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 $\{X_{k},k=1,2,\cdots \}$ 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 $\{X_{k},k=1,2,\cdots \}$ is called an α- series process.
• The threshold geometric process . A stochastic process $\{Z_{n},n=1,2,...\}$ is said to be a threshold geometric process (threshold GP), if there exists real numbers ${a_{i}>0,i=1,2,...,k}$ and integers $\{1=M_{1}<M_{2}<...<M_{k}<M_{k+1}=\infty \}$ such that for each $i=1,...,k$, $\{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n<M_{i+1}\}$ 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 $\{X_{k},k=1,2,\cdots \}$ is called a doubly geometric process (DGP).