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
The gamma process is often written as
The process is a pure-jump increasing Lévy process with intensity measure
Thus jumps whose size lies in the interval
occur as a Poisson process with intensity
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 (
) of the increase per unit time, which is equivalent to
The gamma process is a process which measures the number of occurrences of independent gamma-distributed variables over a span of time.
This image below displays two different gamma processes on from time 0 until time 4.
The red process has more occurrences in the timeframe compared to the blue process because its shape parameter is larger than the blue shape parameter.
We use the Gamma function in these properties, so the reader should distinguish between
(the Gamma function) and
(the Gamma process).
We will sometimes abbreviate the process as
Some basic properties of the gamma process are:[citation needed] The marginal distribution of a gamma process at time
is a gamma distribution with mean
Multiplication of a gamma process by a scalar constant
is again a gamma process with different mean increase rate.
The sum of two independent gamma processes is again a gamma process.
Correlation displays the statistical relationship between any two gamma processes.
The gamma process is used as the distribution for random time change in the variance gamma process.