Gamma process

A gamma process ${\displaystyle \Gamma (t;\gamma ,\lambda )}$ is a random process with independent gamma distributed increments. It is a pure-jump increasing Lévy process with intensity measure ${\displaystyle \nu (x)=\gamma x^{-1}\exp(-\lambda x)}$ for $x > 0$. Thus jumps whose size lies in the interval ${\displaystyle [x,x+dx]}$ occur as a Poisson process with intensity ${\displaystyle \nu (x)dx}$.

Formalization and properties

The main properties are:

• Mean: ${\displaystyle \gamma t/\lambda }$
• Variance ${\displaystyle \gamma t/\lambda ^{2}}$.
• Scaling: ${\displaystyle \alpha \Gamma (t;\gamma ,\lambda )=\Gamma (t;\gamma ,\lambda /\alpha )\,}$
• Adding independent processes: ${\displaystyle \Gamma (t;\gamma _{1},\lambda )+\Gamma (t;\gamma _{2},\lambda )=\Gamma (t;\gamma _{1}+\gamma _{2},\lambda )\,}$
• Moments: ${\displaystyle \mathbb {E} (X_{t}^{n})=\lambda ^{-n}\Gamma (\gamma t+n)/\Gamma (\gamma t),\ \quad n\geq 0}$, where ${\displaystyle \Gamma (z)}$ is the Gamma function.
• Moment generating function: ${\displaystyle \mathbb {E} {\Big (}\exp(\theta X_{t}){\Big )}=(1-\theta /\lambda )^{-\gamma t},\ \quad \theta <\lambda } \mathbb {E} {\Big (}\exp(\theta X_{t}){\Big )}=(1-\theta /\lambda )^{-\gamma t},\ \quad \theta <\lambda
• Correlation: ${\displaystyle \operatorname {Corr} (X_{s},X_{t})={\sqrt {s/t}},\ s<t}$, for any gamma process ${\displaystyle X(t)}$.

See also

Wiener process, Levy process, Markov process, Poisson process