# Doob's martingale convergence theorems

In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph L. Doob.[1]

## Statement of the theorems

In the following, (Ω, FFP), F = (Ft)t ≥ 0, will be a filtered probability space and N : [0, +∞) × Ω → R will be a right-continuous supermartingale with respect to the filtration F; in other words, for all 0 ≤ s ≤ t < +∞,

${\displaystyle N_{s}\geq \operatorname {E} {\big [}N_{t}\mid F_{s}{\big ]}.}$

### Doob's first martingale convergence theorem

Doob's first martingale convergence theorem provides a sufficient condition for the random variables Nt to have a limit as t → +∞ in a pointwise sense, i.e. for each ω in the sample space Ω individually.

For t ≥ 0, let Nt = max(−Nt, 0) and suppose that

${\displaystyle \sup _{t>0}\operatorname {E} {\big [}N_{t}^{-}{\big ]}<+\infty .}$

Then the pointwise limit

${\displaystyle N(\omega )=\lim _{t\to +\infty }N_{t}(\omega )}$

exists and is finite for P-almost all ω ∈ Ω.[2]

### Doob's second martingale convergence theorem

It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any Lp space. In order to obtain convergence in L1 (i.e., convergence in mean), one requires uniform integrability of the random variables Nt. By Chebyshev's inequality, convergence in L1 implies convergence in probability and convergence in distribution.

The following are equivalent:

${\displaystyle \lim _{C\to \infty }\sup _{t>0}\int _{\{\omega \in \Omega \,\mid \,N_{t}(\omega )>C\}}\left|N_{t}(\omega )\right|\,\mathrm {d} \mathbf {P} (\omega )=0;}$
• there exists an integrable random variable N ∈ L1(Ω, PR) such that Nt → N as t → +∞ both P-almost surely and in L1(Ω, PR), i.e.
${\displaystyle \operatorname {E} \left[\left|N_{t}-N\right|\right]=\int _{\Omega }\left|N_{t}(\omega )-N(\omega )\right|\,\mathrm {d} \mathbf {P} (\omega )\to 0{\text{ as }}t\to +\infty .}$

### Corollary: convergence theorem for continuous martingales

Let M : [0, +∞) × Ω → R be a continuous martingale such that

${\displaystyle \sup _{t>0}\operatorname {E} {\big [}{\big |}M_{t}{\big |}^{p}{\big ]}<+\infty }$

for some p > 1. Then there exists a random variable M ∈ Lp(Ω, PR) such that Mt → M as t → +∞ both P-almost surely and in Lp(Ω, PR).

## Discrete-time results

Similar results can be obtained for discrete-time supermartingales and submartingales, the obvious difference being that no continuity assumptions are required. For example, the result above becomes

Let M : N × Ω → R be a discrete-time martingale such that

${\displaystyle \sup _{k\in \mathbf {N} }\operatorname {E} {\big [}{\big |}M_{k}{\big |}^{p}{\big ]}<+\infty }$

for some p > 1. Then there exists a random variable M ∈ Lp(Ω, PR) such that Mk → M as k → +∞ both P-almost surely and in Lp(Ω, PR)

## Convergence of conditional expectations: Lévy's zero–one law

Doob's martingale convergence theorems imply that conditional expectations also have a convergence property.

Let (Ω, FP) be a probability space and let X be a random variable in L1. Let F = (Fk)kN be any filtration of F, and define F to be the minimal σ-algebra generated by (Fk)kN. Then

${\displaystyle \operatorname {E} {\big [}X\mid F_{k}{\big ]}\to \operatorname {E} {\big [}X\mid F_{\infty }{\big ]}{\text{ as }}k\to \infty }$

both P-almost surely and in L1.

This result is usually called Lévy's zero–one law or Levy's upwards theorem. The reason for the name is that if A is an event in F, then the theorem says that ${\displaystyle \mathbf {P} [A\mid F_{k}]\to \mathbf {1} _{A}}$ almost surely, i.e., the limit of the probabilities is 0 or 1. In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be. This sounds almost like a tautology, but the result is still non-trivial. For instance, it easily implies Kolmogorov's zero–one law, since it says that for any tail event A, we must have ${\displaystyle \mathbf {P} [A]=\mathbf {1} _{A}}$ almost surely, hence ${\displaystyle \mathbf {P} [A]\in \{0,1\}}$.

Similarly we have the Levy's downwards theorem :

Let (Ω, FP) be a probability space and let X be a random variable in L1. Let (Fk)kN be any decreasing sequence of sub-sigma algebras of F, and define F to be the intersection. Then

${\displaystyle \operatorname {E} {\big [}X\mid F_{k}{\big ]}\to \operatorname {E} {\big [}X\mid F_{\infty }{\big ]}{\text{ as }}k\to \infty }$

both P-almost surely and in L1.

## Doob's upcrossing inequality

The following result, called Doob's upcrossing inequality or, sometimes, Doob's upcrossing lemma, is used in proving Doob's martingale convergence theorems.[2]
Hypothesis.
Let ${\displaystyle N}$ be a natural number. Let ${\displaystyle X_{n}}$, for ${\displaystyle n=1,\ldots ,N}$, be a martingale with respect to a filtration ${\displaystyle {\mathcal {F}}_{n}}$, for ${\displaystyle n=1,\ldots ,N}$. Let ${\displaystyle a}$, ${\displaystyle b}$ be two real numbers with ${\displaystyle a.
Define the random variables ${\displaystyle U_{n}}$, for ${\displaystyle n=1,\ldots ,N}$, as follows: ${\displaystyle U_{n}=m}$ if and only if ${\displaystyle m}$ is the largest integer such that there exist integers ${\displaystyle j_{1},k_{1},j_{2},k_{2},\ldots ,j_{m}}$, ${\displaystyle k_{m}}$ satisfying 1 ≤ ${\displaystyle j_{1}}$ < ${\displaystyle k_{1} and, for ${\displaystyle i=1,\ldots ,m}$, for each pair ${\displaystyle j_{i},k_{i}}$ the inequalities ${\displaystyle X_{j_{i}} and ${\displaystyle X_{k_{i}}>b}$ are satisfied. Each ${\displaystyle U_{n}}$ is called the number of upcrossings with respect to the interval ${\displaystyle a,b}$ for the martingale ${\displaystyle X_{i}}$, ${\displaystyle i=1,\ldots ,n}$.
Conclusion.

${\displaystyle (b-a)\operatorname {E} [U_{n}]\leq \operatorname {E} [(X_{n}-a)^{-}]}$.[3][4]