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New Upper bounds for KL-divergence Based on Integral Norms

Liuquan Yao, Songhao Liu

Abstract

In this paper, some new upper bounds for Kullback-Leibler divergence(KL-divergence) based on $L^1, L^2$ and $L^\infty$ norms of density functions are discussed. Our findings unveil that the convergence in KL-divergence sense sandwiches between the convergence of density functions in terms of $L^1$ and $L^2$ norms. Furthermore, we endeavor to apply our newly derived upper bounds to the analysis of the rate theorem of the entropic conditional central limit theorem.

New Upper bounds for KL-divergence Based on Integral Norms

Abstract

In this paper, some new upper bounds for Kullback-Leibler divergence(KL-divergence) based on and norms of density functions are discussed. Our findings unveil that the convergence in KL-divergence sense sandwiches between the convergence of density functions in terms of and norms. Furthermore, we endeavor to apply our newly derived upper bounds to the analysis of the rate theorem of the entropic conditional central limit theorem.
Paper Structure (9 sections, 26 theorems, 82 equations)

This paper contains 9 sections, 26 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. New Upper bounds for KL-divergence
  3. The General Case
  4. A Special Case
  5. Conditional Central Limit Theorem
  6. Local Limit Theorem
  7. Convergence rate for $\mathbb{E} D(S_n|y)$
  8. Proof of $\|p_n-p_Y\|_2\to 0 \Rightarrow \|p_n-p_Y\|_1\to 0$
  9. Proof of functionanalysis and functionanalysis

Key Result

Lemma 1.1

If $P\ll Q$ and let $\beta_1,\beta_2\in[0,1]$ be given by Then

Theorems & Definitions (33)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 2.1
  • Corollary 2.1
  • Remark 2.1
  • Corollary 2.2
  • Theorem 2.2
  • Lemma 2.1: Lemma 2.2 in Esseenbound
  • ...and 23 more