Paper VI: Conditional expectation and martingale packets

E6 Conditional expectation and martingale packets

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Let \(\Omega=\{1,\ldots,n\}\) have strictly positive probabilities \(\pi_i\). For a partition \(\mathcal Q\), let \(B(i)\) denote the block containing \(i\). Define

\[ (P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}. \]

Equip \({\mathbb C}^n\) with

\[ {\left\langle f,g\right\rangle}_\pi = \sum_i\pi_if_i\overline{g_i}. \]

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Theorem E6.1: Finite conditional expectation

The operator \(P_\mathcal Q\) is the orthogonal projector onto the functions that are constant on every block of \(\mathcal Q\). In particular,

\[ P_\mathcal Q^2=P_\mathcal Q, \qquad P_\mathcal Q^*=P_\mathcal Q. \]

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Proof of Theorem E6.1

One application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives

\[ {\left\langle P_\mathcal Qf,g\right\rangle}_\pi = \sum_{B\in\mathcal Q} \frac{ \left(\sum_{i\in B}\pi_if_i\right) \overline{\left(\sum_{j\in B}\pi_jg_j\right)} }{\pi(B)} = {\left\langle f,P_\mathcal Qg\right\rangle}_\pi. \]

The range is exactly the block constant subspace.

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Theorem E6.2: Tower law and martingale packets

Let \(P_0,P_1,\ldots,P_r\) come from increasingly fine partitions, so that

\[ {\operatorname{Ran}} P_0\subseteq{\operatorname{Ran}} P_1\subseteq\cdots\subseteq{\operatorname{Ran}} P_r. \]

Define

\[ D_0=P_0, \qquad D_j=P_j-P_{j-1} \quad(1\leq j\leq r). \]

Then

\[ P_iP_j=P_{\min(i,j)}, \qquad D_iD_j=\delta_{ij}D_j, \qquad P_r=\sum_{j=0}^rD_j. \]

Every \(D_j\) is an orthogonal projector, and

\[ {\left\lVert P_rf\right\rVert}_\pi^2 = \sum_{j=0}^r{\left\lVert D_jf\right\rVert}_\pi^2 \]

and

\[ {\left\lVert f\right\rVert}_\pi^2 = \sum_{j=0}^r{\left\lVert D_jf\right\rVert}_\pi^2 + {\left\lVert f-P_rf\right\rVert}_\pi^2. \]

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Proof of Theorem E6.2

A coarse average of a fine average is the coarse average. A fine average leaves a coarse constant function unchanged. Hence \(P_iP_j=P_{\min(i,j)}\). Expanding products of the differences gives the orthogonality relations. The sum telescopes to \(P_r\). Orthogonal Pythagorean identities give the norm formulas.

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Corollary E6.3: Law of total variance

For a real valued \(f\), let \(P_0f={\mathbb E}_\pi[f]{{\boldsymbol{1}}}\). Then

\[ {\operatorname{Var}}_\pi(f) = {\operatorname{Var}}_\pi(P_\mathcal Qf) + {\left\lVert f-P_\mathcal Qf\right\rVert}_\pi^2. \]

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Proof of Corollary E6.3

Subtract the mean and apply the orthogonal decomposition into the range of \(P_\mathcal Q\) and its orthogonal complement.

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Energy carried by the mean packet and three successive martingale difference packets in a finite weighted example.

Standard fact E6.4: General conditional expectations

On a probability space, conditional expectation onto a sub sigma algebra is the orthogonal projector in \(L^2\). Nested sigma algebras satisfy the same tower law. Under the usual hypotheses, martingale convergence gives a limit in \(L^2\) and almost surely. Standard treatments include [Williams1991].

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