# Doob's martingale inequality
**Doob's martingale inequality** is a pair of results in probability theory that bound how large a martingale, or a non-negative submartingale, can become over a finite time interval in terms of its value at the final time alone. The weak form controls the tail probability of the running maximum, and the strong form, Doob's \( L^p \) inequality, controls the \( L^p \) norm of the running maximum by the factor \( p/(p-1) \) times the \( L^p \) norm of the terminal value. The inequalities are named after Joseph L. Doob, who developed martingale theory and stated the results in his 1953 book *Stochastic Processes*.[1](https://celebratio.org/Doob_JL/article/741/)
| Key fact | Detail |
|---|---|
| Weak (submartingale) form | For a non-negative submartingale \( (X_n) \) and \( \lambda > 0 \): \( \lambda\,\mathbb{P}(X_n^* \ge \lambda) \le \mathbb{E}[X_n]\), where \( X_n^* = \max_{0 \le k \le n} X_k \)[2](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf) |
| Strong \( L^p \) form | For a martingale or non-negative submartingale and \( p > 1 \): \( \|X_n^*\|_p \le \frac{p}{p-1}\|X_n\|_p \)[3](https://www.karlin.mff.cuni.cz/~pawlas/2026/MSA405/TP2.pdf) |
| Quadratic case | At \( p = 2 \) the bound reads \( \mathbb{E}[(X_t^*)^2] \le 4\,\mathbb{E}[X_t^2]\), known as Doob's maximal quadratic inequality[4](https://planetmath.org/doobsinequalities) |
| Endpoint \( p = 1 \) | No strong \( L^1 \) inequality holds with any finite constant; only the weak-type estimate survives[5](https://link.springer.com/article/10.1007/s10959-012-0458-8) |
| Sharpness | The constant \( p/(p-1) \) is optimal: for any \( c < p/(p-1) \) there is a cadlag \( L^p \)-integrable martingale with \( X_1^* = c\,X_1 \)[6](https://almostsuremath.com/2016/09/11/the-optimality-of-doobs-maximal-inequality/) |
| Corollary | Kolmogorov's inequality for sums of independent zero-mean random variables follows by applying the weak inequality with \( p = 2 \)[3](https://www.karlin.mff.cuni.cz/~pawlas/2026/MSA405/TP2.pdf) |
| Origin | Doob, *Stochastic Processes*, John Wiley & Sons, New York, 1953, viii + 654 pages[1](https://celebratio.org/Doob_JL/article/741/) |
## Statement
Let \( (X_n) \) be a martingale or a non-negative submartingale with respect to a filtration (increasing sequence of sigma-algebras tracking accumulated information), and write \( X_n^* = \max_{0 \le k \le n} |X_k| \) for the running maximum. Doob's maximal inequalities then give, for every \( n \in \mathbb{N} \):[3](https://www.karlin.mff.cuni.cz/~pawlas/2026/MSA405/TP2.pdf)
1. \[
For p ≥ 1 and a > 0, the probability that |X_k| reaches at least a for some k between 1 and n is bounded above by a^{-p} times the p-th moment of |X_n|.
\]
2. \[
\mathbb{E}\!\left(\max_{k=1,\ldots,n}|X_k|\right)^p \le \left(\frac{p}{p-1}\right)^p \mathbb{E}|X_n|^p, \qquad p > 1.
\]
The first statement is often called Doob's submartingale inequality or the weak maximal inequality; the second is Doob's \( L^p \) inequality. Both resemble [Markov's inequality](https://www.edgepedia.org/markovs-inequality) except that a maximum appears on the left-hand side.[7](https://people.bath.ac.uk/masmdp/mart.bho/notes9.pdf) The weak form is usually stated for a non-negative submartingale \( (X_n) \) as \( \lambda\,\mathbb{P}(X_n^* \ge \lambda) \le \mathbb{E}[X_n]\).[2](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf)
## Proof idea
**Stopping times.** The weak inequality is proved by stopping the process the first time it reaches the level \( \lambda \). Define \( T = \inf\{k \ge 0 : X_k \ge \lambda\} \). Then \( \{T \le n\} = \{X_n^* \ge \lambda\} \), and the optional stopping theorem applied to the bounded stopping time \( T \wedge n \) gives \( \mathbb{E}[X_n] \ge \mathbb{E}[X_{T \wedge n}] \). Splitting this expectation according to whether \( T \le n \) and using \( X_T \ge \lambda \) on \( \{T \le n\} \) yields \( \lambda\,\mathbb{P}(X_n^* \ge \lambda) \le \mathbb{E}[X_n]\).[2](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf) The same argument works for a general submartingale through its Doob decomposition into a martingale and an increasing process.[7](https://people.bath.ac.uk/masmdp/mart.bho/notes9.pdf)
**Layer cake and Hölder.** The \( L^p \) inequality follows from the weak one by integrating over all levels. Using the identity \( x^p = \int_0^x p\,t^{p-1}\,dt \), [Fubini's theorem](https://www.edgepedia.org/fubinis-theorem), and the weak bound, one obtains \( \mathbb{E}[(X_n^*)^p] \le \frac{p}{p-1}\,\mathbb{E}[X_n (X_n^*)^{p-1}] \). [Hölder's inequality](https://www.edgepedia.org/holders-inequality) with conjugate exponents \( p \) and \( p/(p-1) \) then bounds the right side by \( \frac{p}{p-1}\|X_n\|_p\,\|X_n^*\|_p^{p-1} \), and dividing by \( \|X_n^*\|_p^{p-1} \) gives the result.[8](https://www.kunisky.com/static/teaching/2026spring-prob2/prob2-notes-2026.pdf) Truncating the maximum at a level \( C \) and letting \( C \to \infty \) handles integrability.[8](https://www.kunisky.com/static/teaching/2026spring-prob2/prob2-notes-2026.pdf)
## Consequences
**Kolmogorov's inequality.** If \( X_1, X_2, \ldots \) are independent random variables with zero expectation and finite variance and \( S_n = X_1 + \cdots + X_n \), then \( (S_n) \) is a martingale, and applying the weak inequality with \( p = 2 \) gives
\[
\mathbb{P}\!\left(\max_{k=1,\ldots,n}|S_k| \ge a\right) \le a^{-2}\,\mathbb{E}S_n^2 = a^{-2}\sum_{k=1}^n \mathbb{E}X_k^2, \qquad a > 0.
\]
Doob's inequality therefore generalizes this classical bound for independent sums to the martingale setting.[3](https://www.karlin.mff.cuni.cz/~pawlas/2026/MSA405/TP2.pdf)
**Martingale convergence.** The \( L^p \) inequality is the key step in the \( L^p \) martingale convergence theorem: if \( (M_n) \) is a martingale with \( \sup_n \|M_n\|_p < \infty \) for some \( p > 1 \), then \( M_n \) converges almost surely and in \( L^p \) to a limit \( M_\infty \in L^p \). The inequality shows that \( \sup_n \mathbb{E}|M_n^*|^p \le (p/(p-1))^p \sup_n \mathbb{E}|M_n|^p \), which supplies the dominating variable needed for dominated convergence.[8](https://www.kunisky.com/static/teaching/2026spring-prob2/prob2-notes-2026.pdf) In the same way, the weak form shows that any \( L^1 \)-bounded martingale is almost surely bounded.[4](https://planetmath.org/doobsinequalities)
## Continuous time
For a continuous-time process \( (X_t)_{t \ge 0} \) the running maximum \( X_t^* = \sup_{s \le t} |X_s| \) is a supremum over uncountably many random variables and need not be measurable. If the process is right-continuous (cadlag), the supremum can be restricted to the countable set \( \{s \in \mathbb{R}_+ : s/t \in \mathbb{Q}\} \), and the discrete-time inequalities carry over unchanged: for a right-continuous martingale or non-negative submartingale,[4](https://planetmath.org/doobsinequalities)
\[
\lambda\,\mathbb{P}(X_t^* \ge \lambda) \le \mathbb{E}|X_t|, \qquad
\|X_t^*\|_p \le \frac{p}{p-1}\,\|X_t\|_p, \qquad p > 1.
\]
The same statements hold with \( \sup_{s \in [0,t]} \mathbb{E}|X_s| \) on the right when the process is not assumed to have constant conditional expectation over the interval, though the strong bound requires a corresponding \(L^p\) expectation rather than an \(L^1\) expectation.[9](https://keesvs.net/UoM/teaching/martingales/notes/Chapter9.html) • [12](https://www.stats.ox.ac.uk/~etheridg/ctsmartingales.pdf)
## Sharpness and the endpoint p = 1
The constant \( c_p = p/(p-1) \) cannot be improved. For any \( p > 1 \) and any \( 1 \le c < c_p \) one can construct a strictly positive cadlag \( L^p \)-integrable martingale \( (X_t)_{t \in [0,1]} \) with \( X_1^* = c\,X_1 \), so any proposed constant \( \tilde c_p \) must satisfy \( \tilde c_p \ge c \), and letting \( c \uparrow c_p \) proves optimality.[6](https://almostsuremath.com/2016/09/11/the-optimality-of-doobs-maximal-inequality/) The constant diverges as \( p \downarrow 1 \), which reflects the fact that at \( p = 1 \) no strong inequality of the same shape holds with any finite constant; the weak-type \( (1,1) \) estimate is the correct endpoint statement.[5](https://link.springer.com/article/10.1007/s10959-012-0458-8)
## Extensions
The inequality has been generalized in several directions. A Doob-type maximal inequality has been developed for processes satisfying a conditional increment control condition, with applications to time series, Lévy processes, random walks, branching processes, and certain Markov processes including geometric [Brownian motion](https://www.edgepedia.org/brownian-motion).[10](https://arxiv.org/abs/1805.07700) In non-commutative probability, Marius Junge, a mathematician working in operator algebras at the University of Illinois, proved in 2002 a version of Doob's maximal inequality for non-commutative martingales, valid for \( 1 < p \le \infty \), obtained by duality from a Burkholder–Davis–Gundy type inequality in non-commutative \( L_p \) spaces.[11](https://arxiv.org/abs/math/0206062) Related sharp maximal inequalities for differentially subordinated martingales replace the constant \( p/(p-1) \) with \( p^* - 1 \), where \( p^* = \max\{p,\, p/(p-1)\} \), and this constant is also best possible.[5](https://link.springer.com/article/10.1007/s10959-012-0458-8)
## References
1. [J. L. Doob, *Stochastic Processes*, John Wiley & Sons, 1953; Complete Bibliography, Celebratio Mathematica](https://celebratio.org/Doob_JL/article/741/)
2. [Advanced Probability lecture notes, Theorem 2.9 (Doob's maximal inequality), UC Berkeley](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf)
3. [Probability Theory lecture notes, Theorems 3.13–3.14 (Doob's maximal inequalities, Kolmogorov's inequality), Charles University](https://www.karlin.mff.cuni.cz/~pawlas/2026/MSA405/TP2.pdf)
4. [Doob's inequalities, PlanetMath](https://planetmath.org/doobsinequalities)
5. [Maximal Inequalities for Martingales and Their Differential Subordinates, Journal of Theoretical Probability](https://link.springer.com/article/10.1007/s10959-012-0458-8)
6. [The Optimality of Doob's Maximal Inequality, Almost Sure Mathematics (George Lowther)](https://almostsuremath.com/2016/09/11/the-optimality-of-doobs-maximal-inequality/)
7. [Doob's Maximal Inequality, University of Bath martingale notes](https://people.bath.ac.uk/masmdp/mart.bho/notes9.pdf)
8. [Lecture Notes: Probability Theory II, Johns Hopkins University](https://www.kunisky.com/static/teaching/2026spring-prob2/prob2-notes-2026.pdf)
9. [Martingales and friends in continuous time, MATH4/67201 Martingale Theory notes, University of Manchester](https://keesvs.net/UoM/teaching/martingales/notes/Chapter9.html)
10. [A Doob-type maximal inequality and its applications to various stochastic processes, arXiv:1805.07700](https://arxiv.org/abs/1805.07700)
11. [M. Junge, Doob's inequality for non-commutative martingales, arXiv:math/0206062](https://arxiv.org/abs/math/0206062)
12. [stats.ox.ac.uk](https://www.stats.ox.ac.uk/~etheridg/ctsmartingales.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Expectation, moments, and inequalities › Maximal and Kolmogorov-type inequalities*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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