Uniform Integrability and relation to $L^p$ for $p>1$

Question

Let $X_n$ be a martingale.

Then we know that for $p> 1$ the conditions

$\sup_n E[|X_n|^p] < \infty$ and $E[\sup_n |X_n|^p] < \infty$ are equivalent.

For $p=1$ this does not hold, because uniform integrability is stronger than the first and weaker than the second condition.

But my question is: Are the conditions for $p>1$ also equivalent when $X_n$ is not a martingale but arbitrary?

Answer 1

As John Dawkins explained in the comments, we may have $\sup_n\mathbb E\left|X_n\right|^p<\infty$ but $\sup_{n}|X_n|=+\infty$ almost surely when $(X_n)_{n\geqslant 1}$ is an independent identically distributed sequence such that $\mathbb E\left|X_1\right|^p$ is finite and $\mathbb P\{|X_1|\gt x\}\gt 0$ for any $x$.

However, if $X_n=\left(\sum_{j=1}^nY_j/n\right)_{n\geqslant 1}$ where $Y_j$ is a strictly stationary sequence, then for $p\gt 1$, $\sup_n\mathbb E|X_n|^p\lt \infty$ means that $\mathbb E\left|Y_1\right|^p$ which implies, by the maximal ergodic theorem, that $\sup_n|X_n|^p$ is integrable.