ν and f(x) is a p.d.f.

Asequence of random variable(Yn: n = 1;2;:::)is said to converge in probability to another random variable Y, all de ned on Rd if for every ϵ > 0; P(∥Yn Y∥ > ϵ)! For example, by the law of large numbers, the sample variance S2 n! Then 1. The follo wing result (con tin uous mapping theorem) pro vides an answ er to this question in man y problems. Somehow, there are many “Slutsky’s Theorems.” Eugen E. Slutsky, Russia (1880 – 1948) Slutsky’s Theorem ν.

The theorem is stated as: For a continuous function g(X_k) that is not a function of k, SLUTSKY’S THEOREMS 4.2 Slutsky’s theorems Theorem 4.4 (Slutsky’s theorems) Let fX ng n2IN be a sequence of d-dimensional r.v.s with X n!d X. Y nX n!DaX, and 2. In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. Implications. This theorem is also attributed to Harald Cramer (1893–1985). Then it holds (i) For any f: IRd!IRk such that P(X2C(f)) = 1, then f(X n)!d f(X): (ii) Let fY

The theorem is valid also when and are sequences of random matrices (the reason being that random matrices can be thought of as random vectors whose entries have been re-arranged into several columns).

Let X n!DXand Y n!P a, a constant as n!1. Theorem 4.

and thus Slutsky’s theorem together with the fact that nb(X n −a) →d X proves the result. 0: We denote this phenomenon by …

Many results for estimators will be derived from this theorem. 4.2. Basic Probability Theory on Convergence Definition 1 (Convergencein probability). Slutsky's theorem and -metho d T ransformation is an imp ortan t to ol in statistics. If X n con v erges to in some sense, is g the same sense? 2 Slutsky’s Theorem Some useful extensions of the central limit theorem are based on Slutsky’s theorem. w.r.t. Then lim Theorem 1.10. Lecture 7: Convergence of transformations and Slutsky’s theorem Proposition 1.18 (Scheff´e’s theorem) Let {f n} be a sequence of p.d.f.’s on Rk w.r.t. a:s:˙2; the distribution variance as n!1. Suppose that lim n→∞ f n(x) = f(x) a.e. X n+ Y n!DX+ a. a measure ν. This is a very important and useful result.

Slutsky theorem is commonly used to prove the consistency of estimators in Econometrics.