Counting Process¶

Counting process is used in scenarios when we want to count the occurrence of a certain event. \(N_{t}\) denotes the number of events till time \(t\) starting from 0. It is assumed that \(N_{0} = 0\). Formal definition is

A random process \(\{N_{t}, t \in [0, \infty)\}\) is said to be a counting process if \(N_{t}\) is the number of events from time \(t=0\) upto time \(t\). For a counting process, we assume

\(N_{0} = 0\)
\(N_{t} \in \{0, 1, 2, \cdots\}\) for all \(t \in [0, \infty)\)
for \(0 \leq s < t, N_{t} - N_{s}\) shows the number of events that occur in the interval \((s,t]\)

Independent Increments¶

We say that a continuous time counting process \(N_{t}\) has independent increments if for all \(0 \leq t_{1} < t_{2} < \cdots < t_{n}\), the random variables

\[ \begin{aligned} N_{t_{2}} - N_{t_{1}}, \;N_{t_{3}} - N_{t_{2}}, \ldots, \;N_{t_{n}} - N_{t_{n-1}} \end{aligned} \]

are independent.

Note that these differences are nothing but the number of arrivals in a given time interval. Thus, we are equivalently saying that the number of arrivals in any two disjoint intervals are independent.

A very simple consequence of this property is:

Suppose we wise to find the probability of 2 arrivals in the interval \((1,2]\) and 3 arrivals in the interval \((3,5]\). Then,

\[ \begin{aligned} P(2 \text{ arrivals in } (1,2] \text{ and } 3 \text{ arrivals in } (3,5]) = P(2 \text{ arrivals in } (1,2]) P(3 \text{ arrivals in } (3,5])) \end{aligned} \]

since the arrivals in disjoint intervals are independent.

Stationary Increments¶

We say that a continuous time counting process \(N_{t}\) has stationary increments if for all \(t_{2} > t_{1} \geq 0\) and for all \(r > 0\), \(N_{t_{2}} - N_{t_{1}}\) and \(N_{t_{2} + r}\) and \(N_{t_{1} + r}\) are independent.

In other words, the number of arrivals in a given time interval is invariant to it's location. Note that the number of arrivals in the time interval between \(t_{1}\) and \(t_{2}\) is nothing but \(N_{t_{2}} - N_{t_{1}}\). By the above statement, if the process has stationary increments, then this quantity is same as \(N_{t_{2} - t_{1}}\), which is the distribution of the counting process itself.