Counting Process

Definition

A Counting Process is a stochastic process $\{N(t),t\ge 0\}$ where $N(t)$ represents the total number of “events” that occur by time $t$.

The process must satisfy:

1. $N(t)\ge 0$
2. $N(t)$ is integer valued
3. If $s<t$, then $N(s)\le N(t)$ (non-decreasing)
4. For $s<t$, $N(t)-N(s)$ equals the number of events that occur in the interval $(s,t]$

Interpretation

Think of a counter that clicks up each time an event happens. It never goes down, only counts upward in whole numbers. Examples: number of customers entering a store, number of goals scored by a player, number of births in a population.

Examples

• Customers in a store: $N(t)$ = number of persons who enter by time $t$
• Births: $N(t)$ = total number of babies born by time $t$
• Soccer scores: $N(t)$ = number of goals a player makes by time $t$

Related

• Poisson Process — the most important counting process
• Independent and Stationary Increments — properties many counting processes possess