This article is about the stochastic process. For the astrophysical nucleosynthesis process, see Gamma process (astrophysics).
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Also known as the (Moran-)Gamma Process,[1] the gamma process is a random process studied in mathematics, statistics, probability theory, and stochastics. The gamma process is a stochastic or random process consisting of independently distributed gamma distributions where represents the number of event occurrences from time 0 to time . The gamma distribution has shape parameter and rate parameter , often written as .[1] Both and must be greater than 0. The gamma process is often written as where represents the time from 0. The process is a pure-jump increasing Lévy process with intensity measure for all 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 meaning .
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 .
^ abKlenke, Achim, ed. (2008), "The Poisson Point Process", Probability Theory: A Comprehensive Course, London: Springer, pp. 525–542, doi:10.1007/978-1-84800-048-3_24, ISBN 978-1-84800-048-3, retrieved 2023-04-04
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