Gamma process

A gamma process is a random process with independent gamma distributed increments. Often written as Γ ( t ; γ , λ ) {displaystyle Gamma (t;gamma ,lambda )} , it is a pure-jump increasing Lévy process with intensity measure ν ( x ) = γ x − 1 exp ⁡ ( − λ x ) , {displaystyle u (x)=gamma x^{-1}exp(-lambda x),} for positive x {displaystyle x} . Thus jumps whose size lies in the interval [ x , x + d x ) {displaystyle [x,x+dx)} occur as a Poisson process with intensity ν ( x ) d x . {displaystyle u (x)dx.} The parameter γ {displaystyle gamma } controls the rate of jump arrivals and the scaling parameter λ {displaystyle lambda } inversely controls the jump size. It is assumed that the process starts from a value 0 at t=0. A gamma process is a random process with independent gamma distributed increments. Often written as Γ ( t ; γ , λ ) {displaystyle Gamma (t;gamma ,lambda )} , it is a pure-jump increasing Lévy process with intensity measure ν ( x ) = γ x − 1 exp ⁡ ( − λ x ) , {displaystyle u (x)=gamma x^{-1}exp(-lambda x),} for positive x {displaystyle x} . Thus jumps whose size lies in the interval [ x , x + d x ) {displaystyle [x,x+dx)} occur as a Poisson process with intensity ν ( x ) d x . {displaystyle u (x)dx.} The parameter γ {displaystyle gamma } controls the rate of jump arrivals and the scaling parameter λ {displaystyle 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 ( μ {displaystyle mu } ) and variance ( v {displaystyle v} ) of the increase per unit time, which is equivalent to γ = μ 2 / v {displaystyle gamma =mu ^{2}/v} and λ = μ / v {displaystyle lambda =mu /v} . Since we use the Gamma function in these properties, we may write the process at time t {displaystyle t} as X t ≡ Γ ( t ; γ , λ ) {displaystyle X_{t}equiv Gamma (t;gamma ,lambda )} to eliminate ambiguity. Some basic properties of the gamma process are: The marginal distribution of a gamma process at time t {displaystyle t} is a gamma distribution with mean γ t / λ {displaystyle gamma t/lambda } and variance γ t / λ 2 . {displaystyle gamma t/lambda ^{2}.} That is, its density f {displaystyle f} is given by Multiplication of a gamma process by a scalar constant α {displaystyle alpha } is again a gamma process with different mean increase rate.