Gamma process

This article is about the stochastic process. For the astrophysical nucleosynthesis process, see Gamma process (astrophysics).

A gamma process is a random process with independent gamma distributed increments. Often written as $\Gamma (t;\gamma ,\lambda )$, it is a pure-jump increasing Lévy process with intensity measure $\nu (x)=\gamma x^{-1}\exp(-\lambda x),$ for 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.

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 $\lambda =\mu /v$.

Properties

Since we use the Gamma function in these properties, we may write the process at time $t$ as $X_{t}\equiv \Gamma (t;\gamma ,\lambda )$ to eliminate ambiguity.

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

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

$\mathbb {E} (X_{t}^{n})=\lambda ^{-n}\Gamma (\gamma t+n)/\Gamma (\gamma t),\ \quad n\geq 0,$ where $\Gamma (z)$ is the Gamma function.

Moment generating function

$\mathbb {E} {\Big (}\exp(\theta X_{t}){\Big )}=(1-\theta /\lambda )^{-\gamma t},\ \quad \theta <\lambda$

Correlation

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

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

References

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