Assuming countable and dependent choice, countable products of arbitrary probability spaces

Depends on

• Coordinate maps, finite-coordinate cylinders, and the cylinder $\sigma$-algebra
• Finite-coordinate cylinders form a $\pi$-system
• The countable-product cylinder premeasure is countably additive
• Assuming countable choice, a premeasure extends through its induced outer measure
• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
• The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
• Dynkin's pi-lambda theorem

Used by

• Critical-reaction binomial guarded tau-leaping Algorithm
• Dominating thinning for bounded time-varying propensities Algorithm
• Exact uniform sampling from the nonzero residue representatives Algorithm
• Gillespie direct method with horizon and budget statuses Algorithm
• Minimum-intensity coupling of reaction networks Algorithm
• Next-reaction method by internal Poisson clocks Algorithm
• Shor factoring with verified output and bounded outer retries Algorithm
• Birkhoff strong law for iid coordinate shifts Corollary
• Coordinate random elements of a countable product are independent Corollary
• Countably many independent copies of a prescribed law exist Corollary
• Diminishing adaptation and uniform ergodicity need not give a strong law Counterexample
• Quadratic pure-birth rates can explode Counterexample
• Summable untruncated variances are not necessary Counterexample
• Empirical-Bayes credible-ball procedure in the inverse Gaussian sequence model Definition
• A canonical random walk from product increments Example
• A Lindeberg array with no identically distributed row Example
• A nonidentical Bernoulli weak law Example
• Almost sure frequency of heads Example
• An i.i.d. sequence with a prescribed law Example
• Independent but non-identically distributed coordinates Example
• The infinite fair-coin-toss space Example
• The three series impose separate conditions Example
• A universal Martin-Löf test exists Lemma
• Adjacent iid vector functionals admit a finite-moment strong approximation Lemma
• Dynkin identity for a bounded-rate finite stopped reaction chain Lemma
• Finite-moment iid vectors admit a strong Gaussian approximation Lemma
• Geometric Markov chains admit delayed adjacent-iid split tours with spare moments Lemma
• Independent Brownian motion straightens a Poisson clock Lemma
• One-step weak defect of finite-state Poisson tau-leaping Lemma
• Posterior and marginal-likelihood formulas for the Gaussian sequence prior Lemma
• State-space and index-set boundaries of the two extension routes Remark
• A proper Lyapunov bound prevents reaction-network explosion Theorem
• Bounded-error success of Shor factoring Theorem
• Compensators, brackets, and jump-integral isometry for reaction counts Theorem
• Disjoint streams of independent fair bits produce independent uniform variables Theorem
• Dominating thinning has the prescribed reaction law Theorem
• Existence of a locally finite Poisson point process Theorem
• Finite-state Gillespie and next-reaction methods have the specified CTMC law Theorem
• Rejection sampling is uniform, almost surely terminating, and has expected linear bit cost Theorem
• Strong invariance principle for geometrically ergodic chains Theorem

…and 2 more results.

Dependency tree · two levels

37 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

• Kajino, Probability Theory, Theorem 3.65 (standard reference, not scraped)