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 :[math]\displaystyle{ \{X_k,k=1,2, \dots\} }[/math], if they are independent and the cdf of [math]\displaystyle{ X_k }[/math] is given by [math]\displaystyle{ F(a^{k-1}x) }[/math] for [math]\displaystyle{ k=1,2, \dots }[/math], where [math]\displaystyle{ a }[/math] is a positive constant, then [math]\displaystyle{ \{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]\displaystyle{ \{X_k,k=1,2, \dots\} }[/math], if they are independent and the cdf of [math]\displaystyle{ \frac{X_k}{k^a} }[/math] is given by [math]\displaystyle{ F(x) }[/math] for [math]\displaystyle{ k=1,2, \dots }[/math], where [math]\displaystyle{ a }[/math] is a positive constant, then [math]\displaystyle{ \{X_k,k=1,2,\ldots\} }[/math] is called an α- series process.
• The threshold geometric process. A stochastic process [math]\displaystyle{ \{Z_n, n = 1,2, \ldots\} }[/math] is said to be a threshold geometric process (threshold GP), if there exists real numbers [math]\displaystyle{ a_i \gt 0, i = 1,2, \ldots , k }[/math] and integers [math]\displaystyle{ \{1 = M_1 \lt M_2 \lt \cdots \lt M_k \lt M_{k+1} = \infty\} }[/math] such that for each [math]\displaystyle{ i = 1, \ldots , k }[/math], [math]\displaystyle{ \{a_i^{n-M_i}Z_n, M_i \le n \lt M_{i+1}\} }[/math] forms a renewal process.
• The doubly geometric process. Given a sequence of non-negative random variables :[math]\displaystyle{ \{X_k,k=1,2, \dots\} }[/math], if they are independent and the cdf of [math]\displaystyle{ X_k }[/math] is given by [math]\displaystyle{ F(a^{k-1}x^{h(k)}) }[/math] for [math]\displaystyle{ k=1,2, \dots }[/math], where [math]\displaystyle{ a }[/math] is a positive constant and [math]\displaystyle{ h(k) }[/math] is a function of [math]\displaystyle{ k }[/math] and the parameters in [math]\displaystyle{ h(k) }[/math] are estimable, and [math]\displaystyle{ h(k)\gt 0 }[/math] for natural number [math]\displaystyle{ k }[/math], then [math]\displaystyle{ \{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]\displaystyle{ \{X_k, k=1,2,\dots\} }[/math], if [math]\displaystyle{ P\{X_k \lt x|X_{k-1}=x_{k-1}, \dots , X_1=x_1\} = P\{X_k \lt x|X_{k-1}=x_{k-1}\} }[/math] and the marginal distribution of [math]\displaystyle{ X_k }[/math] is given by [math]\displaystyle{ P\{X_k \lt x\}=F_k (x)(\equiv F(a^{k-1} x)) }[/math], where [math]\displaystyle{ a }[/math] is a positive constant, then [math]\displaystyle{ \{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]\displaystyle{ \{Z_k^D,k=1,2, \dots\} }[/math], if they are independent and the cdf of [math]\displaystyle{ Z_k^D }[/math] is given by [math]\displaystyle{ F_k^D(t)=1-\exp\{-\int_0^{t} b_k h(a_k u) du\} }[/math] for [math]\displaystyle{ k=1,2, \dots }[/math], where [math]\displaystyle{ a_k }[/math] and [math]\displaystyle{ b_k }[/math] are positive parameters (or ratios) and [math]\displaystyle{ a_1=b_1=1 }[/math]. We call the stochastic process the double-ratio geometric process (DRGP).