The Living Thing / Notebooks :

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 \(\Gamma (t;\alpha ,\lambda )\sim \operatorname{Exp}(\lambda).\)