Gamma process

A gamma process is a random process with independent gamma distributed increments. Often written as , it is a purejump 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
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
 Lévy Processes and Stochastic Calculus by David Applebaum, CUP 2004, ISBN 0521832632.
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