Paper VI: Conditional expectation as a stochastic projector

6 Conditional expectation as a stochastic projector

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Conditional expectation is a projector in a weighted Hilbert space. Nested partitions therefore split information into orthogonal increments. Keep the underlying measure fixed while reading the projector identities. A change of measure is a different operation.

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Let \(\Omega=\{1,\ldots,n\}\) carry strictly positive probabilities \(\pi_i\), and let \(\mathcal Q=\{B_1,\ldots,B_m\}\) be a partition. Define

\[ (P_{\mathcal Q}f)_i = \frac{\sum_{j\in B(i)}\pi_jf_j}{\sum_{j\in B(i)}\pi_j}, \]

where \(B(i)\) is the block containing \(i\).

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Theorem 6.1: Finite conditional-expectation projector

With the weighted inner product

\[ \langle f,g\rangle_\pi=\sum_i\pi_if_i\overline{g_i}, \]

the operator \(P_{\mathcal Q}\) satisfies

\[ P_{\mathcal Q}^2=P_{\mathcal Q}, \qquad \langle P_{\mathcal Q}f,g\rangle_\pi = \langle f,P_{\mathcal Q}g\rangle_\pi. \]

Its range is the space of functions constant on each partition block.

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Proof of Theorem 6.1

After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block; both equal

\[ \sum_{B\in\mathcal Q} \frac{1}{\pi(B)} \left(\sum_{i\in B}\pi_i f_i\right) \overline{\left(\sum_{j\in B}\pi_j g_j\right)}. \]

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Theorem 6.2: Tower law and martingale packets

If \(\mathcal Q_c\) is coarser than \(\mathcal Q_f\), then

\[ \boxed{ P_cP_f=P_fP_c=P_c. } \]

For a nested tower \(P_0\le P_1\le\cdots\le P_r\), define

\[ D_0=P_0, \qquad D_j=P_j-P_{j-1}. \]

Then

\[ D_iD_j=\delta_{ij}D_j, \qquad P_r=\sum_{j=0}^rD_j. \]

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Proof of Theorem 6.2

A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from \(P_iP_j=P_{\min(i,j)}\).

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Hermite--Gaussian packets, which diagonalize the Ornstein--Uhlenbeck semigroup. The finite numerical release verifies their Gaussian orthogonality through quadrature. The five displayed modes run from n = 0 at the bottom to n = 4 at the top. Vertical offsets separate the curves. — Hermite–Gaussian packets, which diagonalize the Ornstein–Uhlenbeck semigroup. The finite numerical release verifies their Gaussian orthogonality through quadrature. The five displayed modes run from n = 0 at the bottom to n = 4 at the top. Vertical offsets separate the curves.

Classical input 6.3: Gaussian chaos realization

For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein–Uhlenbeck semigroup acts diagonally:

\[ T_tH_n=e^{-nt}H_n. \]

Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.

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