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Entropic Conditional Central Limit Theorem and Hadamard Compression

Zhi-Ming Ma, Liu-Quan Yao, Shuai Yuan, Hua-Zi Zhang

Abstract

We make use of an entropic property to establish a convergence theorem (Main Theorem), which reveals that the conditional entropy measures the asymptotic Gaussianity. As an application, we establish the {\it entropic conditional central limit theorem} (CCLT), which is stronger than the classical CCLT. As another application, we show that continuous input under iterated Hadamard transform, almost every distribution of the output conditional on the values of the previous signals will tend to Gaussian, and the conditional distribution is in fact insensitive to the condition. The results enable us to make a theoretic study concerning Hadamard compression, which provides a solid theoretical analysis supporting the simulation results in previous literature. We show also that the conditional Fisher information can be used to measure the asymptotic Gaussianity.

Entropic Conditional Central Limit Theorem and Hadamard Compression

Abstract

We make use of an entropic property to establish a convergence theorem (Main Theorem), which reveals that the conditional entropy measures the asymptotic Gaussianity. As an application, we establish the {\it entropic conditional central limit theorem} (CCLT), which is stronger than the classical CCLT. As another application, we show that continuous input under iterated Hadamard transform, almost every distribution of the output conditional on the values of the previous signals will tend to Gaussian, and the conditional distribution is in fact insensitive to the condition. The results enable us to make a theoretic study concerning Hadamard compression, which provides a solid theoretical analysis supporting the simulation results in previous literature. We show also that the conditional Fisher information can be used to measure the asymptotic Gaussianity.
Paper Structure (27 sections, 31 theorems, 237 equations)

This paper contains 27 sections, 31 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Entropic Central Limit Theorem
  3. Conditional Central Limit Theorem
  4. Hadamard Compression
  5. Paper Structure and Main Results
  6. Main Theorem and Key Lemma
  7. Statement of Main Theorem
  8. Key Lemma and its Proof
  9. Proof of the Main Theorem \ref{['main theorem']}
  10. Entropic Conditional Central Limit Theorem
  11. Proof of Theorem \ref{['sum theorem']}
  12. Rate of Convergence with Independent Component
  13. Asymptotic Gaussianity Measured by Fisher infomation
  14. Hadamard Compression for Continuous Input
  15. Limit distributions of $a.s.$ polarization pathes
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let $\{(X_n, Y_n)\}_{n=1}^\infty$ be a sequence of random pairs such that the conditional distribution $X_n|Y_n=y_n$ has absolute continuous density for $y_n$ almost surely. Assume the following conditions hold, Denote by $(X_n', Y_n')$ an independent copy of $(X_n, Y_n)$. If and for some $\lambda\in(0,1)$. Then and Furthermore, If $\{h(X_n|Y_n),n\geq 1\}$ is uniformly integrable, then there

Theorems & Definitions (55)

  • Theorem 3.1: Main Theorem
  • Remark 3.1
  • Lemma 3.2: Key Lemma
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Definition 3.6
  • Definition 3.7
  • Proposition 3.8
  • Proposition 3.9
  • ...and 45 more