Gamma process

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.

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Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
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Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
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