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

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