Example for uniformly integrable $\mathbb{L}^2$-bounded sequences

Question

How can I construct an example to show that the sequence $\{X_n\}_{n \ge 1}$ can be $\mathbb{L}^2$-bounded, however it has $\mathbb{E}\left[\sup\limits_{n\ge 1} |X_n|\right]=\infty$. Basically I need to show that if a sequence satisfies $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|^2] < \infty$, it can still have $\mathbb{E}\left[\sup\limits_{n\ge 1} |X_n|\right]=\infty$. I know that in general if $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|^p] < \infty$ for some $p>1$, the sequence is uniformly integrable. Hence, I am trying to construct a uniformly integrable sequence with $\mathbb{E}\left[\sup\limits_{n\ge 1} |X_n|\right]=\infty$. Any hints would be appreciated.

Answer 1

Consider a sequence of disjoint set $(A_n)_{n\geqslant 1}$ and a sequence of positive real numbers $(c_n)_{n\geqslant 1}$. Define $X_n:=c_n\mathbf 1(A_n)$ and $S:=\sup_{i\geqslant 1}\left|X_i\right|$. Notice that $S\mathbf 1(A_n)=c_n$ hence $\mathbb E[S]\geqslant \sum_{n\geqslant 1}c_np_n$, where $p_n:=\mathbb P(A_n)$. Since $\mathbb E\left[X_n^2\right]=c_n^2p_n$, we are looking for sequences of positive real numbers $(c_n)_{n\geqslant 1}$ and $(p_n)_{n\geqslant 1}$ such that $$\sum_{n\geqslant 1}p_n\leqslant 1,\quad \sum_{n\geqslant 1}c_np_n=+\infty\mbox{ and }\quad \sup_{n\geqslant 1}c_n^2p_n<\infty.$$ Such a selection is possible: take $p_n:=1/n^2$ and $c_n=n$.

(in order to construct measurable set with the wanted probability, one can work with the unit interval endowed with the Borel $\sigma$-algebra and the Lebesgue measure)