Gamma process


A gamma process is a random process with independent gamma distributed increments. Often written as, it is a pure-jump increasing Lévy process with intensity measure for positive. Thus jumps whose size lies in the interval occur as a Poisson process with intensity The parameter controls the rate of jump arrivals and the scaling parameter inversely controls the jump size. It is assumed that the process starts from a value 0 at t=0.
The gamma process is sometimes also parameterised in terms of the mean and variance of the increase per unit time, which is equivalent to and.

Properties

Since we use the Gamma function in these properties, we may write the process at time as to eliminate ambiguity.
Some basic properties of the gamma process are:

Marginal distribution

The marginal distribution of a gamma process at time is a gamma distribution with mean and variance
That is, its density is given by

Scaling

Multiplication of a gamma process by a scalar constant is again a gamma process with different mean increase rate.

Adding independent processes

The sum of two independent gamma processes is again a gamma process.

Moments

Moment generating function

Correlation

The gamma process is used as the distribution for random time change in the variance gamma process.