Kolmogorov's Three-Series Theorem gives a necessary and sufficient condition for an infinite series of independent random variables to converge almost surely, stating that this holds exactly when three associated series each converge: one governing the probability the terms exceed a fixed truncation level, one of the truncated terms' expectations, and one of the truncated terms' variances. Named for Andrey Kolmogorov, it is a foundational criterion of probability theory for almost-sure convergence of random series.
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StatementFor independent random variables, the series of X_n converges almost surely if for some A > 0, and only if for any A > 0, three series converge: the probabilities that |X_n| is at least A, the expectations of the truncated variables, and their variances. 1 Classification
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Source Kolmogorov's three-series theorem (Wikipedia)
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Source Kolmogorov's three-series theorem (Wikipedia)
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1. Kolmogorov's three-series theorem (Wikipedia)
Statement of the Theorem
Let ( X n ) n ∈ N be independent random variables.
In Branch: Probability and Statistics, Lead sentence
In probability theory, Kolmogorov's three-series theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure con
Proved By: Andrey Kolmogorov, Lead section
Kolmogorov's three-series theorem, named after Andrey Kolmogorov
In Group: Probability Theory, Lead section
In probability theory, Kolmogorov's three-series theorem, named after
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