What is the distribution of inter-arrival time of a non-homogenous Poisson Process?
In other words, if $T_n$ is a non-homogeneous Poisson process with intensity $\lambda(s)$, and $S_{n+1} = T_{n+1} - T_n$, what is $$ \mathbb{P}[S_{n+1} > t]? $$
What I got:
$$\mathbb{P}[S_1\leq t] = 1 - \mathbb{P}[S_1> t]=\mathbb{P}[N_t=0]=1-e^{m_t} $$ with $m_t=\int_0^t\lambda(s)ds$
Therefore, $f_{S_1}(t)=\lambda(t)e^{-m_t}$
$$\mathbb{P}[S_2\leq t] = \int_0^\infty\mathbb{P}[S_2\leq t| S_1=s]f_{S_1}(t)ds=\int_0^\infty\lambda(s)e^{-m_{t+s}}ds$$
Is this correct? If yes, what would be: $$ \mathbb{P}[S_{n+1} > t] $$