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Hoeffding-Style Concentration Bounds for Exchangeable Random Variables

Nina Maria Gottschling, Michele Caprio

Abstract

We establish Hoeffding-type concentration inequalities for the low and high tail bounds of sums of exchangeable random variables. Our results exhibit an anti-symmetry in such tail bounds due to the assumption of exchangeability, a generalization of the i.i.d. setting. In contrast to the existing literature on this problem, our result provides an upper tail bound with respect to the largest mean of a distribution in the support of the de Finetti mixing measure, and not the population mean. Equivalently, we establish a lower tail bound with respect to the smallest mean of a distribution in the support of the de Finetti mixing measure. This bridges the gap between finite sample and population means of exchangeable random variables, and distributional means.

Hoeffding-Style Concentration Bounds for Exchangeable Random Variables

Abstract

We establish Hoeffding-type concentration inequalities for the low and high tail bounds of sums of exchangeable random variables. Our results exhibit an anti-symmetry in such tail bounds due to the assumption of exchangeability, a generalization of the i.i.d. setting. In contrast to the existing literature on this problem, our result provides an upper tail bound with respect to the largest mean of a distribution in the support of the de Finetti mixing measure, and not the population mean. Equivalently, we establish a lower tail bound with respect to the smallest mean of a distribution in the support of the de Finetti mixing measure. This bridges the gap between finite sample and population means of exchangeable random variables, and distributional means.
Paper Structure (7 sections, 6 theorems, 48 equations)

This paper contains 7 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Hoeffding's Inequality
  4. de Finetti's Theorem
  5. Hoeffding-type bounds for exchangeable random variables
  6. Proof
  7. Conclusion

Key Result

Theorem 2.1

Let $\{X_m\}_{m=1}^M$ be i.i.d. random variables, such that for all $m \in \{1, \ldots, M\}$ we have that $0 \leq X_m \leq 1$ with probability 1. Then, for $0 < t < 1 - \mu$, we have that

Theorems & Definitions (12)

  • Theorem 2.1: Hoeffding's Inequality
  • Definition 2.2: Exchangeably-Distributed Random Variables
  • Theorem 2.3: de Finetti's Theorem
  • Lemma 3.1: Hoeffding-Type Bounds for Exchangeable Random Variables
  • Corollary 3.2: Hoeffding's Inequality as a Consequence of Lemma \ref{['lem:exchhoeff']}
  • proof : Proof of Corollary \ref{['corr:recover']}
  • Remark 3.3
  • Lemma 4.1: Lemma 1 in hoeffding1994probability
  • proof
  • Proposition 4.2: Simple Bound
  • ...and 2 more