- Properties Marginal distribution Scaling Adding independent processes Moments Moment generating function Correlation
- References
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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 
.
Properties
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:{{citation needed|date=February 2012}}
Marginal distribution
The marginal distribution of a gamma process at time is a gamma distribution with mean 
and variance 
That is, its density 
is given by 
Scaling
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.
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|0-521-83263-2}}.
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