Uniform Integrability

Uniform Integrability

Let $T$ be an index set. A family $\{X_t : t \in T\} \subset L^1$ is uniformly integrable (ui) if

$$\lim_{a \to \infty} \sup_{t \in T} E\big[|X_t| \mathbf{1}\{|X_t| > a\}\big] = 0$$

• If $X \in L^1$, $\{X\}$ is ui.
• If $|X_t| \le Y$ for all $t$ with $Y \in L^1$, $\{X_t\}$ is ui.
• Every finite family in $L^1$ is ui.
• If $|X_t| \le |Y_t|$ for all $t$ and $\{Y_t\}$ is ui, then $\{X_t\}$ is ui.
• If $p > 0$ and $\sup_t E|X_t|^{p + \delta} < \infty$ for some $\delta > 0$, then $\{|X_t|^p\}$ is ui.

Bounded in $L^1$ is not enough. For $U \sim \mathrm{Unif}(0,1)$ and $X_n := n \mathbf{1}\{U \le \frac{1}{n}\}$, $E X_n = 1$ for all $n$ but $\{X_n\}$ is not ui as $E[X_n \mathbf{1}\{X_n > a\}] = 1$ for $n > a$.

Characterization $\{X_t\} \subset L^1$ is ui iff

1. for all $\varepsilon > 0$ there is $\delta > 0$ such that $\sup_t E[|X_t| \mathbf{1}_A] < \varepsilon$ whenever $A \in \mathcal{F}$ and $P(A) < \delta$, and
2. $\sup_t E|X_t| < \infty$.

$L^1$ and $L^p$ Convergence

Scheffé If $X_n, X \in L^1$, then $X_n \xrightarrow{L^1} X$ iff

$$\sup_{A \in \mathcal{F}} \Big| \int_A (X_n - X) \, dP \Big| \to 0$$

If $X_n \in L^1$ for all $n$, the following are equivalent

1. $X_n \xrightarrow{L^1} X$ for some $X \in L^1$,
2. $(X_n)$ is Cauchy in $L^1$,
3. $\{X_n\}$ is ui and $X_n \xrightarrow{P} X$ for some $X$.

The limits in 1 and 3 agree a.s.

If $1 < p < \infty$ and $X_n \in L^p$ for all $n$, the following are equivalent

1. $X_n \xrightarrow{L^p} X$ for some $X \in L^p$,
2. $(X_n)$ is Cauchy in $L^p$,
3. $\{|X_n|^p\}$ is ui and $X_n \xrightarrow{P} X$ for some $X$.