Increasing Martingale Theorem

Theorem

Let $\struct {X, \Sigma, \mu}$ be a probability space.

Let $f$ be a $\mu$-integrable function.

Given sub-$\sigma$-algebra $\CC \subseteq \Sigma$, let $\expect {f \mid \CC}$ denote the conditional expectation of $f$ on $\CC$.

Let $\sequence {\FF_n}_{n \mathop \in \N}$ be a filtration of $\Sigma$.

Let $\FF_\infty$ be the limit of $\sequence {\FF_n}_{n \mathop \in \N}$.

Then:

$\ds \lim_{n \mathop \to \infty} \expect {f \mid \FF_n} = \expect {f \mid \FF_\infty}$

holds in $L^1$ norm, and $\mu$-almost surely.

Proof

By Tower Property of Conditional Expectation:

$\expect {f \mid \FF_n} = \expect {\expect {f \mid \FF_\infty} \mid \FF_n}$

Let $\tilde f := \expect {f \mid \FF_\infty}$.

Then we need to show that:

$\ds \lim_{n \mathop \to \infty} \expect {\tilde f \mid \FF_n} = \tilde f$

holds in $L^1$ norm, and $\mu$-almost surely.

To this end, let $\epsilon > 0$.

By Filtration's Lp Spaces are Dense in Limit Filtration's Lp Space, there exist:

$N \in \N$

$g \in \map {L^1} {X, \FF_N, \mu}$

such that:

$(1):\quad \norm {\tilde f - g}_1 \le \epsilon$

Then, for all $n \ge N$:

\(\ds \norm {\expect {\tilde f \mid \FF_n} - \tilde f}_1\)

\(=\)

\(\ds \norm {\expect {\tilde f - g \mid \FF_n} - \paren {\tilde f - g} }_1\)

\(\ds \)

\(\le\)

\(\ds \norm {\expect {\tilde f - g \mid \FF_n} }_1 + \norm {\tilde f - g}_1\)

\(\ds \)

\(\le\)

\(\ds 2 \norm {\tilde f - g}_1\)

\(\ds \)

\(\le\)

\(\ds 2 \epsilon\)

Thus:

$\ds \lim_{n \mathop \to \infty} \norm {\expect {\tilde f \mid \FF_n} - \tilde f}_1 = 0$

$\Box$

On the other hand:

\(\ds \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > \sqrt \epsilon}\)

\(=\)

\(\ds \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f - g \mid \FF_n} - \paren {\tilde f - g} } > \sqrt \epsilon}\)

\(\ds \)

\(\le\)

\(\ds \map \mu {\set {\sup_{n \mathop \in \N} \size {\expect {\tilde f - g \mid \FF_n} } > \frac {\sqrt \epsilon} 2 } \cup \set { \size { \tilde f - g} > \frac {\sqrt \epsilon} 2 } }\)

\(\ds \)

\(\le\)

\(\ds \map \mu {\sup_{n \mathop \in \N} \size {\expect {\tilde f - g \mid \FF_n} } > \frac {\sqrt \epsilon} 2 } + \map \mu { \size { \tilde f - g} > \frac {\sqrt \epsilon} 2 }\)

\(\text {(2)}: \quad\)

\(\ds \)

\(\le\)

\(\ds \map \mu {\sup_{n \mathop \in \N} \size {\expect {\tilde f - g \mid \FF_n} } > \frac {\sqrt \epsilon} 2 } + \frac 2 {\sqrt \epsilon} \norm {\tilde f - g}_1\)

Markov's Inequality

Observe:

\(\ds \map \mu { \sup_{n \in \N} \size{ \expect { {\tilde f - g} \mid \FF_n} } \ge \frac {\sqrt \epsilon} 2}\)

\(\le\)

\(\ds \map \mu { \sup_{n \in \N} \expect {\size {\tilde f - g} \mid \FF_n} \ge \frac {\sqrt \epsilon} 2}\)

\(\ds \)

\(=\)

\(\ds \lim_{m \to \infty} \map \mu { \sup_{0 \le n \le m} \expect {\size {\tilde f - g} \mid \FF_n} \ge \frac {\sqrt \epsilon} 2}\)

Measure of Limit of Increasing Sequence of Measurable Sets

\(\ds \)

\(\le\)

\(\ds \lim_{m \to \infty} \frac 2 {\sqrt \epsilon} \expect {\expect {\size {\tilde f - g} \mid \FF_m} }\)

Doob's Maximal Inequality

\(\ds \)

\(=\)

\(\ds \frac 2 {\sqrt \epsilon} \expect {\size {\tilde f - g} }\)

Definition of Conditional Expectation on Sigma-Algebra

\(\text {(3)}: \quad\)

\(\ds \)

\(=\)

\(\ds \frac 2 {\sqrt \epsilon} \norm {\tilde f - g}_1\)

Definition of $L^1$ norm

From $(2)$ and $(3)$:

\(\ds \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > \sqrt \epsilon}\)

\(\le\)

\(\ds \frac 4 {\sqrt \epsilon} \norm {\tilde f - g}_1\)

\(\ds \)

\(\le\)

\(\ds 4 \sqrt \epsilon\)

by $(1)$

Therefore:

\(\ds \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > 0}\)

\(=\)

\(\ds \lim_{\epsilon \mathop \to 0} \map \mu {\limsup_{n \mathop \to \infty} \size {\expect {\tilde f \mid \FF_n} - \tilde f} > \sqrt \epsilon}\)

Measure of Limit of Increasing Sequence of Measurable Sets

\(\ds \)

\(\le\)

\(\ds \lim_{\epsilon \mathop \to 0} 4 \sqrt \epsilon\)

\(\ds \)

\(=\)

\(\ds 0\)

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Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $5.2:$ Martingales