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