Search results
Results From The WOW.Com Content Network
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]
By Slutsky's theorem and continuous mapping theorem these results can be combined to establish consistency of estimator ...
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 ).
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:
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.
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.
Demonstration of this result is fairly straightforward under the assumption that () is differentiable near the neighborhood of and ′ is continuous at with ′ ().To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):
Slutsky's later work was principally in probability theory and the theory of stochastic processes. He is generally credited for the result known as Slutsky's theorem. In 1928 he was an Invited Speaker of the ICM in Bologna. [8]