Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Correlated Binomial Process

Moïse Blanchard, Doron Cohen, Aryeh Kontorovich

Abstract

Cohen and Kontorovich (COLT 2023) initiated the study of what we call here the Binomial Empirical Process: the maximal absolute value of a sequence of inhomogeneous normalized and centered binomials. They almost fully analyzed the case where the binomials are independent, and the remaining gap was closed by Blanchard and Voráček (ALT 2024). In this work, we study the much more general and challenging case with correlations. In contradistinction to Gaussian processes, whose behavior is characterized by the covariance structure, we discover that, at least somewhat surprisingly, for binomial processes covariance does not even characterize convergence. Although a full characterization remains out of reach, we take the first steps with nontrivial upper and lower bounds in terms of covering numbers.

Correlated Binomial Process

Abstract

Cohen and Kontorovich (COLT 2023) initiated the study of what we call here the Binomial Empirical Process: the maximal absolute value of a sequence of inhomogeneous normalized and centered binomials. They almost fully analyzed the case where the binomials are independent, and the remaining gap was closed by Blanchard and Voráček (ALT 2024). In this work, we study the much more general and challenging case with correlations. In contradistinction to Gaussian processes, whose behavior is characterized by the covariance structure, we discover that, at least somewhat surprisingly, for binomial processes covariance does not even characterize convergence. Although a full characterization remains out of reach, we take the first steps with nontrivial upper and lower bounds in terms of covering numbers.
Paper Structure (40 sections, 21 theorems, 192 equations, 2 figures, 1 algorithm)

This paper contains 40 sections, 21 theorems, 192 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Our contributions.
  4. Notation.
  5. Main results
  6. Open problem.
  7. Open problem.
  8. Proof of Theorem \ref{['thm:neg-corr']} (Decoupling from below)
  9. Notation.
  10. Bernoulli case.
  11. Relaxing pairwise independence.
  12. General positive random variables.
  13. Relaxing pairwise independence.
  14. Proof of Theorem \ref{['thm:2procs']} (Covariance does not characterize )
  15. Construction of the example.
  16. ...and 25 more sections

Key Result

Theorem 1

Let $\mu$ be a probability measure on $\left\{ 0,1 \right\}^\mathbb{N}$ with negatively correlated coordinates (i.e., $X\sim\mu$ verifies $\mathbb{E}[X_i X_j]\le \mathbb{E}[X_i]\mathbb{E}[X_j]$ for $i,j\in\mathbb{N}$) and $\tilde{\mu}$ its product version. Then

Figures (2)

  • Figure 1: Two examples of skeleton trees constructed by Algorithm \ref{['alg:construct_tree']}. The components of the distribution $\mu$ for Proposition \ref{['prop:example_covering_nb']} are associated to leaves (or rather paths) of this infinite tree so that the distance $\xi$ between leaves coincides with the natural tree metric (up to a constant factor $2$). On the left we represent the simpler case when $N_{k+1}-N_k=1$ for $k\geq 1$, that is, covering numbers $N(\varepsilon)$ grow by one at a time as $\varepsilon\to 0$. In this case, the constructed tree is exactly a binary tree, constructed according to the exact ordering given by $Order(l)$ for $l\geq 1$. On the right, we represent a general case when covering numbers can grow via jumps $N_{k+1}-N_k\geq 1$. In the specific example, we have $(N_i,i\leq 7)=(1,2,7,10,13,14,15)$. Although the tree is not formally a complete binary tree, the ordering choice balances all subtrees evenly. We represent with dashed lines, all levels $\varepsilon$ which complete a layer of the constructed binary tree.
  • Figure 2: Tree corresponding to the distribution $\mu$ constructed in Proposition \ref{['prop:ex_fast_converging']}. As in Figure \ref{['fig:skeleton_tree']}, the leaves of the tree represent the components of the distribution, and the distance $\xi$ coincides exactly with the natural tree metric (up to a constant factor 2). In this example, the covering numbers are $(N_k,k\leq 5)=(1,3,4,7,8)$.

Theorems & Definitions (22)

  • Theorem 1: Decoupling from below
  • Corollary 1
  • Theorem 2: Covariance does not characterize $\Delta_n$
  • Theorem 3
  • Proposition 1
  • Theorem 4
  • Proposition 2
  • Proposition 3
  • Theorem 5
  • Proposition 4
  • ...and 12 more