An Elementary Proof of the Order-Statistics Characterization of the Poisson Process

Abstract

A continuous-time renewal process (N(t), t ≥ 0) is a Poisson process if and only if the conditional distribution of an interarrival time given N(t) = n is the same as that of the minimum order statistic corresponding to n independent random variables identically and uniformly distributed on the interval [0, t). Similar characterization is also given for a nonhomogeneous Poisson process.

Key Words:

• Arrival times
• Interarrival times
• Nonhomogeneous Poisson process
• Renewal process