# Geometric process

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 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} such that for each $i=1,\ldots ,k$ , $\{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n forms a renewal process.
• The doubly geometric process. Given a sequence of non-negative random variables :$\{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} and the marginal distribution of $X_{k}$ is given by $P\{X_{k} , where $a$ is a positive constant, then $\{X_{k},k=1,2,\dots \}$ is called a semi-geometric process