# Gamma processes

### Sound more exotic than Brownian motion but really aren't

A simple univariate Lévy process of a particular structure. (monotonic increasing, real-valued.)

Notes to myself on its care and feeding.

We write it as $$\Gamma (t;\alpha ,\lambda )$$ to facilitate copy-pasting to and from Wikipedia, because I’m editing that article concurrently with this. According to that same source, a Gamma process is a pure jump process with jump intensity

$$\nu (x)=\alpha x^{-1}\exp(-\lambda x).$$

That is, the Poisson rate, with respect to “time” $$t$$, of jumps whose size is in $$[x, x+dx)$$, is $$\nu(x)dx.$$ This only makes sense on the first reading if you are already familiar with Lévy processes, or terribly clever.

The marginal distribution of the an increment of duration $$t$$ is given by the Gamma distribution

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

This is the shape-rate parameterisation, with rate $$\lambda$$ and shape $$\alpha.$$

Note that if $$\alpha t=1,$$ then $$\Gamma (t;\alpha ,\lambda )\sim \operatorname{Exp}(\lambda).$$