Written By: Qingyang Xu (website)

Date Created: July 17, 2023

Last Modified: January 22, 2024

• Chapter summary of “All of Statistics” by Larry Wasserman (2004)
• Skip first 4 chapters as they are preliminary materials covered in probability textbooks

Chapter 5. Convergence of Random Variables

Types of Convergence

• Convergence in probability (i.p.) if $\forall \epsilon >0, \mathbb{P}(|X_n-X|>\epsilon) \rightarrow0$
• Convergence in distribution (d) if $F_n(t) \rightarrow F(t)$ for all $t$ where $F$ is continuous
• Convergence in quadratic mean (q.m. or L2) if $\mathbb{E}(X_n-X)^2 \rightarrow0$

Convergence under continous transformations

Theorem: Let $g$ be a continuous function

• If $X_n \xrightarrow d X, Y_n \xrightarrow d c$ then $X_n+ Y_n \xrightarrow d X+c$ and $X_n Y_n \xrightarrow d Xc$
• If $X_n \xrightarrow p X$ then $g(X_n) \xrightarrow p g(X)$ and similarly for d
• If $X_n \xrightarrow p X, Y_n \xrightarrow p Y$ then $X_n+ Y_n \xrightarrow p X+Y$ and similarly for q.m.
• If $X_n \xrightarrow p X, Y_n \xrightarrow p Y$ then $X_n Y_n \xrightarrow p XY$

Delta Method