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 W L ( θ ; μ , c ) = ∑ n = − ∞ ∞ c 2 π e − c / 2 ( θ + 2 π n − μ ) ( θ + 2 π n − μ ) 3 / 2 {\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 θ + 2 π n − μ ≤ 0 {\displaystyle \theta +2\pi n-\mu \leq 0} , c {\displaystyle c} is the scale factor and μ {\displaystyle \mu } is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:
f W L ( θ ; μ , c ) = 1 2 π ∑ n = − ∞ ∞ e − i n ( θ − μ ) − c | n | ( 1 − i sgn n ) = 1 2 π ( 1 + 2 ∑ n = 1 ∞ e − c n cos ( n ( θ − μ ) − c n ) ) {\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 z = e i θ {\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:
⟨ z n ⟩ = ∫ Γ e i n θ f W L ( θ ; μ , c ) d θ = e i n μ − c | n | ( 1 − i sgn ( n ) ) . {\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 Γ {\displaystyle \Gamma \,} is some interval of length 2 π {\displaystyle 2\pi } . The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:
… excerpt ends here. Continue reading the full article.
