics

References

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{{DEFAULTSORT:Wrapped Levy distribution
Continuous distributions
Directional statistics