Let be $(\xi_i)_{i\geq 1}$ random variables IID with exponential distribution of parameter $\lambda$,. Define the inter-arrival times by $T_0=0$, $T_n=\sum_{i=1}^n \xi_i$ for $n\geq 1$ and the asociated counting process $N_t=\sum_{k \geq 1}1_{T_k\leq t}$ for $t\geq 0$.
I know tha this defines a poisson process respect to the filtration induced by the process, usually called natual filtration. But now define another filtration as $G_n=\sigma(T_1,...,T_n)$ for $n\geq 1$ and $F_t:=G_{N_t+1}=\{A \in G_{\infty}: \forall j\geq 1\ \ A\cap \{N_t+1=j\}\in G_j\}$.
I have to prove that the process $(N_t)_{t\geq 0}$ is also poisson process but respecto to the filtration $(F_t)_{t\geq 0}$. I have al ready prove that the process is adapted to this filtration, but I cannot se how to show that the increment $N_{t+s}-N_s$ conditioned to $F_s$ has poisson distribution with parameter $\lambda t$.
What I have so far: $P(N_{t+s}-N_s=m|F_s)=P(\sum_{k\geq 1}1_{(s,t+s]}(T_k)=m|F_s)$ and taking s fixed there exist $i\in\mathbb{N}$ such that $T_{i-1}\leq s < T_i$ wich is equivalent to $N_s=i-1$. So the sum is only respecto to $m$ inter-arrival times $T_i,T_{i+1},...,T_{i+m-1}$. But I dont know how to proceed. Thanks for any help.