I want to find an example where $X_n$ is uniformly integrable, $N$ is a stopping time, but $X_n^N = X_{\min\{n,N\}}$ in not uniformly integrable.
There is a theorem saying that if $M_n$ is a uniform integrable martingale, $N$ is a stopping time, then $M_n^N = M_{\min\{n,N\}}$ in also a uniform integrable martingale. So, what I know at least is that $X_n$ cannot be a martingale. Any suggestion?
Example: Let $\{X_n\}$ be i.i.d. with the unit rate exponential distribution on $[0,\infty)$. This is a uniformly integrable sequence because each $X_n$ is integrable and they all have the same distribution. Let $Y$ be a positive random variable independent of $\{X_n\}$ with $\Bbb E[Y]=\infty$. Let $\mathcal F_n:=\sigma(X_1,\ldots,X_n,Y)$ and define an $\{\mathcal F_n\}$ stopping time $N$ by $N:=\inf\{n:X_n\ge Y\}$. Then $\Bbb P[N<\infty]=1$ and $\Bbb E[X_N]\ge\Bbb E[Y]=\infty$, so $\{X^N_n\}$ cannot be uniformly integrable.