Geometric process

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

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

The GP has been widely applied in reliability engineering

Below are some of its extensions.

• The α- series process. Given a sequence of non-negative random variables:$\{X_{k},k=1,2,\dots \}$, if they are independent and the cdf of ${\frac {X_{k}}{k^{a}}}$ is given by $F(x)$ for $k=1,2,\dots$, where $a$ is a positive constant, then $\{X_{k},k=1,2,\ldots \}$ is called an α- series process.
• The threshold geometric process. A stochastic process $\{Z_{n},n=1,2,\ldots \}$ is said to be a threshold geometric process (threshold GP), if there exists real numbers $a_{i}>0,i=1,2,\ldots ,k$ and integers $\{1=M_{1}<M_{2}<\cdots <M_{k}<M_{k+1}=\infty \}$ such that for each $i=1,\ldots ,k$, $\{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n<M_{i+1}\}$ forms a renewal process.
• The doubly geometric process. Given a sequence of non-negative random variables :$\{X_{k},k=1,2,\dots \}$, if they are independent and the cdf of $X_{k}$ is given by $F(a^{k-1}x^{h(k)})$ for $k=1,2,\dots$, where $a$ is a positive constant and $h(k)$ is a function of $k$ and the parameters in $h(k)$ are estimable, and $h(k)>0$ for natural number $k$, then $\{X_{k},k=1,2,\ldots \}$ is called a doubly geometric process (DGP).
• The semi-geometric process. Given a sequence of non-negative random variables $\{X_{k},k=1,2,\dots \}$, if $P\{X_{k}<x|X_{k-1}=x_{k-1},\dots ,X_{1}=x_{1}\}=P\{X_{k}<x|X_{k-1}=x_{k-1}\}$ and the marginal distribution of $X_{k}$ is given by $P\{X_{k}<x\}=F_{k}(x)(\equiv F(a^{k-1}x))$, where $a$ is a positive constant, then $\{X_{k},k=1,2,\dots \}$ is called a semi-geometric process
• The double ratio geometric process. Given a sequence of non-negative random variables $\{Z_{k}^{D},k=1,2,\dots \}$, if they are independent and the cdf of $Z_{k}^{D}$ is given by $F_{k}^{D}(t)=1-\exp\{-\int _{0}^{t}b_{k}h(a_{k}u)du\}$ for $k=1,2,\dots$, where $a_{k}$ and $b_{k}$ are positive parameters (or ratios) and $a_{1}=b_{1}=1$. We call the stochastic process the double-ratio geometric process (DRGP).