Kolmogorov's early work on convergence theory and foundations
type: Item
instance of: scholarly article
title: Kolmogorov's early work on convergence theory and foundations
zbMATH Open document ID: 0687.01008
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: In a chronological attempt to enumerate Kolmogorov's early work on convergence theory and foundations, his joint paper with Khintchine to the problem of convergence of infinite series of discretely distributed independent random variables (1925) must be mentioned. In 1928, Kolmogorov dropped the hypothesis of discretely distributed summands and proved that an infinite series of independent mean 0 random variables converges almost surely if the series of summands variances converge, proving furthermore, his famous Three Series Theorem. Generalizing Chebyshev's inequality, at present called Kolmogorov's inequality for maxima of partial sums of mean 0 independent random variables, he uses it as a basic tool. Around 1930, it was generally understood that the basic manipulation of mathematical probabilities were the same as those of measure theory, yet the relation between the two had not been given a usable formulation. By a 1933 monograph, Kolmogorov transformed the character of the calculus of probability, showing that there is a probability measure defined on the smallest \(\sigma\) algebra of subsets of \(R^ T\), making every coordinate function measurable, assigning the prescribed joint distributions to the finite sets of coordinate functions. However, there are differences between Kolmogorov's measure conventions and the presently accepted ones, which may lead to unexpected contradictions.
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 60-03
zbMATH DE Number: 4125353
zbMATH Keywords: convergence of infinite series
zbMATH Keywords: random variables
zbMATH Keywords: discretely
zbMATH Keywords: distributed summands
zbMATH Keywords: three series theorem
zbMATH Keywords: probability measure
zbMATH Keywords: measure convention
zbMATH Keywords: Chebyshev's inequality
reviewed by: Cristina Irimia
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991247
OpenAlex ID: W2093095322
DOI: 10.1214/AOP/1176991247
author: Joseph Leo Doob
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description: scientific article; zbMATH DE number 4125353