Proof about an Inhomogeneous Poisson Process

[i_a_n]

We know that an inhomogeneous Poisson process is a process with a rate function $\lambda(t)$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \}=\frac{\text{exp}(-s)s^k}{k!}$, where $s=\int_{t}^{t+\Delta t}\lambda(t)dt$.

And Here is the problem:Assume $s=(s_1,\cdots,s_n)$ is a simulation of an inhomogeneous Poisson process with rate function $\lambda(t)$, $0≤t≤T$. Let $\gamma$ be a mapping from $[0, T]$ to $[0, T]$ which satisfies: 1) $\gamma(0)=0$; 2) $\gamma(T)=T$; 3) $0<\gamma'<\infty$ ($\forall t\in [0,T]$) where $\gamma'$ denotes the derivative of $\gamma(t)$ with respect to $t$. Prove that $\gamma^{-1}(s)=(\gamma^{-1}(s_1),\cdots,\gamma^{-1}(s_n))$ is also an inhomogeneous Poisson process, with rate function $\lambda(\gamma (t))\gamma'(t)$.My question is do we need to use the expression of $\gamma^{-1}(s)$ or we can just use the definition from inhomogeneous Poisson process?

Thanks in advance!

 

[mmwave]

you should always strive to use the most precise and accurate information available in your work. In this case, it would be beneficial to use the expression of $\gamma^{-1}(s)$ in your proof. This will provide a more thorough and specific explanation, as it takes into account the specific mapping function $\gamma$ that is being used. However, if you are able to prove the statement using the definition of an inhomogeneous Poisson process, then that would also be acceptable. It ultimately depends on the level of detail and precision required for your work.