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Also known as the(Moran-)Gamma Process,^[1] thegamma process is a random process studied inmathematics,statistics,probability theory, andstochastics. The gamma process is astochastic or random process consisting of independently distributedgamma distributions where${\displaystyle N(t)}$ represents the number of event occurrences from time 0 to time${\displaystyle t}$. Thegamma distribution has shape parameter${\displaystyle \gamma }$ and rate parameter${\displaystyle \lambda }$, often written as${\displaystyle \mathrm{\Gamma }(\gamma ,\lambda )}$.^[1] Both${\displaystyle \gamma }$ and${\displaystyle \lambda }$ must be greater than 0. Thegamma process is often written as${\displaystyle \mathrm{\Gamma }(t,\gamma ,\lambda )}$ where${\displaystyle t}$ represents the time from 0. The process is a pure-jumpincreasingLévy process with intensity measure${\displaystyle \nu (x)=\gamma x^{-1}\exp (-\lambda x),}$ for all positive${\displaystyle x}$. Thus jumps whose size lies in the interval${\displaystyle [x,x+dx)}$ occur as aPoisson process with intensity${\displaystyle \nu (x)dx.}$ The parameter${\displaystyle \gamma }$ controls the rate of jump arrivals and the scaling parameter${\displaystyle \lambda }$ inversely controls the jump size. It is assumed that the process starts from a value 0 at t = 0 meaning${\displaystyle N(0)=0}$.  

The gamma process is sometimes also parameterised in terms of the mean (${\displaystyle \mu }$) and variance (${\displaystyle v}$) of the increase per unit time, which is equivalent to${\displaystyle \gamma ={\mu }^2/v}$ and${\displaystyle \lambda =\mu /v}$.

Thegamma process is a process which measures the number ofoccurrences of independentgamma-distributed variables over a span oftime. 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 itsshape parameter is larger than the blue shape parameter.

We use theGamma function in these properties, so the reader should distinguish between${\displaystyle \mathrm{\Gamma }(\cdot )}$ (the Gamma function) and${\displaystyle \mathrm{\Gamma }(t;\gamma ,\lambda )}$ (the Gamma process). We will sometimes abbreviate the process as${\displaystyle X_t\equiv \mathrm{\Gamma }(t;\gamma ,\lambda )}$.

Some basic properties of the gamma process are:^{[citation needed]}

Themarginal distribution of a gamma process at time${\displaystyle t}$ is agamma distribution with mean${\displaystyle \gamma t/\lambda }$ and variance${\displaystyle \gamma t/{\lambda }^2.}$

That is, the probability distribution${\displaystyle f}$ of the random variable${\displaystyle X_t}$ is given by the density$${\displaystyle f(x;t,\gamma ,\lambda )=\frac{{\lambda }^{\gamma t}}{\mathrm{\Gamma }(\gamma t)}{x}^{\gamma t-1}{e}^{-\lambda x}.}$$

Multiplication of a gamma process by a scalar constant${\displaystyle \alpha }$ is again a gamma process with different mean increase rate.

${\displaystyle \alpha \mathrm{\Gamma }(t;\gamma ,\lambda )\simeq \mathrm{\Gamma }(t;\gamma ,\lambda /\alpha )}$

The sum of two independent gamma processes is again a gamma process.

${\displaystyle \mathrm{\Gamma }(t;{\gamma }_1,\lambda )+\mathrm{\Gamma }(t;{\gamma }_2,\lambda )\simeq \mathrm{\Gamma }(t;{\gamma }_1+{\gamma }_2,\lambda )}$

Themoment function helps mathematicians find expected values, variances, skewness, and kurtosis.

${\displaystyle \mathrm{E}(X_t^n)={\lambda }^{-n}\cdot \frac{\mathrm{\Gamma }(\gamma t+n)}{\mathrm{\Gamma }(\gamma t)},n\ge 0,}$ where${\displaystyle \mathrm{\Gamma }(z)}$ is theGamma function.

Themoment generating function is the expected value of${\displaystyle \exp (tX)}$ where X is therandom variable.

Correlation displays the statistical relationship between any two gamma processes.

, for any gamma process${\displaystyle X(t).}$

The gamma process is used as the distribution for random time change in thevariance gamma process.

• Lévy Processes and Stochastic Calculus by David Applebaum, CUP 2004,ISBN 0-521-83263-2.

• Lévy processes

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