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 , … } {\displaystyle \{X_{k},k=1,2,\dots \}} , if they are independent and the cdf of X k {\displaystyle X_{k}} is given by F ( a k − 1 x ) {\displaystyle F(a^{k-1}x)} for k = 1 , 2 , … {\displaystyle k=1,2,\dots } , where a {\displaystyle a} is a positive constant, then { X k , k = 1 , 2 , … } {\displaystyle \{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 , … } {\displaystyle \{X_{k},k=1,2,\dots \}} , if they are independent and the cdf of X k k a {\displaystyle {\frac {X_{k}}{k^{a}}}} is given by F ( x ) {\displaystyle F(x)} for k = 1 , 2 , … {\displaystyle k=1,2,\dots } , where a {\displaystyle a} is a positive constant, then { X k , k = 1 , 2 , … } {\displaystyle \{X_{k},k=1,2,\ldots \}} is called an α- series process. The threshold geometric process. A stochastic process { Z n , n = 1 , 2 , … } {\displaystyle \{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 , … , k {\displaystyle a_{i}>0,i=1,2,\ldots ,k} and integers { 1 = M 1 < M 2 < ⋯ < M k < M k + 1 = ∞ } {\displaystyle \{1=M_{1} 0 {\displaystyle h(k)>0} for natural number k {\displaystyle k} , then { X k , k = 1 , 2 , … } {\displaystyle \{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 , … } {\displaystyle \{X_{k},k=1,2,\dots \}} , if P { X k < x | X k − 1 = x k − 1 , … , X 1 = x 1 } = P { X k < x | X k − 1 = x k − 1 } {\displaystyle P\{X_{k}

Geometric process

Geometric process