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{{About|the stochastic process|the astrophysical nucleosynthesis process|Gamma process (astrophysics)}}

{{Expert-subject|Mathematics|date=November 2008}}
{{Context|date=March 2010}}

A '''gamma process''' is a [[random process]] with [[Statistical independence|independent]] [[Gamma distribution|gamma distributed]] increments. Often written as <math>\Gamma(t;\gamma,\lambda)</math>, it is a pure-jump [[increasing]] [[Lévy process]] with intensity measure <math>\nu(x)=\gamma x^{-1}\exp(-\lambda x)</math>, for positive <math>x</math>. Thus jumps whose size lies in the interval <math>[x,x+dx]</math> occur as a [[Poisson process]] with intensity <math>\nu(x)dx.</math> The parameter <math>\gamma</math> controls the rate of jump arrivals and the scaling parameter <math>\lambda</math> 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 (<math>\mu</math>) and variance (<math>v</math>) of the increase per unit time, which is equivalent to <math>\gamma = \mu^2/v</math> and <math>\lambda = \mu/v</math>.

==Properties==

Some basic properties of the gamma process are:{{citation needed|date=February 2012}}

;marginal distribution
The [[marginal distribution]] of a gamma process at time <math>t</math>, is a [[gamma distribution]] with mean <math>\gamma t/\lambda</math> and variance <math>\gamma t/\lambda^2.</math>

;scaling
:<math>\alpha\Gamma(t;\gamma,\lambda) = \Gamma(t;\gamma,\lambda/\alpha)\,</math>

;adding independent processes

:<math>\Gamma(t;\gamma_1,\lambda) + \Gamma(t;\gamma_2,\lambda) = \Gamma(t;\gamma_1+\gamma_2,\lambda)\,</math>

;moments

:<math>\mathbb{E}(X_t^n) = \lambda^{-n}\Gamma(\gamma t+n)/\Gamma(\gamma t),\ \quad n\geq 0 ,</math> where <math>\Gamma(z)</math> is the [[Gamma function]].

;moment generating function

:<math>\mathbb{E}\Big(\exp(\theta X_t)\Big) = (1-\theta/\lambda)^{-\gamma t},\ \quad \theta<\lambda</math>
;correlation
:<math>\operatorname{Corr}(X_s, X_t) = \sqrt{s/t},\ s<t</math>, for any gamma process <math>X(t) .</math>

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

== References ==

* ''Lévy Processes and Stochastic Calculus'' by David Applebaum, CUP 2004, ISBN 0-521-83263-2.

{{Stochastic processes}}

[[Category:Stochastic processes]]

{{probability-stub}}

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