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Entropy Theory for Random Walks on Lie Groups

Samuel Kittle, Constantin Kogler

Abstract

We develop entropy and variance results for the product of independent identically distributed random variables on Lie groups. Our results apply to the study of stationary measures in various contexts.

Entropy Theory for Random Walks on Lie Groups

Abstract

We develop entropy and variance results for the product of independent identically distributed random variables on Lie groups. Our results apply to the study of stationary measures in various contexts.
Paper Structure (12 sections, 21 theorems, 121 equations)

This paper contains 12 sections, 21 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Basic Properties of Entropy on Lie groups
  3. Entropy and Basic Properties
  4. Kullback-Leibler Divergence
  5. Regular Conditional Distributions
  6. Conditional Entropy
  7. Entropy Growth for Stopped Random Walks
  8. Entropy and Trace on Lie groups
  9. Entropy and Trace
  10. Conditional Trace and Entropy between Scales
  11. Proof of Theorem \ref{['MainEntropyIncrease']}
  12. From Entropy Gap to Trace Sum

Key Result

Theorem 1.2

Let $\mu$ be a finitely supported probability measure on $G$. Let $\eta = (\eta_n)_{n \geq 1}$ be a sequence of stopping times satisfying the large deviation principle and write $L_n = \mathbb{E}[\eta_n]$ for $n\geq 1$. Let $a \geq 1$, $\varepsilon > 0$ and let $r_n > 0$ be a sequence satisfying for for a constant $c_G > 0$ depending only on $G$. Then for all $n \geq 1$,

Theorems & Definitions (42)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 32 more