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Finite-length Analysis on Tail probability for Markov Chain and Application to Simple Hypothesis Testing

Shun Watanabe, Masahito Hayashi

TL;DR

Using terminologies of information geometry, upper and lower bounds of the tail probability of the sample mean are derived and another simple proof of central limit theorem for Markov chain is obtained.

Abstract

Using terminologies of information geometry, we derive upper and lower bounds of the tail probability of the sample mean. Employing these bounds, we obtain upper and lower bounds of the minimum error probability of the 2nd kind of error under the exponential constraint for the error probability of the 1st kind of error in a simple hypothesis testing for a finite-length Markov chain, which yields the Hoeffding type bound. For these derivations, we derive upper and lower bounds of cumulant generating function for Markov chain. As a byproduct, we obtain another simple proof of central limit theorem for Markov chain.

Finite-length Analysis on Tail probability for Markov Chain and Application to Simple Hypothesis Testing

TL;DR

Using terminologies of information geometry, upper and lower bounds of the tail probability of the sample mean are derived and another simple proof of central limit theorem for Markov chain is obtained.

Abstract

Using terminologies of information geometry, we derive upper and lower bounds of the tail probability of the sample mean. Employing these bounds, we obtain upper and lower bounds of the minimum error probability of the 2nd kind of error under the exponential constraint for the error probability of the 1st kind of error in a simple hypothesis testing for a finite-length Markov chain, which yields the Hoeffding type bound. For these derivations, we derive upper and lower bounds of cumulant generating function for Markov chain. As a byproduct, we obtain another simple proof of central limit theorem for Markov chain.

Paper Structure

This paper contains 20 sections, 34 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Cumulant generating function
  4. Tail probability
  5. Simple hypothesis testing
  6. Geometric structure for transition matrices
  7. Preparations
  8. Exponential family
  9. Relation with Legendre transform
  10. Information processing inequality
  11. Cumulant generating function
  12. Asymptotic variance
  13. Tail probability
  14. Simple hypothesis testing
  15. Conclusion
  16. ...and 5 more sections

Key Result

Lemma 3.1

HW14-1 Consider an irreducible and ergodic transition matrix $W$ over ${\cal X}$ and a real-valued function $g$ on ${\cal X} \times {\cal X}$. Then, we define the support ${\cal X}^2_W:=\{(x,\bar{x}) \in {\cal X}^2| W(x|\bar{x})>0\}$. Define $\phi(\theta)$ as the logarithm of the Perron-Frobenius ei Then, the function $\phi(\theta)$ is convex. Further, the following conditions are equivalent.

Theorems & Definitions (50)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 5.1
  • proof
  • ...and 40 more