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In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. [1] The theorem was named after Eugen Slutsky. [2] Slutsky's theorem is also attributed to Harald Cramér. [3]
The proof can be found in Page 126 (Theorem 5.3.4) of the book by Kai Lai Chung. [13] However, for a sequence of mutually independent random variables, convergence in probability does not imply almost sure convergence. [14] The dominated convergence theorem gives sufficient conditions for almost sure convergence to imply L 1-convergence:
Proof: We will prove this statement using the portmanteau lemma, part A. First we want to show that ( X n , c ) converges in distribution to ( X , c ). By the portmanteau lemma this will be true if we can show that E[ f ( X n , c )] → E[ f ( X , c )] for any bounded continuous function f ( x , y ).
There are two parts of the Slutsky equation, namely the substitution effect and income effect. In general, the substitution effect is negative. Slutsky derived this formula to explore a consumer's response as the price of a commodity changes. When the price increases, the budget set moves inward, which also causes the quantity demanded to decrease.
For example, the formulae ... by assumption, it follows immediately from appeal to Slutsky's theorem that [() ... This concludes the proof.
Śleszyński–Pringsheim theorem (continued fraction) Slutsky's theorem (probability theory) Smn theorem (recursion theory, computer science) Sobolev embedding theorem (mathematical analysis) Sokhatsky–Weierstrass theorem (complex analysis) Solèr's theorem (mathematical logic) Solutions of a general cubic equation
Suppose (^) (,).Then, by Slutsky's theorem and by the properties of the normal distribution, multiplying by R has distribution: (^) = (^) (, ′)Recalling that a quadratic form of normal distribution has a Chi-squared distribution:
The original proof that the Hausdorff–Young inequality cannot be extended to > is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic. [1] The first construction of a Salem set was probabilistic. [2] Only in 1981 did Kaufman give a deterministic construction.