Abstract
Random elements and processes; distributions and expectation; independence; zero-one laws; Borel-Cantelli lemma; Bernoulli sequences and existence; moments and continuity of paths.
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The use of countably additive probability measures dates back to Borel (1909), who constructed random variables as measurable functions on the Lebesgue unit interval and proved Theorem 3.18 for independent events. Cantelli (1917) noticed that the “easy” part remains true without the independence assumption. Lemma 3.5 was proved by Jensen (1906) after Holder had obtained a special case.
The modern framework, with random variables as measurable functions on an abstract probability space (Ω,A, P) and with expected values as P-integrals over Ω, was used implicitly by Kolmogorov from (1928) on and was later formalized in Kolmogorov (1933). The latter monograph also contains Kolmogorov’s zero-one law, discovered long before Hewitt and Savage (1955) obtained theirs.
Early work in probability theory deals with properties depending only on the finite-dimensional distributions. Wiener (1923) was the first author to construct the distribution of a process as a measure on a function space. The general continuity criterion in Theorem 3.23, essentially due to Kol-Mogorov, was first published by Slutsky (1937), with minor extensions later added by Loève (1978) and Chentsov (1956). The general search for regularity properties was initiated by Doob (1937, 1947). Soon it became clear, especially through the work of Lévy (1934–35, 1954), Doob (1951, 1953), and Kinney (1953), that most processes of interest have right-continuous versions with left-hand limits.
More detailed accounts of the material in this chapter appear in many textbooks, such as in Billingsley (1995), Itô (1984), and Williams (1991). Further discussions of specific regularity properties appear in Loève (1977) and Cramer and Leadbetter (1967). Earlier texts tend to give more weight to distribution functions and their densities, less weight to measures and σ-fields.
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© 2002 Springer Science+Business Media New York
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Kallenberg, O. (2002). Processes, Distributions, and Independence. In: Foundations of Modern Probability. Probability and Its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4015-8_3
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