For we have

(using (2.1))

i.e.,

But as , hence:

(2.2)

Using mathematical induction we have:

 In general ,  and both are not independent of . This type of process can be said to be an evolutionary process, thus Poisson process is not a stationary process but rather an evolutionary process.

It is important to understand that , and it gives us the same meaning when we assume . Another important fact worth mentioning is that a Poisson process is a Markov Process with the transition probability being independent of the past.