Inhomogeneous Poisson process

An inhomogeneous Poisson process with locally integrable intensity $\lambda (t)\ge 0$ has independent increments andfor $0\le s<t$. Its cumulative intensity is $\Lambda (t)={\int }_0^t\lambda (u)du$.

If $\lambda >0$ and $g={\Lambda }^{-1}$, then $M_t=N_{g(t)}$ is a rate-one Poisson process, becauseConversely, $N_t=M_{\Lambda (t)}$, so $N_t$ has a Poisson distribution of mean $\Lambda (t)$.