Normal variance-mean mixture

In probability theory and statistics, a normal variance-mean mixture with mixing probability density $$g$$ is the continuous probability distribution of a random variable $$Y$$ of the form


 * $$Y=\alpha + \beta V+\sigma \sqrt{V}X,$$

where $$\alpha$$, $$\beta$$ and $$\sigma > 0$$ are real numbers, and random variables $$X$$ and $$V$$ are independent, $$X$$ is normally distributed with mean zero and variance one, and $$V$$ is continuously distributed on the positive half-axis with probability density function $$g$$. The conditional distribution of $$Y$$ given $$V$$ is thus a normal distribution with mean $$\alpha + \beta V$$ and variance $$\sigma^2 V$$. A normal variance-mean mixture can be thought of as the distribution of a certain quantity in an inhomogeneous population consisting of many different normal distributed subpopulations. It is the distribution of the position of a Wiener process (Brownian motion) with drift $$\beta$$ and infinitesimal variance $$\sigma^2$$ observed at a random time point independent of the Wiener process and with probability density function $$g$$. An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution.

The probability density function of a normal variance-mean mixture with mixing probability density $$g$$ is


 * $$f(x) = \int_0^\infty \frac{1}{\sqrt{2 \pi \sigma^2 v}} \exp \left( \frac{-(x - \alpha - \beta v)^2}{2 \sigma^2 v} \right) g(v) \, dv$$

and its moment generating function is


 * $$M(s) = \exp(\alpha s) \, M_g \left(\beta s + \frac12 \sigma^2 s^2 \right),$$

where $$M_g$$ is the moment generating function of the probability distribution with density function $$g$$, i.e.


 * $$M_g(s) = E\left(\exp( s V)\right) = \int_0^\infty \exp(s v) g(v) \, dv.$$