Probability and Measure

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

Type

Book

Intellectual base

Textbook

Year

1995

Link

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

Cited from

mathematics, physics

Cited by

13 citations across 13 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.
• 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..
• 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.

• 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

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Links previously used in the corpus

Before the registry existed this work was also linked 2 other ways.

• 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