Is a compensated Poisson process uniformly integrable

Question

Let $(N_t)_t$ be a Poisson process with intensity $\lambda$. Define $$ \bar{N}_t = N_t - \lambda t $$ which is clearly a martingale. My question is: is $\bar{N}$ uniformly integrable? I strongly suspect the answer is NO, but if I try to compute $\mathbb{E}(\bar{N}_{T_i})$ I always get $0$. Here $T_i$ is the $i$-th jump time of $N$.
In other words I am not able to give a simple example of a stopping time $T$ such that $\mathbb{E}(\bar{N}_T) \neq \mathbb{E}(\bar{N}_0)$

Thanks in advance

Tom

Answer 1

By the central limit theorem, $\bar N_t/\sqrt{\lambda t}$ converges in distribution to a standard normal random variable $Z$ hence $P[|\bar N_t|\geqslant\sqrt{\lambda t}]\to P[|Z|\geqslant1]\gt0$ when $t\to\infty$.

In particular, $E[|\bar N_t|]\geqslant\sqrt{\lambda t}\cdot P[|\bar N_t|\geqslant\sqrt{\lambda t}]\to\infty$ hence $(\bar N_t)$ is not uniformly integrable.