Slutsky's Theorem states that if a sequence of random variables converges in distribution to some limit and a second sequence converges in probability to a constant, then the sum, product and ratio of the two sequences converge in distribution to the corresponding combination of the limit and the constant. Named for Evgeny Slutsky, it is a standard tool in statistics for establishing the asymptotic distribution of estimators built by combining simpler pieces.
Facts
StatementSlutsky's theorem states that if one sequence of random variables converges in distribution to a limit and a second sequence converges in probability to a constant, then the sum, product and ratio of the two sequences converge in distribution to the corresponding combination of the limit and the constant. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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Source Slutsky's Theorem (Wikipedia)
Sources
1. Slutsky's Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, first sentence
In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.
Proved By: Harald Cramér, Lead paragraph
The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed to Harald Cramér.
In Group: Probability Theory, Lead section
In probability theory, Slutsky's theorem extends some properties of
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