Uniform integrability

A family of integrable random variables is uniformly integrable whenAlmost-sure or probabilistic convergence together with uniform integrability upgrades to $L^1$ convergence.

On a finite time horizon, an adapted process is of class D if its values at all stopping times in that horizon form a uniformly integrable family. Deterministic boundedness implies this property. The stopping-time uniformity is stronger than having an integrable value at each fixed time.

An adapted process is of class DL when, for every finite $T$, the family $\{X_{\tau }:\tau \le T\text{ a stopping time}\}$ is uniformly integrable. A local martingale is a true martingale exactly when it has this property. This is a local-in-time condition, not uniform integrability over the whole half-line.

// Target: probability-and-statistics.bigb

For $p>1$, ${\sup }_n\mathbb{E}|X_n{|}^p=C<\infty$ gives ${\sup }_n\mathbb{E}[|X_n|1_{|X_n|>M}]\le C{M}^{1-p}\to 0$. Thus an $L^p$ bound implies uniform integrability; a mere $L^1$ bound does not.

For integrable random variables $Z,Y$,Apply this with $Z=X_n$ and $Y=X$ to prove that convergence in L1 implies uniform integrability; control finitely many early terms separately.

For one integrable random variable $Y$, the family $Z_{\mathcal{G}}=\mathbb{E}[Y\mid \mathcal{G}]$, over arbitrary sigma-algebras $\mathcal{G}$, has uniform integrability. On $A=\{|Z_{\mathcal{G}}|>K\}$, the defining property of conditional expectation gives $\mathbb{E}[|Z_{\mathcal{G}}|1_A]\le \mathbb{E}[|Y|1_A]$, while $\mathbb{P}(A)\le \mathbb{E}|Y|/K$ by the Markov inequality. The integrable random variable $Y$ has uniformly small absolute integrals over events of sufficiently small probability.

If ${\sup }_n\mathbb{E}[X_n^2]<\infty$, then $(X_n)$ is uniformly integrable. Indeed, the Cauchy-Schwarz inequality and Markov inequality give