# Counting process with independent, stationary increments is Poisson

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?

• Not an answer, but you may be interested in reading about Lévy processes. – angryavian Feb 24 at 18:51
• Take a look at Theorem 8.1 here – saz Feb 24 at 19:09