A Gamma process is a Lévy process with independent Gamma increments. Often written as Γ(t;γ,λ), it is a pure-jump increasing Levy process with intensity measure ν(x) = γx ^{− 1}exp( − λx), for positive x. Thus jumps whose size lies in the interval [x,x + dx] occur as a Poisson process with intensity ν(x)dx.The parameter γ controls the rate of jump arrivals and the scaling parameter λ inversely controls the jump size. In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that has stationary independent increments -- this phrase will be explained below. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... A simple Poisson process, named after the French mathematician Siméon-Denis Poisson (1781 - 1840), is a stochastic process which is defined in terms of the occurrences of events. ...

The marginal distribution of a Gamma process at time t, is a Gamma distribution with mean γt / λ and variance γt / λ². In probability theory and statistics, the gamma distribution is a continuous probability distribution. ...

The Gamma process is sometimes also parameterised in terms of the mean (μ) and variance (v) per unit time, which is equivalent to γ = μ² / v and λ = μ / v.

Some basic properties of the Gamma process are:

(scaling)

(adding independent processes)

(moments), where Γ(z) is the Gamma function.

(moment generating function)

, for any Gamma process X(t)

A good reference for Levy processes, including the Gamma process, is Lévy Processes and Stochastic Calculus by David Applebaum, CUP 2004, ISBN 0-521-83263-2. The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...