A uniformly integrable sequence that is not uniformly integrable in $\mathbb{L}^p$ for $p>1$

Question

I need to construct an example to show that if for the sequence $\{X_n\}_{n \ge 1}$ we have that $\mathbb{E}\left[\sup\limits_{n \ge 1} |X_n|\right] < \infty$, the sequence in not necessarily uniformly integrable in $\mathbb{L}^p$ for any $p>1$. I know that I need to find a sequence for which $\sup\limits_{n \ge 1} \mathbb{E}\left[|X_n|^p\right] = \infty$ even though $\mathbb{E}\left[\sup\limits_{n \ge 1} |X_n|\right] < \infty$. I would appreciate any ideas on what family of sequences I should think about.

Answer 1

Take $X_n=X$ for each $n$, where $X$ is integrable random variable such that $\mathbb E\left|X\right|^p=+\infty$ for each $p>1$ (for example a non-negative random variable $X$ such that $\mathbb P(X\gt t)$ behaves like $1/\left(t\left(\log t\right)^2\right)$ as $t$ goes to infinity).