Normal variance-mean mixture

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

where ${\displaystyle \alpha }$, ${\displaystyle \beta }$ and ${\displaystyle \sigma >0}$ are real numbers, and random variables ${\displaystyle X}$ and ${\displaystyle V}$ are independent, ${\displaystyle X}$ is normally distributed with mean zero and variance one, and ${\displaystyle V}$ is continuously distributed on the positive half-axis with probability density function ${\displaystyle g}$. The conditional distribution of ${\displaystyle Y}$ given ${\displaystyle V}$ is thus a normal distribution with mean ${\displaystyle \alpha +\beta V}$ and variance ${\displaystyle \sigma ^{2}V}$. 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 ${\displaystyle \beta }$ and infinitesimal variance ${\displaystyle \sigma ^{2}}$ observed at a random time point independent of the Wiener process and with probability density function ${\displaystyle g}$. An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution.

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