Foundations of the Theory of Probability¶
Kolmogorov, A. N. (1956). Foundations of the Theory of Probability. Grundbegriffe der Wahrscheinlichkeitsrechnung.
Cited by¶
3 citations across 3 artifacts.
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Primes¶
- Disjointness
- In set theory and probability, disjoint events satisfy P(A ∪ B) = P(A) + P(B), and partitions are families of pairwise disjoint sets that cover a whole.
This sourceAxiomatizes probability with finite/countable additivity over disjoint (mutually exclusive) events, P(A ∪ B) = P(A) + P(B), distinguishing mutual exclusivity from statistical independence.
- In set theory and probability, disjoint events satisfy P(A ∪ B) = P(A) + P(B), and partitions are families of pairwise disjoint sets that cover a whole.
- Statistical Independence
- Its precise signature is a factorization: the joint distribution equals the product of the marginals, written \(P(A \cap B) = P(A)\,P(B)\), and in the conditional form factoring given a separator set.
This sourceGives the measure-theoretic axiomatization of probability and the definition of independence as the factorization P(A∩B)=P(A)P(B).
- Its precise signature is a factorization: the joint distribution equals the product of the marginals, written \(P(A \cap B) = P(A)\,P(B)\), and in the conditional form factoring given a separator set.
- Stochastic Process
- In mathematics and probability theory it is the foundational object: Kolmogorov's extension theorem constructs processes from their finite-dimensional distributions, and the whole edifice — martingales, stationary processes, ergodic theory, the classification into Markov, Gaussian, Lévy, and point processes — is the study of stochastic processes and their dependence structures.
This sourceContains the extension theorem constructing a process from a consistent family of finite-dimensional distributions.
- In mathematics and probability theory it is the foundational object: Kolmogorov's extension theorem constructs processes from their finite-dimensional distributions, and the whole edifice — martingales, stationary processes, ergodic theory, the classification into Markov, Gaussian, Lévy, and point processes — is the study of stochastic processes and their dependence structures.
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