Counting Process

Description:

• A Stochastic Process $\{N(t),t\ge 0\}$ is said to be a counting process if $N(t)$ represents the total number of events that occur by time $t$, it must satisfy:
  1. $N(t)\ge 0$
  2. $N(t)$ is integer valued
  3. If $s<t$ then $N(s)\le N(t)$, ever increasing
  4. For $s<t,N(t)-N(s)$ equals the number of events that occur in the interval $(s,t]$
• Define a function $o(x)$ and function $f$ is said to be $o(x)$ if ${\lim }_{x\to 0}\frac{f(x)}{x}=0$
  • ex: $f(x)=o(x)=x^2$

Independent increments:

• A counting process possess is independenbt increment if the numbers of events that occur in disjoint time intervals are independent

Stationary increments:

• A counting process possess stationary increments if the distribution of the number of events that occur in any interval of time depends only on the length of the time interval.