Counterexamples for expressions regarding uniform integrability

Question

I have two questions regarding uniform integrability.

1- Is there any example such that $E[\sup_{n \geq 1} \vert X_n \vert] < \infty$ but $\sup_{n \geq1} E [\vert X_n \vert^p] =\infty$ for any $p >1$?

2- Is there any example such that $\sup_{n \geq1} E [\vert X_n \vert^2] < \infty$ but $E[\sup_{n \geq 1} \vert X_n \vert] = \infty$?

Any idea?

Answer 1

In examples below all RV's are nonnegative, so no need for absolute value.

1. Yes. $X_1=X_2=\dots$ with density

$$f(x) = \begin{cases} \frac{c}{x^2 \ln (1+x)^{2}}& x>1 \\ x=0 & \mbox{elsewhere}.\end{cases}$$

Now $\sup_{n} X_n = X_1 \in L^1$, but $X_1 \not \in L^p$ for any $p>1$.

2. Yes. Let $A_1,A_2,\dots$ be a partition of the probability space (disjoint, and union is entire space), such that $$P(A_j)=\frac{c}{j^2}\mbox{ for all }j.$$ Set $$X_j = \frac{1}{\sqrt{P(A_j)}} {\bf 1 }_{A_j}.$$

Then $$E [X_j^2] = \frac{1}{P(A_j)}P(A_j)=1.$$

However, $\sup_j X_j = \sum_{j=1}^\infty X_j$, because the events $A_j$ are disjoint. And so $$E [\sup_j X_j ] = \sum_{j=1}^\infty \sqrt{P(A_j)}=\sqrt{c} \sum_{j=1}^\infty \frac{1}{j}.$$

Note that a slight modification can give you a sequence such that $\lim_{j\to\infty} E X_j^2 =0$ yet $E \sup_{j} X_j =\infty$, and we can push the power from $2$ to any $1+\epsilon$.