Gamma process
In mathematics and probability theory, a gamma process, also known as (Moran-)Gamma subordinator, is a random process with independent gamma distributed increments. Often written as formula_1, it is a pure-jump increasing Lévy process with intensity measure formula_2 for positive formula_3. Thus jumps whose size lies in the interval formula_4 occur as a Poisson process with intensity formula_5 The parameter formula_6 controls the rate of jump arrivals and the scaling parameter formula_7 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 (formula_8) and variance (formula_9) of the increase per unit time, which is equivalent to formula_10 and formula_11. Properties. Since we use the Gamma function in these properties, we may write the process at time formula_12 as formula_13 to eliminate ambiguity. Some basic properties of the gamma process are: Marginal distribution. The marginal distribution of a gamma process at time formula_12 is a gamma distribution with mean formula_15 and variance formula_16 That is, its density formula_17 is given by formula_18 Scaling. Multiplication of a gamma process by a scalar constant formula_19 is again a gamma  process with different mean increase rate. Adding independent processes. The sum of two independent gamma processes is again a gamma process. Correlation. The gamma process is used as the distribution for random time change in the variance gamma process.