Gamma process

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This article is about the stochastic process. For the astrophysical nucleosynthesis process, see Gamma process (astrophysics).

­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[edit]

Some basic properties of the gamma process are:[citation needed]

marginal distribution

The marginal distribution of a gamma process at time , is a gamma distribution with mean and variance

scaling
adding independent processes
moments
where is the Gamma function.
moment generating function
correlation
, for any gamma process

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

References[edit]

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