Convergence in $L^1$ of martingale related to a Poisson process

Question

I was solving Problem 3.21 b) of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Essentially, the problem says the following:

Let $N_t$ be a Poisson random variable with parameter $\lambda t$. Define the process $$X_t=\exp(N_t-\lambda t(e-1)).$$ Does $X$ converges in $L_1$?

I know $(X_t)_t$ is a martingale, and I tried to prove that $(X_t)_t$ is uniformly integrable, but I couldn't do it. Any hisnts would be appreciated. Thanks!

Answer 1

By the law of large numbers we have $N_t \sim \lambda t$ as $t \to \infty$, thus $\lim_{t \to \infty} X_t =0$ almost surely. Since $X$ is a martingale, $\mathbb{E}[X_t] = \mathbb{E}[X_0] = 1$ so $X_t$ cannot converge in $L^1$.