In mathematics and

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, a gamma process, also known as (Moran-)Gamma subordinator, is a

random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

with

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...

gamma distributed increments. Often written as

\Gamma(t;\gamma,\lambda)

, it is a pure-jump

increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

Lévy process In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...

with intensity measure

\nu(x)=\gamma x^ \exp(-\lambda x),

for positive

x

. Thus jumps whose size lies in the interval

,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 mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

in these properties, we may write the process at time

t

X_t\equiv\Gamma(t;\gamma, \lambda)

to eliminate ambiguity. Some basic properties of the gamma process are:

Marginal distribution

The

marginal distribution In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...

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  x^e^.

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

\operatorname E(X_t^n) = \lambda^ \cdot \frac,\ \quad n\geq 0 ,

where

\Gamma(z)

is the

Moment generating function

\operatorname E\Big(\exp(\theta X_t)\Big) = \left(1- \frac\theta\lambda\right)^,\  \quad \theta<\lambda

Correlation

\operatorname(X_s, X_t) = \sqrt,\ s, for any gamma process X(t) . The gamma process is used as the distribution for random time change in the variance gamma process .

Literature

* ''Lévy Processes and Stochastic Calculus'' by David Applebaum, CUP 2004, .

References

Lévy processes {{probability-stub