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 $\{X_k,k=1,2,\dots \}$ 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 $\{X_k,k=1,2,\dots \}$ is called an α- series process.
• The threshold geometric process. A stochastic process $\{Z_n,n=1,2,\dots \}$ is said to be a threshold geometric process (threshold GP), if there exists real numbers $a_i>0,i=1,2,\dots ,k$ and integers $\{1=M_1<M_2<\cdots <M_k<M_{k+1}=\mathrm{\infty }\}$ such that for each $i=1,\dots ,k$, $\{a_i^{n-M_i}{Z}_n,M_i\le n<M_{i+1}\}$ 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 $\{X_k,k=1,2,\dots \}$ 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 $F_k^D(t)=1-\exp \{-\underset{0}{\overset{t}{\int }}{b}_{k}h(a_{k}u)du\}$ 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).