Martingale Convergence Theorems

The martingale convergence theorems are fundamental results stating that, under appropriate conditions, martingales converge almost surely to a limit. These theorems underpin the theory of stochastic processes, providing conditions for long-run convergence of random sequences.

Doob's martingale convergence theorem

Theorem3.1Doob's martingale convergence theorem

Let $(M_n)_{n \geq 0}$ be a submartingale (or supermartingale) with $\sup_n \mathbb{E}[|M_n|] < \infty$ . Then there exists a random variable $M_\infty$ with $\mathbb{E}[|M_\infty|] < \infty$ such that:

$M_n \to M_\infty \quad \text{almost surely as } n \to \infty.$

For martingales: If $(M_n)$ is a martingale with $\sup_n \mathbb{E}[M_n^2] < \infty$ (bounded in $L^2$ ), then $M_n \to M_\infty$ a.s.

Intuition: A martingale with bounded variation cannot "wander off" indefinitely — it must converge to a limit. The key is that the expected absolute value (or second moment) is uniformly bounded, preventing the process from escaping to infinity.

ExamplePólya urn

An urn initially contains 1 red ball and 1 blue ball. At each step, draw a ball uniformly at random, then return it along with one additional ball of the same color. Let $R_n$ be the number of red balls after $n$ draws and $X_n = R_n/(n+2)$ the proportion of red balls.

Then $(X_n)$ is a martingale: $\mathbb{E}[X_{n+1} \mid \mathcal{F}_n] = X_n$ . Moreover, $0 \leq X_n \leq 1$ , so $\mathbb{E}[X_n^2] \leq 1$ for all $n$ . By Doob's theorem, $X_n \to X_\infty$ a.s. for some random variable $X_\infty$ .

Remarkably, $X_\infty \sim \text{Uniform}(0, 1)$ : the limiting proportion of red balls is uniformly distributed on $[0, 1]$ . This is a classic example of exchangeability and the de Finetti theorem.

$L^2$ convergence

Theorem3.2$L^2$ martingale convergence

If $(M_n)$ is a martingale with $\sup_n \mathbb{E}[M_n^2] < \infty$ , then:

$M_n \to M_\infty$ almost surely.
$M_n \to M_\infty$ in $L^2$ : $\mathbb{E}[(M_n - M_\infty)^2] \to 0$ .
$\mathbb{E}[M_\infty^2] \leq \liminf_{n \to \infty} \mathbb{E}[M_n^2]$ .

$L^2$ convergence is stronger than almost sure convergence: it implies convergence of second moments and allows us to pass limits through expectations.

ExampleSimple random walk up to a stopping time

Let $X_n$ be a simple random walk and $\tau = \inf\{n : X_n \notin (-a, a)\}$ the exit time from $(-a, a)$ . Define $M_n = X_{n \wedge \tau}$ (the stopped process). Then:

$(M_n)$ is a martingale (stopped processes preserve the martingale property).
$|M_n| \leq a$ for all $n$ , so $\mathbb{E}[M_n^2] \leq a^2$ .

By the convergence theorem, $M_n \to M_\infty$ a.s. and in $L^2$ . Since $\tau < \infty$ a.s., $M_\infty = X_\tau \in \{-a, a\}$ .

Uniform integrability

Definition3.1Uniformly integrable

A family of random variables $\{X_\alpha\}_{\alpha \in A}$ is uniformly integrable (UI) if:

$\lim_{K \to \infty} \sup_{\alpha \in A} \mathbb{E}[|X_\alpha| \mathbf{1}_{|X_\alpha| > K}] = 0.$

Equivalently, $\sup_\alpha \mathbb{E}[|X_\alpha|] < \infty$ and for any $\varepsilon > 0$ , there exists $\delta > 0$ such that $\mathbb{P}(A) < \delta$ implies $\mathbb{E}[|X_\alpha| \mathbf{1}_A] < \varepsilon$ for all $\alpha$ .

Theorem3.3$L^1$ convergence

If $(M_n)$ is a martingale and the family $\{M_n\}$ is uniformly integrable, then:

$M_n \to M_\infty$ almost surely.
$M_n \to M_\infty$ in $L^1$ : $\mathbb{E}[|M_n - M_\infty|] \to 0$ .
$\mathbb{E}[M_\infty] = \mathbb{E}[M_0]$ .

Uniform integrability is the key to $L^1$ convergence. It ensures that the "tails" of the distribution do not contribute significantly, allowing us to pass limits through expectations.

ExampleBounded martingales are UI

If $|M_n| \leq C$ a.s. for all $n$ , then $\{M_n\}$ is uniformly integrable (trivially: $\mathbb{E}[|M_n| \mathbf{1}_{|M_n| > K}] = 0$ for $K \geq C$ ). Hence, bounded martingales converge in $L^1$ .

Doob's upcrossing inequality

Theorem3.4Upcrossing inequality

Let $(X_n)$ be a submartingale and $U_n(a, b)$ the number of upcrossings of the interval $[a, b]$ by $X_0, X_1, \ldots, X_n$ (i.e., the number of times the process crosses from below $a$ to above $b$ ). Then:

$\mathbb{E}[U_n(a, b)] \leq \frac{\mathbb{E}[(X_n - a)^+]}{b - a}.$

The upcrossing inequality is the key tool in proving Doob's convergence theorem. If $\mathbb{E}[(X_n - a)^+]$ is uniformly bounded, then the expected number of upcrossings is finite, which implies that $X_n$ cannot oscillate indefinitely — it must converge.

Backwards martingales

Definition3.2Backwards martingale

A sequence $(M_n)_{n \in \mathbb{Z}}$ indexed by negative integers is a backwards martingale if:

$\mathbb{E}[M_n \mid M_{n-1}, M_{n-2}, \ldots] = M_{n-1}.$

Equivalently, with reversed indexing, $M_{-n}$ is a forwards martingale.

Theorem3.5Backwards martingale convergence

Every backwards martingale converges almost surely and in $L^1$ : there exists $M_{-\infty}$ such that $M_n \to M_{-\infty}$ a.s. as $n \to -\infty$ .

No integrability condition is needed for backwards martingales — they always converge! This is because moving backwards in time provides more and more information, naturally leading to a limit.

ExampleExchangeable sequences and de Finetti

Let $X_1, X_2, \ldots$ be an exchangeable sequence of $\{0, 1\}$ -valued random variables. Define $M_n = \mathbb{E}[X_1 \mid X_n, X_{n+1}, \ldots]$ . Then $(M_n)$ is a backwards martingale (as $n$ decreases, we condition on more information). By the backwards convergence theorem, $M_n \to M_{-\infty}$ a.s.

The de Finetti theorem states that $M_{-\infty} = \theta$ for some random variable $\theta$ , and conditionally on $\theta$ , the $X_i$ are i.i.d. Bernoulli $(\theta)$ . This is the foundation of Bayesian statistics: exchangeable sequences arise from a mixture of i.i.d. sequences.

Summary

Martingale convergence theorems provide conditions for almost sure and $L^p$ convergence:

Doob's theorem: If $\sup_n \mathbb{E}[|M_n|] < \infty$ , then $M_n \to M_\infty$ a.s.
$L^2$ convergence: If $\sup_n \mathbb{E}[M_n^2] < \infty$ , then $M_n \to M_\infty$ in $L^2$ .
$L^1$ convergence: If $\{M_n\}$ is uniformly integrable, then $M_n \to M_\infty$ in $L^1$ .
Backwards martingales: Always converge (no integrability needed).

These results are central to probability theory, with applications to ergodic theory, statistical inference, and stochastic analysis.