Day 7 : Gamma Process

Day 7 : Gamma Process#

The Gamma process, also known as the Moran-Gamma Process is a random process that is studied in probability theory and statistics. It is a fundamental stochastic process with significant applications in fields like finance, engineering, and reliability theory.

Definition#

We let \(G(\alpha,\beta)\) denote the gamma distribution with density
$\(f (x)= \frac{1}{\beta^{\alpha}\Gamma(\alpha)} x^{\alpha -1} e^{-\frac{x}{\beta}}, \qquad x>0,\)\( where \)\Gamma\( is the usual Gamma function, and we refer to \)\alpha\( and \)\beta$ as the shape and scale parameters, respectively.

The gamma process \(X = \{ X(t; \mu,\nu) : t \geq 0\}\) with mean parameter \(\mu\) and variance parameter \(\nu\) is a continuous-time process with stationary, independent gamma increments such that for any \(h > 0\),
$\(X(t + h; \mu, \nu)− X(t; \mu, \nu) \sim G\left( \frac{\mu^2 h}{\nu}, \frac{\nu}{\mu} \right).\)$

🔔 Random Facts 🔔#

  • The Gamma process was formally introduced by mathematician Abdel-Hameed in 1975, in the context of reliability engineering and maintenance modelling. In his paper he called this stochastic process the “Gamma wear process.”

  • Abdel-Hameed seminal work leveraged the mathematical structure of the Gamma process to describe the progressive deterioration of systems or components over time. This marked a significant step in applied probability, as it provided engineers and researchers with a robust framework to model cumulative wear and failure

  • The Gamma process is a pure-jump Lévy process. Lévy processes can have three components: a drift term, a Brownian motion component, and a jump component. In the case of the Gamma process, the Brownian motion and drift components are absent, leaving only the jumps.

  • By definition, the distribution of the increments depends on the length \(h\) of the time increment but not on the time \(t\). Moreover, an increment of length \(h\) has mean \(\mu h\) and variance \(\nu h\).

  • The sum of two independent gamma processes is again a gamma process

  • Gamma process are also used to model the harvested energy for an Energy Harvesting System (EHS). Energy harvesting refers to harnessing energy from the environment or other energy sources and converting it to electrical energy.

More to Read#

  • M. Abdel-Hameed, “A Gamma Wear Process,” in IEEE Transactions on Reliability, vol. R-24, no. 2, pp. 152-153, June 1975, doi: 10.1109/TR.1975.5215123.

  • Avramidis, L’ecuyer and Tremblay, “Efficient simulation of gamma and variance-gamma processes,” Proceedings of the 2003 Winter Simulation Conference, 2003., New Orleans, LA, USA, 2003, pp. 319-326 Vol.1, doi: 10.1109/WSC.2003.1261439.

  • J.M. van Noortwijk, “A survey of the application of gamma processes in maintenance”, Reliability Engineering & System Safety, Volume 94, Issue 1, 2009, Pages 2-21, ISSN 0951-8320, https://doi.org/10.1016/j.ress.2007.03.019.

P.s. If you are curious about probability distributions visit the Advent Calendar 2023 ✨

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