Английская Википедия:Geometric process

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Шаблон:Technical 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 :<math> \{X_k,k=1,2, \dots\} </math>, if they are independent and the cdf of <math> X_k </math> is given by <math>F(a^{k-1}x)</math> for <math> k=1,2, \dots </math>, where <math>a </math> is a positive constant, then <math>\{X_k,k=1,2,\ldots\}</math> 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:<math> \{X_k,k=1,2, \dots\} </math>, if they are independent and the cdf of <math> \frac{X_k}{k^a} </math> is given by <math>F(x)</math> for <math> k=1,2, \dots </math>, where <math>a </math> is a positive constant, then <math>\{X_k,k=1,2,\ldots\}</math> is called an α- series process.
• The threshold geometric process. A stochastic process <math>\{Z_n, n = 1,2, \ldots\}</math> is said to be a threshold geometric process (threshold GP), if there exists real numbers <math>a_i > 0, i = 1,2, \ldots , k</math> and integers <math>\{1 = M_1 < M_2 < \cdots < M_k < M_{k+1} = \infty\}</math> such that for each <math>i = 1, \ldots , k</math>, <math>\{a_i^{n-M_i}Z_n, M_i \le n < M_{i+1}\}</math> forms a renewal process.
• The doubly geometric process. Given a sequence of non-negative random variables :<math> \{X_k,k=1,2, \dots\} </math>, if they are independent and the cdf of <math> X_k </math> is given by <math>F(a^{k-1}x^{h(k)})</math> for <math> k=1,2, \dots </math>, where <math>a </math> is a positive constant and <math>h(k)</math> is a function of <math>k </math> and the parameters in <math>h(k)</math> are estimable, and <math>h(k)>0</math> for natural number <math>k</math>, then <math>\{X_k,k=1,2,\ldots\}</math> is called a doubly geometric process (DGP).
• The semi-geometric process. Given a sequence of non-negative random variables <math> \{X_k, k=1,2,\dots\} </math>, if <math> 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}\} </math> and the marginal distribution of <math> X_k </math> is given by <math> P\{X_k < x\}=F_k (x)(\equiv F(a^{k-1} x)) </math>, where <math> a </math> is a positive constant, then <math>\{X_k, k=1,2,\dots\}</math> is called a semi-geometric process
• The double ratio geometric process. Given a sequence of non-negative random variables <math>\{Z_k^D,k=1,2, \dots\}</math>, if they are independent and the cdf of <math> Z_k^D </math> is given by <math>F_k^D(t)=1-\exp\{-\int_0^{t} b_k h(a_k u) du\}</math> for <math>k=1,2, \dots</math>, where <math> a_k</math> and <math> b_k </math> are positive parameters (or ratios) and <math> a_1=b_1=1</math>. We call the stochastic process the double-ratio geometric process (DRGP).