Suppose that $L_t$ is a counting process, i.e. $$ L_t= \sum_{i \in \mathbb N} 1_{T_i \le t} $$ for a collection $(T_i) _{i \in \mathbb N}$ of stopping times.
I have often seen the claim that if $L_t$ has stationary and independent increments, then it is Poisson, that is there exists $\lambda>0$ s.t. $$ \mathbb P(L_t=k) =e^{- \lambda t} \frac{(\lambda t) ^k }{k!} $$
Is this claim correct or is it missing some assumptions? Could someone prove this fact or point to a reference for a proof?