Doob’s Maximal Inequalities
1 Doob’s Submartingale Inequality
Doob’s martingale inequality, also known as Kolmogorov’s submartingale inequality, gives a bound on the probability that a submartingale exceeds a given value over a given interval of time. Informally, the expected value of the process at some final time controls the probability that a sample path reaches above any particular value beforehand.
The proof uses very direct reasoning, so it does not require any restrictive assumptions on the underlying filtration or on the process itself. In the continuous-time setting, right-continuity (or left-continuity) of the sample paths is required, but only so that the supremum of a sample path equals the supremum over a countable dense subset of times.
Suppose that \((X_n, \mathcal{F}_n)_{n\ge 0}\) is a submartingale. Then for every \(\lambda>0\) and \(n\ge 0\), \[ \lambda\,\mathbb{P}\left( \max_{0\le k\le n}X_k \ge \lambda \right) \le \mathbb{E}\left[ X_n \mathbb{1}_{\{ \max_{0\le k\le n}X_k \ge \lambda \}} \right] \le \mathbb{E}[X_n^+]. \]
After rearranging, the result is similar in form to the Markov Inequality, \[ \mathbb{P}\left( \max_{0\le k\le n}X_k \ge \lambda \right) \le \frac{\mathbb{E}[X_n^+]}{\lambda}. \]
\begin{proof} We define a stopping time \[
\tau:=\inf\{k \geq 0:X_{k}\geq \lambda\}
\] (where \(\tau= +\infty\) is \(X_{k}<\lambda\) for all \(k \geq 0\)). Furthermore, we define event \[
A:=\{ \max_{0\leq k\leq n}X_{k}\geq \lambda \}=\{ \tau\leq n \}\in\mathcal{F}_{\tau \wedge n}\subseteq \mathcal{F}_{\tau}.
\] Then, for every \(k=0,1,\dots,n\) we have \(A\subset \{ \tau=k \}\in \mathcal{F}_{k}\) and also \(A \cap \{ \tau\geq n+1 \}\in \mathcal{F}_{n}\). This implies \[
\begin{align}
\mathbb{E}[X_{\tau \wedge n}\cdot \mathbb{1}_{A}] & = \mathbb{E}\left[ \sum_{k=0}^\infty X_{\tau \wedge n}\cdot \mathbb{1}_{A\cap \{ \tau=k \}} \right] =\mathbb{E}\left[ \sum_{k=0}^\infty X_{k}\cdot \mathbb{1}_{A\cap \{ \tau=k \}}+X_{n}\cdot \mathbb{1}_{A \cap \{ \tau \geq n+1 \}} \right] \\
& \leq \mathbb{E}\left[ \sum_{k=0}^n X_{n}\cdot \mathbb{1}_{A\cap \{ \tau=k \}}+X_{n}\cdot \mathbb{1}_{A\cap \{ \tau \geq n+1 \}} \right]=\mathbb{E}[X_{n}\cdot \mathbb{1}_{A}],
\end{align}
\] where the inequality follows from the submartingale property. Using this inequality, since \(X_{\tau}\geq \lambda\) on the set \(A\), we obtain \[
\begin{align}
\lambda \cdot \mathbb{P}\left( \max_{0\leq k\leq n} X_{k}\geq \lambda \right) & = \lambda \cdot \mathbb{P}(A) = \mathbb{E}[\lambda \mathbb{1}_{A}]\leq \mathbb{E}[X_{\tau}\cdot \mathbb{1}_{A}]=\mathbb{E}[X_{\tau \wedge n}\cdot \mathbb{1}_{\tau\leq n}] \\
& \leq \mathbb{E}[X_{n}\cdot \mathbb{1}_{\tau \leq n}] = \mathbb{E}[X_{n}\cdot \mathbb{1}_{\max_{0\leq k\leq n}(X_{k})\geq \lambda}]\leq \mathbb{E}[X_{n}^+].
\end{align}
\] \end{proof}
2 Further Maximal Inequalities
Let \((X_n, \mathcal{F}_n)\) be a submartingale. Then
- \(L^p\) inequality: for \(p>1\), \[ \mathbb{E}\left[ \left( \max_{0\le m\le n}X_m^+ \right)^p \right] \le \left( \frac{p}{p-1} \right)^p \mathbb{E}\left[ (X_n^+)^p \right]. \]
- Minimal inequality: for \(\lambda>0\), \[ \lambda\,\mathbb{P}\left( \min_{0\le m\le n}X_m \le -\lambda \right) \le \mathbb{E}[X_n^+] - \mathbb{E}[X_0]. \]
- Doob-Kolmogorov inequality: if \((X_n, \mathcal{F}_n)\) is a martingale with \(\mathbb{E}[X_n^2]<\infty\) for all \(n\), then for \(\lambda>0\) \[ \mathbb{P}\left( \max_{0\le m\le n}|X_m| \ge \lambda \right) \le \frac{\mathbb{E}[X_n^2]}{\lambda^2}. \]
\begin{proof} For (3), \(X_n^2\) is a submartingale by Jensen’s inequality (see Martingale Theory), so applying the submartingale inequality to \(X_n^2\) with level \(\lambda^2\) gives \[
\lambda^2\,\mathbb{P}\left( \max_{0\le m\le n}X_m^2 \ge \lambda^2 \right)
\le \mathbb{E}[X_n^2].
\] For (1) and (2) see [[files-notes-pstat213bc-lecture-notes-raya-pdf|PSTAT213BC Lecture Notes (Raya), page 102]]. \end{proof}
3 Applications
- Kolmogorov’s maximal inequality for sums follows from the Doob-Kolmogorov inequality applied to \(S_n - \mathbb{E}S_n\).
- The inequalities are used to prove the Martingale Convergence Theorems.