Confusion on Uniform Integrability of Random Variables

Question

We have the definition that a random process, $X_n$, is (1st power) uniformly integrable if

$$\lim_{M\to\infty}\sup_n\mathbb{E}(|X_n| ; |X_n|\geq M)=0$$.

My question is whether the following four statements are true and (intuitively) why this is or isn't the case:

1. "$|X_n|$ is bounded for all $n$ $\implies$ $X_n$ is 1st power uniformly integrable."

My guess: it's true since if it's bounded there must exist some $M\in \mathbb{R}$ such that $M>|X_n|$, and so $\lim_{M\to \infty}\mathbb{I}\{|X_n|\geq M\}=0 $

2. "$|X_n| < \infty$ $\implies$ $X_n$ is 1st power uniformly integrable."

My guess: intuitively to me it feels like this should be true by similar reasoning to (1); if $|X_n|$ is finite then as $M\to \infty$ we get $\mathbb{I}\{|X_n|\geq M\}=0$? And hence $\lim_{M\to\infty}\sup_n\mathbb{E}(|X_n| ; |X_n|\geq M)=0$? I think I'm wrong, however.

3. "$\mathbb{E}|X_n| < \infty$ $\implies$ $X_n$ is 1st power uniformly integrable."

My guess: I think this is wrong because $\mathbb{E}|X_n| < \infty$ doesn't imply that $|X_n|$ is finite and so $\lim_{M\to \infty}\mathbb{I}\{|X_n|\geq M\}$ is not necessarily $0$?

4. "If $\forall \epsilon>0, \exists\delta>0 $ such that $\mathbb{P}(|X_n|\geq M)<\delta $ implies $\mathbb{E}(|X_n| ; |X_n|\geq M)<\epsilon$ $\implies$ $X_n$ is 1st power uniformly integrable."

My guess: I think this is not true since I think that this doesn't necessarily mean that we can find $\delta$ such that $\sup_n\mathbb{E}(|X_n| ; |X_n|\geq M)<\epsilon$? My question would also be - if we add in either the condition that $\mathbb{E}|X_n|<\infty$ or that $|X_n|<\infty$ then does this imply uniform integrability?

Answer 1

1. It depends actually whether the bound is allowed to depend on $n$ or not. If $\sup_{n\geqslant 1}\lvert X_n\rvert$ is finite, then your arguement works. But if we only assume that $\lvert X_n(\omega)\rvert\leqslant c_n$ for almost every $\omega$, uniform integrability may fail: consider $X_n(\omega)=n$ if $0<\omega<1/n$ and $0$ otherwise on $(0,1)$ endowed with Lebesgue measure.

2. and 3. are covered by the previous counterexample.

For 4., the thing is that there are probability spaces such that $\mathbb P(A)$ is either $0$ or bigger than some $\delta_0$, like $\{a,b\}$ with $\mathbb P(\{a\})=\mathbb P(\{b\})=1/2$. In this case, the implication will be always satisfied with $\delta<1/2$, because the only way to have $\mathbb{P}(|X_n|\geq M)<\delta$ is that $\{|X_n|\geq M\}$ is empty. Hence a non-uniformly integrable sequence can also satisfy the implication.