# Geometric process

In probability, statistics and related fields, the geometric process is a counting process, introduced by Lam in 1988.[1] It is defined as

The geometric process. Given a sequence of non-negative random variables :${\displaystyle \{X_{k},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle X_{k}}$ is given by ${\displaystyle F(a^{k-1}x)}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a}$ is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}$ is called a geometric process (GP).

The GP has been widely applied in reliability engineering [2]

Below are some of its extensions.

• The α- series process.[3] Given a sequence of non-negative random variables :${\displaystyle \{X_{k},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle {\frac {X_{k}}{k^{a}}}}$ is given by ${\displaystyle F(x)}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a}$ is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}$ is called an α- series process.
• The threshold geometric process.[4] A stochastic process ${\displaystyle \{Z_{n},n=1,2,\ldots \}}$ is said to be a threshold geometric process (threshold GP), if there exists real numbers ${\displaystyle a_{i}>0,i=1,2,\ldots ,k}$ and integers ${\displaystyle \{1=M_{1} such that for each ${\displaystyle i=1,\ldots ,k}$, ${\displaystyle \{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n forms a renewal process.
• The doubly geometric process.[5] Given a sequence of non-negative random variables :${\displaystyle \{X_{k},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle X_{k}}$ is given by ${\displaystyle F(a^{k-1}x^{h(k)})}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a}$ is a positive constant and ${\displaystyle h(k)}$ is a function of ${\displaystyle k}$ and the parameters in ${\displaystyle h(k)}$ are estimable, and ${\displaystyle h(k)>0}$ for natural number ${\displaystyle k}$, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}$ is called a doubly geometric process (DGP).
• The semi-geometric process.[6] Given a sequence of non-negative random variables ${\displaystyle \{X_{k},k=1,2,\dots \}}$, if ${\displaystyle P\{X_{k} and the marginal distribution of ${\displaystyle X_{k}}$ is given by ${\displaystyle P\{X_{k}, where ${\displaystyle a}$ is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\dots \}}$ is called a semi-geometric process

## References

1. ^ Lam, Y. (1988). Geometric processes and replacement problem. Acta Mathematicae Applicatae Sinica. 4, 366–377
2. ^ Lam, Y. (2007). Geometric process and its applications. World Scientific, Singapore MATH. ISBN 978-981-270-003-2.
3. ^ Braun, W. J., Li, W., & Zhao, Y. Q. (2005). Properties of the geometric and related processes. Naval Research Logistics (NRL), 52(7), 607–616.
4. ^ Chan, J.S., Yu, P.L., Lam, Y. & Ho, A.P. (2006). Modelling SARS data using threshold geometric process. Statistics in Medicine. 25 (11): 1826–1839.
5. ^ Wu, S. (2017). Doubly geometric processes and applications. Journal of the Operational Research Society, 1–13. doi:10.1057/s41274-017-0217-4.
6. ^ Wu, S., Wang, G. (2017). The semi-geometric process and some properties. IMA J Management Mathematics, 1–13.