Prerequisites & Notation

Before You Begin

This chapter requires a solid grasp of channel coding theory (capacity, achievability, converse) and Gaussian channel fundamentals. The shift from asymptotic to finite-blocklength thinking is conceptually significant.

DMC capacity theorem: achievability (random coding) and converse (Fano's inequality)(Review ch09)
Self-check: Can you state and prove both directions of Shannon's channel coding theorem?
Gaussian channel capacity $C = \frac{1}{2}\log(1 + \text{SNR})$ (Review ch10)
Self-check: Can you derive the AWGN capacity and explain the role of the power constraint?
Typicality and the AEP(Review ch03)
Self-check: Can you state the weak AEP and explain how it enables random coding arguments?
Hypothesis testing (Neyman-Pearson lemma)
Self-check: Can you state the Neyman-Pearson lemma and compute the optimal test for two Gaussians?
Central limit theorem and Berry-Esseen bounds
Self-check: Can you state the CLT and the Berry-Esseen bound on the approximation error?

Notation for This Chapter

Symbols introduced in this chapter. The finite-blocklength framework introduces several quantities that have no direct analog in classical information theory.

Symbol	Meaning	Introduced
$R^*(n, \epsilon)$	Maximum coding rate at blocklength $n$ and error probability $\epsilon$	s01
$V$	Channel dispersion (variance of the information density)	s01
$\iota(x; y)$	Information density: $\log \frac{p_{Y\|X}(y\|x)}{p_Y(y)}$	s01
$Q^{-1}(\cdot)$	Inverse of the Gaussian Q-function	s01
$\kappa_\beta$	The $\kappa\beta$ bound (meta-converse parameter)	s02
$\beta_{1-\epsilon}(P, Q)$	Minimum type-II error in hypothesis testing $P$ vs $Q$ at significance $\epsilon$	s02
$\text{RCU}(n, M)$	Random coding union bound on error probability	s02
$T$	Third absolute moment of information density (Berry-Esseen parameter)	s01

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