Poisson process has independent and stationary increments

Question

Being $N_t$ a Poisson process, defined as $$N_t:=\sum_{n\geq 1} n \mathbb{1}_{[T_n,T_{n+1}[}(t)$$ where $T_n$ are sums of independent exponential random variables, how can I prove it has stationary and independent increments? It seems a natural consequence of "memoryless" property of exponential distribution, $$P(T>t+s|T>s)=\frac{e^{-\lambda(t+s)}}{e^{-\lambda t}}=e^{-\lambda s}$$ involving some induction...

Answer 1

What about a look at the literature? Sato proves this result in the first chapter of his monograph Lévy processes and infinitely divisible distributions (Theorem 3.2) using elementary properties of conditional probability.

A more elegant proof can be based on Campbell's formula, but then we need some results on stochastic integration with respect to jump processes (or martingales).