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 ${\displaystyle \{1=M_{1}<M_{2}<\cdots <M_{k}<M_{k+1}=\infty \}}$ such that for each ${\displaystyle i=1,\ldots ,k}$, ${\displaystyle \{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n<M_{i+1}\}}$ 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 ${\displaystyle P\{X_{k}<x|X_{k-1}=x_{k-1},\dots ,X_{1}=x_{1}\}=P\{X_{k}<x|X_{k-1}=x_{k-1}\}}$ and the marginal distribution of ${\displaystyle X_{k}}$ is given by ${\displaystyle P\{X_{k}<x\}=F_{k}(x)(\equiv F(a^{k-1}x))}$, where ${\displaystyle a}$ is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\dots \}}$ is called a semi-geometric process
• The double ratio geometric process.^[7] Given a sequence of non-negative random variables ${\displaystyle \{Z_{k}^{D},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle Z_{k}^{D}}$ is given by ${\displaystyle F_{k}^{D}(t)=1-\exp\{-\int _{0}^{t}b_{k}h(a_{k}u)du\}}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a_{k}}$ and ${\displaystyle b_{k}}$ are positive parameters (or ratios) and ${\displaystyle a_{1}=b_{1}=1}$. We call the stochastic process the double-ratio geometric process (DRGP).