0.1 Slutsky’s Lemma
In probability theory, Slutsky’s lemma extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.
Let \(X_{n}\) and \(Y_{n}\) be sequences of random variables where \(X_{n}\stackrel{\mathcal{D}}{\to}X\) and \(Y_{n}\stackrel{\mathbb{P}}{\to}c\) then the following results hold: 1. \(X_{n}+Y_{n}\stackrel{\mathcal{D}}{\to}X+c\) 2. \(X_{n}Y_{n}\stackrel{\mathcal{D}}{\to}Xc\) 3. \(\frac{X_{n}}{Y_{n}}\stackrel{\mathcal{D}}{\to} \frac{X}{c}\) given c is invertible.