Definition (Poisson Process)

A random process $\{X_{t}:t\in[0,\infty)\}$ is called a Poisson process with parameter $\lambda>0$ if the holding times are iid exponentially distributed i.e. $$S_{i}\stackrel{iid}\sim\mbox{Exp}(\lambda), \ i\in I$$and the jump chain is defined as $$Y_{n}=n, \ \mbox{for }n=0,1,2,\ldots$$

Definition (Little o notation)

Let $f:\mathbb{R}\to\mathbb{R}$ be a function. We say $f(h)=o(h)$ if $$\lim_{h\to0} \frac{f(h)}{h}=0$$i.e. $f(h)$ vanishes faster than $h$.

Definition (Increment)

For a Poisson Process, $\{X_{t}:t\ge0\}$, for $0\le s<t$ we call $$X_{t}-X_{s}$$the increment over $(s,t]$ (i.e. the number of arrivals over $(s,t]$).

Definition (Independent increments)

We say that a continuous-time process, $\{X_{t}:t\ge0\}$, has independent increments if its’ increments over any finite collection of disjoint intervals are independent, (partition process anyway you like, all increments are independent).

Definition (Stationary increments)

We say that a continuous-time process, $\{X_{t}:t\ge0\}$, has stationary increments if the distribution of $$X_{h+s}-X_{s}$$does not depend on $s$ for any $h>0$ (i.e. irregardless of where we take the increment, every increment of size $h$ has an identical distribution).