Independent and Stationary Increments

A counting process can possess two important properties regarding how events accumulate over time.

Independent Increments

The numbers of events that occur in disjoint time intervals are independent. For example, $N(10)$ (events by time 10) is independent of $N(15)-N(10)$ (events between time 10 and 15).

When reasonable: Customer arrivals at a store where each arrival is unrelated to previous ones.

When unreasonable: Births in a population — if $N(t)$ is very large, many people are alive, making future births more likely (dependence).

Stationary Increments

The distribution of the number of events in any interval depends only on the length of the interval, not on when it starts. The number of events in $(s,s+t)$ has the same distribution for all $s$.

When reasonable: Events occurring at a constant underlying rate with no time-of-day effects.

When unreasonable: Store customers with rush hours, or a soccer player who scores more in their prime years than later.

Key Insight

Not all counting processes possess both properties. The Poisson Process is the canonical example that has both.