Show that nonhomogeneous poisson process through this operation turns into a homogeneous poisson process.

Question

I wonder how to prove this statement:

Suppose $\{N(t),t\geq 0\}$ is a nonhomogeneous poisson process with $\{\lambda(s)>0,s\geq 0\}$. Let $m(t)=\int_{0}^{t}\lambda(s)ds$ and $m^{-1}(t)$ be the inverse of $m(t)$, i.e. $$m^{-1}(u)=\inf\{t:t>0,m(t)\geq u, u\geq 0\}$$ Further denote $M(u)=N(m^{-1}(u))$. Then $\{M(u),u\geq 0\}$ is a homogeneous Poisson process.

Thanks a lot.

Answer 1

Clearly you have independent increments, and number of occurrences in an interval has Poisson distribution. All you have to do is compute the mean of that Poisson distribution.