Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $\left(Z_{n}\right)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with $\mathbb{P}\left(Z_{n}=1\right)=\mathbb{P}\left(Z_{n}=-1\right)=1 / 2$. For $t \in \mathbb{R}^{+}$ we define $\mathcal{F}_{t}=\sigma\left(Z_{p} : p \in \mathbb{N}, p \leq t\right)$ and ${X_{t}=\sum_{1 \leq p \leq | t \rfloor} Z_{p}}$.
I have shown that it is a cadlag martingale but how to show it is not uniformly integrable by apply central limit theorem and equivalence of uniform integrability of cadlag martingale instead of setting a stopping time?
By the central limit theorem, $X_t/\sqrt t$ converges in distribution to a standard normal law. Moreover, the family $\{X_t^2/t,t\gt 0\}$ is bounded in $\mathbb L^1$ hence $\{\left\lvert X_t\right\rvert/\sqrt t,t\gt 0\}$ is uniformly integrable. It follows that $\mathbb E\left[\left\lvert X_t\right\rvert/\sqrt t\right]\to\mathbb E\left[\left\lvert N\right\rvert\right]$, where $N$ has a standard normal distribution, hence $\{\left\lvert X_t\right\rvert,t\gt 0\}$ is not even bounded in $\mathbb L^1$.