A note on the characteristics of Poisson processes (Q1063331)

Two characteristic theorems of Poisson processes are given. If \(\{\) N(t); \(t\geq 0\}\) is a renewal process, \(U_ t\), \(V_ t\) are, respectively, the time since the last renewal and the time to the next renewal at t, \(Z_ t=U_ t+V_ t\), then a Poisson process can be characterized by the limiting independence of the joint distribution of \((U_ t,V_ t)\) when \(t\to \infty\), or \(EZ_ t\), or the distribution of \(Z_ t\).