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Conditional expectation as an ae class
Definition
Assume AC for the supplied existence theorem. Write for the unique class in consisting of conditional-expectation versions of . A chosen real -measurable representative is a version. Equalities and inequalities involving these classes mean almost-sure equalities and inequalities.
Existence is Conditional expectation exists by radon nikodym and uniqueness is Conditional expectation is unique almost surely, applied to Conditional expectation given a sigma algebra. The quotient is The space as the quotient by null functions. If almost surely, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree gives identical event integrals, so the output class is independent of the input representative. The The Axiom of Choice assumption is inherited from RN existence.
Definition references
Source notes
Durrett §4.1, printed p.206, version/uniqueness convention; van der Vaart Definition 1.1 and Theorem 1.3, printed pp.1–2.
Depends on
- Conditional expectation given a sigma algebra
- Conditional expectation exists by radon nikodym
- Conditional expectation is unique almost surely
- The space $L^p(\mu)$ as the quotient by null functions
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- The Axiom of Choice
Used by
- Inverse probability weighting for a causal mean Algorithm
- Standardization for a causal mean Algorithm
- A conditionally superuniform stage-2 p-value preserves conditional error Corollary
- Conditional cauchy schwarz inequality Corollary
- Heat-semigroup martingales Corollary
- One-dimensional Brownian motion is recurrent Corollary
- Square-integrable Brownian terminal variables have Ito representations Corollary
- The exponential Brownian martingale Corollary
- Vector Levy characterization Corollary
- A submartingale need not have increasing sample paths Counterexample
- A supermartingale supremum threshold need not be attained Counterexample
- A version can fail a pointwise identity on a null set Counterexample
- An unbounded predictable transform may lose integrability Counterexample
- An unbounded stopped exponential martingale needs uniform integrability Counterexample
- Conditioning does not preserve strict inequalities Counterexample
- Strong Markov fails at a nonstopping random time Counterexample
- Taking out an unbounded factor needs integrability Counterexample
- Average and conditional average treatment effects Definition
- Conditional expectation for nonnegative variables Definition
- Conditional independence given a sigma-algebra Definition
- Conditional probability given a sigma algebra Definition
- Conditional variance Definition
- Conditionally sub-Gaussian increments Definition
- Continuous-time adapted processes and martingales Definition
- Continuous-time filtrations and all-pairs martingales Definition
- Filtration-adapted stochastic-gradient oracles and Robbins–Monro updates Definition
- Martingale difference sequence Definition
- Martingale submartingale and supermartingale Definition
- Positivity for discrete treatment Definition
- Predictable quadratic variation in discrete time Definition
- Reverse filtration and reverse martingale Definition
- Simple counting-process intensity and compensator contract Definition
- Square-integrable martingale-difference array and variance clock Definition
- Calibrated selected sample size and an uncalibrated weight change Example
- Conditional expectation given a discrete random variable Example
- Conditioning on a finite partition Example
- Constant-step SGD can have a nonzero noise floor Example
- Delayed entry with a predictable risk process Example
- Dyadic conditional expectation martingale Example
- L² best prediction by conditional expectation Example
…and 86 more results.
Dependency tree · two levels
20 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)