Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Generalized Chernoff-Stein Lemma, Applications and Examples

Jihad Fahs, Ibrahim Abou Faycal, Ibrahim Issa

TL;DR

The paper generalizes Chernoff–Stein analysis to continuous and correlated settings by introducing $\delta$-typicality for entropy and relative entropy and an $\varepsilon$-goodness framework. It proves a generalized Chernoff–Stein lemma applicable to dependent data and continuous quantities, and demonstrates the approach on Gaussian hypothesis testing with potential correlations. A key result is the per-letter KL divergence rate for correlated Gaussian processes, given by a spectral integral $C_s$, and the corresponding exponential decay of the type-II error, $-\log \beta^{[n]}_{\tau} \approx C_s n$. The methodology combines δ-typical sets, CLT-based goodness criteria, and spectral-covariance analysis to extend classical results to broad dependent scenarios with practical implications for hypothesis testing in correlated data settings.

Abstract

In this manuscript we define the notion of "$δ$-typicality" for both entropy and relative entropy, as well as a notion of $ε$-goodness and provide an extension to Stein's lemma for continuous quantities as well as correlated setups. We apply the derived results on the Gaussian hypothesis testing problem where the observations are possibly correlated.

The Generalized Chernoff-Stein Lemma, Applications and Examples

TL;DR

The paper generalizes Chernoff–Stein analysis to continuous and correlated settings by introducing -typicality for entropy and relative entropy and an -goodness framework. It proves a generalized Chernoff–Stein lemma applicable to dependent data and continuous quantities, and demonstrates the approach on Gaussian hypothesis testing with potential correlations. A key result is the per-letter KL divergence rate for correlated Gaussian processes, given by a spectral integral , and the corresponding exponential decay of the type-II error, . The methodology combines δ-typical sets, CLT-based goodness criteria, and spectral-covariance analysis to extend classical results to broad dependent scenarios with practical implications for hypothesis testing in correlated data settings.

Abstract

In this manuscript we define the notion of "-typicality" for both entropy and relative entropy, as well as a notion of -goodness and provide an extension to Stein's lemma for continuous quantities as well as correlated setups. We apply the derived results on the Gaussian hypothesis testing problem where the observations are possibly correlated.
Paper Structure (15 sections, 10 theorems, 120 equations)

This paper contains 15 sections, 10 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. $\delta$-Typical Sets
  3. The Dependent Gaussian Case
  4. Relative Entropy $\delta$-Typical Sets
  5. Generalized Chernoff–Stein Lemma
  6. Testing the Correlated Gaussian Hypotheses
  7. Equivalent Formulation
  8. Main Result
  9. Asymptotics of Functions of Eigenvalues of Covariance Matrices
  10. Asymptotically Equivalent Sequences of Matrices
  11. Some Asymptotic Equivalence of Covariance Matrices
  12. Asymptotic Behavior of Relative Entropy
  13. Relative Entropy between Correlated and Independent Gaussians
  14. Relative Entropy between Correlated Gaussians
  15. Sublinear Growth of $D(p^{[n]\space}||q^{[n]\space})$

Key Result

Lemma 1

Let $B^{[n]} \subset \mathcal{X}^n$ be any subset of sequences $\mathbf{x} \in \mathcal{X}^n$ such that $p^{[n]\space} \! \left( B^{[n]} \right) \geq \left(1 - \varepsilon^{[n]\space}\right)$ when $n$ is large enough, and where $\varepsilon^{[n]\space}$ is a family of positive scalars less than one. whenever $n$ is large enough.

Theorems & Definitions (29)

  • Example : IID
  • Definition 1
  • Example : IID
  • Definition 2
  • Example 1: IID
  • Lemma 1
  • proof
  • Example : IID
  • Definition 3
  • Example : IID
  • ...and 19 more