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.
Facts
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 Connections
Named After
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Sources
1. Kolmogorov's three-series theorem (Wikipedia)
Statement of the TheoremQuote, Statement of the Theorem
Let ( X n ) n ∈ N be independent random variables.
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