Voigt Profile - Properties

Properties

The Voigt profile is normalized:

\int_{-\infty}^\infty V(x;\sigma,\gamma)\,dx = 1

since it is the convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth) and so the moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The characteristic function for the (centered) Voigt profile will then be the product of the two:

\varphi_f(t;\sigma,\gamma) = E(e^{ixt}) = e^{-\sigma^2t^2/2 - \gamma |t|}.

Since both the normal and the Cauchy distribution are stable distributions, they are closed under convolution and it follows that the Voigt distribution will also be closed under convolution.

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