User:Studentstudent84/Gamma Process

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

In addition to this, the gamma processes parameters ($$\gamma,\lambda$$) can also be measured by its mean ($$\mu$$) and variance ($$\upsilon$$) of the increase per unit time. In this case, $$\gamma=\mu^2/\upsilon$$ and $$\lambda=\mu/\upsilon$$.

Properties
Since we use the Gamma function in these properties, we may write the process at time  as  to eliminate ambiguity.

Some basic properties of the gamma process are:

Applications
The gamma process has many applications throughout mathematics. One of these applications is in queueing theory which details the process of waiting in lines and arrival times.