Normal Variance-Mean Mixtures and z Distributions

A survey is given of general properties of normal variance-mean mixtures, including various new results. In particular, it is shown that the class of self-reciprocal normal variance mixtures is rather wide, and some Tauberian results are established from which relations between the tail behaviour of a normal variance-mean mixture and its mixing distribution may be deduced. The generalized hyperbolic distributions and the modulated normal distributions provide examples of normal variance-mean mixtures whose densities can be given in terms of well-known functions, and it is proved that also the z distributions, i.e. the class of distributions generated from the beta distribution through logistic transformation followed by introduction of location and scale parameters, are normal variance-mean mixtures. (The z distributions include the hyperbolic cosine distribution and the logistic distribution.) Some properties of the associated mixing distributions are derived, and the z distributions are shown to be self-decomposable.

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