Doob’s inequalities

Doob’s inequalitiesMathworldPlanetmath place bounds on the maximum value attained by a martingaleMathworldPlanetmath in terms of the terminal value. We consider a process (Xt)t𝕋 defined on the filtered probability space (Ω,,()t𝕋,). The associated maximum process (Xt*) is


The notation p for the Lp-norm ( of a random variableMathworldPlanetmath will be used. In discrete-time or, more generally whenever the index setMathworldPlanetmathPlanetmath 𝕋 is countableMathworldPlanetmath, then Doob’s inequalities are as follows.

Theorem 1 (Doob).

Let (Xt)tT be a submartingale with countable index set T. Then,

(supstXsK)K-1𝔼[(Xt)+] (1)

If X is either a martingale or nonnegative submartingale then,

(Xt*K)K-1𝔼[|Xt|], (2)
Xt*ppp-1Xtp. (3)

for every K>0 and p>1.

In particular, (3) shows that the maximum of any Lp-bounded martingale is itself Lp-bounded and, martingales Xn converge to X in the Lp-norm if and only if (Xn-X)*0 in the Lp-norm. The special case where p=2 gives


which is known as Doob’s maximal quadratic inequality.

Similarly, (2) shows that any L1-bounded martingale is almost surely bounded and that convergence in the L1-norm implies ucp convergence. Inequality (1) is also known as Kolmogorov’s submartingale inequality.

Doob’s inequalities are often applied to continuous-time processes, where 𝕋=+. In this case, Xt*=supst|Xs| is a supremum of uncountably many random variables, and need not be measurable. Instead, it is typically assumed that the processes are right-continuous, in which case, for any t>0 the supremum may be restricted to the countable set


Putting this into Theorem 1 gives the following continuous-time version of the inequalities.

Theorem 2 (Doob).

Let (Xt)tR+ be a right-continuous submartingale. Then,


for every K>0. If X is right-continuous and either a martingale or nonnegative submartingale then,


for every K>0 and p>1.

Title Doob’s inequalities
Canonical name DoobsInequalities
Date of creation 2013-03-22 18:39:52
Last modified on 2013-03-22 18:39:52
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Theorem
Classification msc 60G46
Classification msc 60G44
Classification msc 60G42
Synonym Doob’s inequality
Related topic KolmogorovsMartingaleInequality
Defines Doob’s maximal quadratic inequality