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The Inversion of a Characteristic Function

One of the interesting and important properties of the transformation that generates the characteristic function is that if it is applied a second time it generates the original function; i.e., the characteristic function of a characteristic function is essentially the original function. More precisely stated:

if p(z) is a probability density function
then its characteristic function is
Φ(ω) = ∫-∞+∞exp(iωz)p(z)dz
however it is also true that
p(z) = (1/2π) ∫-∞+∞exp(-iωz)Φ(ω)dω

Note the factor of (1/2π) and the difference in the sign of the argument of the exponential function, -ωz instead of +ωz.

The problem of the numerical approximation of the above inversion formula is not trivial. The range of the numerical integration must be finite rather than infinite and the integration over a continuous variable must be replaced by summation over a discrete variable. Nevertheless a simple implementation of an algorithm for the approximation of the inversion formula gives reasonable results. Below is the case for the the normal distribution, which happens to have the same functional form as the original distribution. Only the positive axis portion is shown. The distribution is for a normal variable with mean zero and standard deviation equal to 1.0.

The numerical inversion algorithm can be used to find the general shape of a Lévy stable distribution for particular values of the parameters.

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