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 :, if they are independent and the cdf of is given by for, where is a positive constant, then is called a geometric process.
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:, if they are independent and the cdf of is given by for, where is a positive constant, then is called an α- series process.
  • The threshold geometric process. A stochastic process is said to be a threshold geometric process, if there exists real numbers and integers such that for each, forms a renewal process.
  • The doubly geometric process. Given a sequence of non-negative random variables :, if they are independent and the cdf of is given by for, where is a positive constant and is a function of and the parameters in are estimable, and for natural number, then is called a doubly geometric process.
  • The semi-geometric process. Given a sequence of non-negative random variables, if and the marginal distribution of is given by, where is a positive constant, then is called a semi-geometric process
  • The double ratio geometric process. Given a sequence of non-negative random variables, if they are independent and the cdf of is given by for, where and are positive parameters and. We call the stochastic process the double-ratio geometric process.