21 — Doob’s $L^p$ Inequality, BDG Inequality, and Uniform Integrability
Doob’s $L^p$ Inequality ($p>1$)
Let ${X_n, \mathcal{F}n}{n\ge 0}$ be an $L^p$ martingale with
\(E(|X_n|^p) < \infty,\quad p>1.\)
Then \(E(|X_n|^p) \;\le\; E\!\left(\max_{1\le k\le n} |X_k|^p\right) \;\le\; C_p\, E(|X_n|^p),\) where $C_p = \left(\frac{p}{p-1}\right)^p$.
This is the standard Doob $L^p$ maximal inequality.
Burkholder–Davis–Gundy (BDG) Inequality ($p\ge 1$)
Let $D_k = X_k - X_{k-1}$ be the martingale differences.
Then for $p\ge 1$,
\(\tilde C_p\,
E\!\left( \left( \sum_{k=1}^n D_k^2 \right)^{p/2} \right)
\;\le\;
E\!\left( \max_{1\le k\le n} |X_k|^p \right)
\;\le\;
C_p\,
E\!\left( \left( \sum_{k=1}^n D_k^2 \right)^{p/2} \right).\)
Special case $p=2$: \(E\left(\sum_{k=1}^n D_k^2\right) = E(X_n^2),\) because martingale differences are orthogonal: \(E(D_k D_j)=0,\quad k\ne j.\)
Section 5: $p=1$ — Uniform Integrability and $L^1$ Convergence
(Durrett §5.5, p. 220)
Definition: Uniform Integrability (UI)
A family ${X_n}$ is uniformly integrable if
\(\lim_{\lambda\to\infty} \sup_n E(|X_n|\,; |X_n|>\lambda) = 0.\)
Equivalent condition:
-
$\sup_n E X_n < \infty$, and - For every $\varepsilon>0$ there exists $\delta>0$ such that
\(P(A) < \delta \quad \Rightarrow\quad \sup_n E(|X_n|; A) < \varepsilon.\)
Theorem (B): Conditional Expectations Form a UI Martingale
Let $X$ be an integrable r.v. on $(\Omega,\mathcal{F},P)$.
Let $\mathcal{F}n \uparrow \mathcal{F}\infty \subseteq \mathcal{F}$.
Then
\(X_n := E(X \mid \mathcal{F}_n)\)
is a UI martingale.
Examples
-
If $\sup_n E X_n ^p < \infty$ for some $p>1$, then ${X_n}$ is UI. -
If $ X_n \le Z$ for some integrable $Z$, then ${X_n}$ is UI.
Key Equivalence Theorem (TFAE)
Assume $X_n \xrightarrow{a.s.} X$ and $E|X_n|<\infty$.
The following are equivalent:
- ${X_n}$ is uniformly integrable.
-
$X_n \xrightarrow{L^1} X$ and $E X <\infty$. - There exists an integrable $X$ such that $X_n = E(X \mid \mathcal{F}_n)$ for all $n$
(i.e., ${X_n}$ is the Doob martingale of $X$).
Sketch of implications
- (1 ⇒ 2): Standard result: UI + $X_n\to X$ a.s. implies $X_n\to X$ in $L^1$.
- (2 ⇒ 3): For any $A\in \mathcal{F}_n$:
\(E(X_n; A) = E(X; A)\) so $X_n = E(X \mid \mathcal{F}_n)$. - (3 ⇒ 1): Follows from Theorem (B).
Additional Notes
- The lecture indeed continues directly from Lecture 21, expanding the Uniform Integrability discussion and proving the UI ⇔ $L^1$ convergence ⇔ Doob-martingale-of-$X$ equivalence.
- On page 1 of the handwritten notes, the transition from Doob $L^p$ inequality to BDG is explicit.
- Pages 2 and 3 include the complete UI argument and the TFAE theorem.
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