Gamma process
# Gamma process

Also known as the (Moran-)Gamma Process, 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 $N(t)$ represents the number of event occurrences from time 0 to time $t$. The gamma distribution has scale parameter $\gamma$ and shape parameter $\lambda$, often written as $\Gamma (\gamma ,\lambda )$. Both $\gamma$ and $\lambda$ must be greater than 0. The gamma process is often written as $\Gamma (t,\gamma ,\lambda )$ where $t$ represents the time from 0. The process is a pure-jump increasing Lévy process with intensity measure $\nu (x)=\gamma x^{-1}\exp(-\lambda x),$ for all positive $x$. Thus jumps whose size lies in the interval $[x,x+dx)$ occur as a Poisson process with intensity $\nu (x)\,dx.$ The parameter $\gamma$ controls the rate of jump arrivals and the scaling parameter $\lambda$ inversely controls the jump size. It is assumed that the process starts from a value 0 at t = 0 meaning $N(0)=0$.

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

## Plain English definition

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.

## Properties

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

Some basic properties of the gamma process are:

### Marginal distribution

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

That is, the probability distribution $f$ of the random variable $X_{t}$ is given by the density

$f(x;t,\gamma ,\lambda )={\frac {\lambda ^{\gamma t}}{\Gamma (\gamma t)}}x^{\gamma t\,-\,1}e^{-\lambda x}.$

### Scaling

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

    $\alpha \Gamma (t;\gamma ,\lambda )\simeq \Gamma (t;\gamma ,\lambda /\alpha )$

### Adding independent processes

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

    $\Gamma (t;\gamma _{1},\lambda )+\Gamma (t;\gamma _{2},\lambda )\simeq \Gamma (t;\gamma _{1}+\gamma _{2},\lambda )$

### Moments

    The moment function helps mathematicians find expected values, variances, skewness, and kurtosis.
    $\operatorname {E} (X_{t}^{n})=\lambda ^{-n}\cdot {\frac {\Gamma (\gamma t+n)}{\Gamma (\gamma t)}},\ \quad n\geq 0,$ where $\Gamma (z)$ is the Gamma function.

### Moment generating function

    The moment generating function is the expected value of $\exp(tX)$ where X is the random variable.
    $\operatorname {E} {\Big\exp(\theta X_{t}){\Big )}=\left(1-{\frac {\theta }{\lambda }}\right)^{-\gamma t},\ \quad \theta <\lambda$

### Correlation

Correlation displays the statistical relationship between any two gamma processes.

    $\operatorname {Corr} (X_{s},X_{t})={\sqrt {\frac {s}{t}}},\ s<t$, for any gamma process $X(t).$

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

## Literature

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