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 :${\displaystyle \{X_{k},k=1,2,\dots \}}${\displaystyle \{X_{k},k=1,2,\dots \}}, if they are independent and the cdf of ${\displaystyle X_{k}}${\displaystyle X_{k}} is given by ${\displaystyle F(a^{k-1}x)}${\displaystyle F(a^{k-1}x)} for ${\displaystyle k=1,2,\dots }${\displaystyle k=1,2,\dots }, where ${\displaystyle a}${\displaystyle a} is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}${\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:${\displaystyle \{X_{k},k=1,2,\dots \}}${\displaystyle \{X_{k},k=1,2,\dots \}}, if they are independent and the cdf of ${\displaystyle {\frac {X_{k}}{k^{a}}}}${\displaystyle {\frac {X_{k}}{k^{a}}}} is given by ${\displaystyle F(x)}${\displaystyle F(x)} for ${\displaystyle k=1,2,\dots }${\displaystyle k=1,2,\dots }, where ${\displaystyle a}${\displaystyle a} is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}${\displaystyle \{X_{k},k=1,2,\ldots \}} is called an α- series process.
• The threshold geometric process.^[4] A stochastic process ${\displaystyle \{Z_{n},n=1,2,\ldots \}}${\displaystyle \{Z_{n},n=1,2,\ldots \}} is said to be a threshold geometric process (threshold GP), if there exists real numbers ${\displaystyle a_{i}>0,i=1,2,\ldots ,k}${\displaystyle a_{i}>0,i=1,2,\ldots ,k} and integers ${\displaystyle \{1=M_{1}<M_{2}<\cdots <M_{k}<M_{k+1}=\infty \}}${\displaystyle \{1=M_{1}<M_{2}<\cdots <M_{k}<M_{k+1}=\infty \}} such that for each ${\displaystyle i=1,\ldots ,k}${\displaystyle i=1,\ldots ,k}, ${\displaystyle \{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n<M_{i+1}\}}${\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 :${\displaystyle \{X_{k},k=1,2,\dots \}}${\displaystyle \{X_{k},k=1,2,\dots \}}, if they are independent and the cdf of ${\displaystyle X_{k}}${\displaystyle X_{k}} is given by ${\displaystyle F(a^{k-1}x^{h(k)})}${\displaystyle F(a^{k-1}x^{h(k)})} for ${\displaystyle k=1,2,\dots }${\displaystyle k=1,2,\dots }, where ${\displaystyle a}${\displaystyle a} is a positive constant and ${\displaystyle h(k)}${\displaystyle h(k)} is a function of ${\displaystyle k}${\displaystyle k} and the parameters in ${\displaystyle h(k)}${\displaystyle h(k)} are estimable, and ${\displaystyle h(k)>0}${\displaystyle h(k)>0} for natural number ${\displaystyle k}${\displaystyle k}, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}${\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 ${\displaystyle \{X_{k},k=1,2,\dots \}}${\displaystyle \{X_{k},k=1,2,\dots \}}, if ${\displaystyle P\{X_{k}<x|X_{k-1}=x_{k-1},\dots ,X_{1}=x_{1}\}=P\{X_{k}<x|X_{k-1}=x_{k-1}\}}${\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 ${\displaystyle X_{k}}${\displaystyle X_{k}} is given by ${\displaystyle P\{X_{k}<x\}=F_{k}(x)(\equiv F(a^{k-1}x))}${\displaystyle P\{X_{k}<x\}=F_{k}(x)(\equiv F(a^{k-1}x))}, where ${\displaystyle a}${\displaystyle a} is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\dots \}}${\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 ${\displaystyle \{Z_{k}^{D},k=1,2,\dots \}}${\displaystyle \{Z_{k}^{D},k=1,2,\dots \}}, if they are independent and the cdf of ${\displaystyle Z_{k}^{D}}${\displaystyle Z_{k}^{D}} is given by ${\displaystyle F_{k}^{D}(t)=1-\exp\{-\int _{0}^{t}b_{k}h(a_{k}u)du\}}${\displaystyle F_{k}^{D}(t)=1-\exp\{-\int _{0}^{t}b_{k}h(a_{k}u)du\}} for ${\displaystyle k=1,2,\dots }${\displaystyle k=1,2,\dots }, where ${\displaystyle a_{k}}${\displaystyle a_{k}} and ${\displaystyle b_{k}}${\displaystyle b_{k}} are positive parameters (or ratios) and ${\displaystyle a_{1}=b_{1}=1}${\displaystyle a_{1}=b_{1}=1}. We call the stochastic process the double-ratio geometric process (DRGP).

• Point processes
• Markov processes
• Poisson point processes

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