Probability and Measure

Billingsley, P. (1995). Probability and Measure. Wiley.

Type

Book

Intellectual base

Textbook

Year

1995

Pages cited

6-37

Link

https://www.wiley.com/en-us/Probability+and+Measure%2C+3rd+Edition-p-9780471007104

Cited from

mathematics, physics

Cited by

15 citations across 15 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Central Limit Theorem
  • The classical Lindeberg–Lévy form requires only that the contributions be independent, identically distributed, and have finite variance; generalizations relax the identical-distribution and independence assumptions.

    This sourceRigorous statement and proof of the Lindeberg–Lévy and Lindeberg–Feller central limit theorems, the finite-variance and Lindeberg conditions, and the 1/√n convergence rate of the sample mean.

• Law of Large Numbers
  • The claim is about the normalized aggregate and nothing else: the division by n is not bookkeeping but the whole content of the theorem, since it is what converts an accumulating quantity into a settling one.

    This sourceStates the law of large numbers for the normalized partial sum under a finite expectation with no finite-variance requirement, and extends convergence of averages to ergodic stationary sequences, martingale differences, and independent non-identically distributed terms.

• Measure
  • Conditional probability, change-of-variable, importance sampling, stratified surveying, and risk-adjusted return are all the same structural move — re-weighting against a different base measure — repeated in different clothing.

    This sourceTreats the Radon–Nikodym derivative / change of measure as the single operation underlying conditional probability, change of variable, and importance/re-weighting.

• Universality
  • Mathematics: the central limit theorem makes the Gaussian the universal limit of sums of many independent contributions almost regardless of their distribution; random matrix theory shows identical eigenvalue statistics emerging across vast classes of matrix ensembles.

    This sourceStandard graduate text proving the central limit theorem — the Gaussian as the universal limit of sums of independent contributions with finite variance, the probabilistic instance of universality.

Domain-specific

• Asymptotic theory (statistics)
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Borel Set
  • They support probability distributions on \(\mathbb R^n\) and function spaces, Borel measures, measurable maps, stochastic processes, dynamical systems, harmonic analysis, descriptive set theory, and standard Borel models in statistics and economics.
• Brownian Skorokhod Embedding
  • In the classical centered finite-variance setting, suitable constructions have finite expected duration; Billingsley's first embedding theorem gives one with $E[T]=\int x^2\,\mu(dx)$.
• Empirical process
  • … population law, index class, empirical measure, centering and scaling, function-space topology and measurability, entropy or complexity conditions and convergence mode are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Murphy's Law
  • The empirical part is a corollary of the Borel–Cantelli observation: for any failure mode with non-zero per-trial probability, occurrence across sufficiently many independent trials approaches certainty — every permitted failure state will eventually be visited.

    This sourceBillingsley's Probability and Measure states the second Borel-Cantelli lemma (occurrence infinitely often with probability one when the sum of the probabilities diverges); the text checked does not itself state the independence condition the sentence relies on.

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Probability Mass Function
  • Measure-theoretically, it is the Radon–Nikodym density of the law of \(X\) with respect to counting measure.
• Probability measure
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Radon–Nikodym theorem
  • If ν is absolutely continuous with respect to a suitable measure μ, the theorem provides a measurable derivative dν/dμ such that ν(A)=∫A(dν/dμ)dμ for every measurable set.
• Random Variable
  • Sequences of random variables converge in multiple distinct senses — almost surely, in probability, in distribution, and in Lp — and the choice of convergence mode matters for the law of large numbers, the central limit theorem, and consistency results in statistics

    This sourceDevelops the distinct modes of convergence for sequences of random variables and their role in the law of large numbers and the central limit theorem; the statistical notion of consistency is outside its scope.

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Tightness of measures
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Total variation
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Verification

Does it exist? Not checked yet. This entry carries no identifier to resolve. It was extracted from the citation as written in the article, normalized, and deduplicated against the rest of the registry.

Does it back the claim? Read against the text for 2 of 15 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Links previously used in the corpus

Before the registry existed this work was also linked 3 other ways.

• https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf ×1
• https://www.wiley.com/en-us/Probability+and+Measure%2C+Anniversary+Edition-p-9781118122372 ×1
• https://www.worldcat.org/isbn/9780471007104 ×1

Registry ID ref:108665861969 · see in the full table