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Tower property of conditional expectation
Statement
Assume AC for existence. If and , then and almost surely. Also, if a version of is -measurable, it is a version of .
Facts & Assumptions
Given: AC, real integrable X, and ; for the last clause a G-conditional version is H-measurable.
Conditional versions are measurable and integrable and have the defining event integrals. (Conditional expectation as an ae class)
Versions for the same input and sigma-algebra agree almost surely. (Conditional expectation is unique almost surely)
Proof
Set and take a version . For , . Since is -measurable and integrable, [F2] identifies it with .
A version is already -measurable and integrable. It has its own event integrals on every , so it is a version of ; [F2] gives the second identity. Finally if itself is -measurable, its event identities restrict to , so [F2] gives the additional assertion.
Source notes
Durrett Theorems 4.1.12–4.1.13, printed p.212; van der Vaart Lemma 1.9(v), printed p.4.
Depends on
Used by
- Heat-semigroup martingales Corollary
- Square-integrable Brownian terminal variables have Ito representations Corollary
- The exponential Brownian martingale Corollary
- Vector Levy characterization Corollary
- Reverse filtration and reverse martingale Definition
- A reverse martingale and the tail sigma-algebra Example
- Characteristic exponential for a continuous local martingale with deterministic clock Lemma
- Concentration for adaptively sampled MDP observations Lemma
- Conditional CLT for a random subset of an IID finite population Lemma
- Conditional expectation process is a martingale Lemma
- Conditional independence under kernel extension and conditioning Lemma
- Conditional-independence equivalences and preservation Lemma
- Elementary Ito integrals do not depend on step representation Lemma
- Finite-horizon Brownian change of drift and relative entropy Lemma
- Geometric ergodicity and reversibility give exponential rho-mixing Lemma
- Multistep martingale characterization Lemma
- Predictably weighted bounded martingale differences have a convergent series Lemma
- Azuma-Hoeffding inequality Theorem
- Brownian-filtration martingale representation Theorem
- Chapman-Kolmogorov equations Theorem
- Communicating MDPs have a common optimal average reward Theorem
- Conditional-class standardization identifies a causal mean Theorem
- Conditional-error budget for two-stage adaptation Theorem
- Discrete-treatment potential-outcome means are identifiable Theorem
- Doob maximal bound for the Ito integral Theorem
- Empirical-Bernstein confidence sequence under adaptive sampling Theorem
- Escape-event reduction for optimistic greedy learning Theorem
- Expected-average stationarity under aligned stochastic directions Theorem
- Finite-dimensional laws of a Markov chain Theorem
- Front-door adjustment identifies an interventional mean Theorem
- Future-path Markov property Theorem
- Inverse probability weighting identifies a causal mean Theorem
- Inverse response-probability weighting identifies the mean under MAR Theorem
- Ito isometry for elementary integrands Theorem
- Levy characterization of Brownian motion Theorem
- Levy downward convergence of conditional expectations Theorem
- Martingale central limit theorem Theorem
- Nonconvex SGD weighted random-iterate stationarity Theorem
- Optional stopping under uniform integrability Theorem
- Quadratic covariation of Brownian Ito processes Theorem
…and 5 more results.
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Durrett, Probability: Theory and Examples, 5th ed. (standard reference, not scraped)