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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 .
Since we use the Gamma function in these properties, we may write the process at time as to eliminate ambiguity.
Some basic properties of the gamma process are:
That is, its density is given by
Multiplication of a gamma process by a scalar constant is again a gamma process with different mean increase rate.
Adding independent processes
The sum of two independent gamma processes is again a gamma process.
- where is the Gamma function.
Moment generating function
- , for any gamma process
The gamma process is used as the distribution for random time change in the variance gamma process.
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