Prerequisites & Notation

This chapter builds on several tools developed earlier in the book. If any item feels unfamiliar, revisit the linked chapter before proceeding.

Random variables, CDF, PDF, PMF(Review ch05)
Self-check: Can you write down the CDF of an exponential and a Bernoulli random variable?
Expectation, variance, covariance(Review ch09)
Self-check: Can you compute $\text{Var}(\bar{X}_n)$ for i.i.d. $X_i$ with variance $\sigma^2$ ?
Characteristic functions and the Levy continuity theorem(Review ch10)
Self-check: Can you state why $\phi_X(u) = 1 - u^2/2 + o(u^2)$ when $\mathbb{E}[X]=0$ , $\text{Var}(X)=1$ ?
Chebyshev's inequality(Review ch09)
Self-check: Can you bound $\mathbb{P}(|X - \mu| \geq \epsilon)$ using only the variance?
Borel-Cantelli lemmas(Review ch04)
Self-check: If $\sum_n \mathbb{P}(A_n) < \infty$ , what can you conclude about $\mathbb{P}(A_n \text{ i.o.})$ ?

Symbols introduced or heavily used in this chapter. See also the NGlobal Notation Table master table.

Symbol	Meaning	Introduced
$\bar{X}_n$	Sample mean: $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$	s01
$X_n \xrightarrow{\text{a.s.}} X$	Almost sure convergence	s01
$X_n \xrightarrow{P} X$	Convergence in probability	s01
$X_n \xrightarrow{L^r} X$	Convergence in $r$ -th mean ( $L^r$ )	s01
$X_n \xrightarrow{d} X$	Convergence in distribution	s01
$\phi_X(u)$	Characteristic function of a random variable	s01
$\Phi(x)$	CDF of the standard normal: $\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-t^2/2} dt$	s04
$Q(x)$	Gaussian tail probability: $Q(x) = 1 - \Phi(x)$	s04
$\boldsymbol{\Sigma}$	Covariance matrix of a random vector	s05