Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy.
Norbert Wiener first proved Wiener's 1/f theorem,[1] see Wiener's theorem. It states that if has absolutely convergent Fourier series and is never zero, then its inverse also has an absolutely convergent Fourier series.
Paul Levy generalized Wiener's result,[2] showing that
Let
F(\theta)=
| infty | |
| \sum\limits | |
| k=-infty |
ckeik\theta, \theta\in[0,2\pi]
\|F\|=
| infty | |
| \sum\limits | |
| k=-infty |
|ck|<infty.
The values of
F(\theta)
C
H(t)
C
H[F(\theta)]
The proof can be found in the Zygmund's classic book Trigonometric Series.[3]
Let
H(\theta)=ln(\theta)
F(\theta)=
| infty | |
| \sum\limits | |
| k=0 |
pkeik\theta
| infty | |
| ,(\sum\limits | |
| k=0 |
pk=1
F(\theta)
F(\theta)
H[F(\theta)]=ln\left(
| infty | |
| \sum\limits | |
| k=0 |
pkeik\theta\right)=
| infty | |
| \sum | |
| k=0 |
ckeik\theta,
\|H\|=
| infty | |
| \sum\limits | |
| k=0 |
|ck|<infty.
The statistical application of this example can be found in discrete pseudo compound Poisson distribution[4] and zero-inflated model.
If a discrete r.v.
X
\Pr(X=i)=Pi
i\inN
P(z)=
| infty | |
| \sum\limits | |
| i=0 |
Pizi=\exp
| infty | |
| \left\{\sum\limits | |
| i=1 |
\alphaiλ(zi-1)\right\},z=eik\theta
where
| infty | |
| \sum\limits | |
| i=1 |
\alphai=1
\sum
| infty | |
| \limits | |
| i= 1 |
\left|\alphai\right|<infty
\alphai\in R
λ>0
X
We denote it as
X\simDPCP({\alpha1}λ,{\alpha2}λ, … )