Details

The probability density function for the variance-gamma distribution is given by

,

where is the modified Bessel function of the second kind. It has mean and variance .

As , the probability density decays exponentially like . This is intermediate between the behavior of the normal distribution, which decays more rapidly (like ), and the more extreme "fat tail" behavior of power-law distributions. For this reason, it is sometimes referred to as a "semi-heavy tailed" distribution.

Snapshot 1: as increases, the distribution becomes more rounded around its peak value

Snapshot 2: for non-zero , the distribution becomes skewed, that is, asymmetric

Snapshot 3: as increases, the tails drop off more steeply

See the Wikipedia article on Variance-gamma distribution.