On some properties of Poisson processes (Q1106563)

The Poisson process, i.e., the simple stream, is defined by \textit{A. Ya. Khintchine} [Mathematical methods in the theory of queueing (1955; Zbl 0068.120)] as a stationary, orderly and finite stream without aftereffects. A necessary and sufficient condition for a stream to be a simple stream is that the interarrival times are independent random variables with identical exponential distributions. This paper gives a simple and rigorous proof of the necessary and sufficient condition, and discusses the other necessary and sufficient conditions for a renewal process to be a Poisson process.