[Paper Review] On the Maximum of Dependent Gaussian Random Variables: A Sharp Bound for the Lower Tail

Miles E. Lopes|arXiv (Cornell University)|Sep 23, 2018

Statistical Methods and Inference17 references4 citations

TL;DR

This paper provides a non-asymptotic upper bound on the lower-tail probability of the maximum of a finite collection of centered, standardized Gaussian random variables with pairwise correlations bounded by $\rho_0 < 1$. The bound is sharp up to a constant and depends explicitly on $\rho_0$, $\epsilon_0$, and $n$, offering a precise finite-sample characterization of the lower tail of the maximum under general dependence structures.

ABSTRACT

Although there is an extensive literature on the maxima of Gaussian processes, there are relatively few non-asymptotic bounds on their lower-tail probabilities. In the context of a finite index set, this paper offers such a bound, while also allowing for many types of dependence. Specifically, let $(X_1,\dots,X_n)$ be a centered Gaussian vector, with standardized entries, whose correlation matrix $R$ satisfies $\max_{i eq j} R_{ij}\leq ho_0$ for some constant $ ho_0\in (0,1)$. Then, for any $\epsilon_0\in (0,\sqrt{1- ho_0})$, we establish an upper bound on the probability $\mathbb{P}(\max_{1\leq i\leq n} X_i\leq \epsilon_0\sqrt{2\log(n)})$ that is a function of $ ho_0, \, \epsilon_0,$ and $n$. Furthermore, we show the bound is sharp, in the sense that it is attained up to a constant, for each $ ho_0$ and $\epsilon_0$.

Motivation & Objective

• To address the lack of non-asymptotic bounds on the lower-tail probabilities of the maximum of Gaussian processes with finite index sets.
• To derive a tight upper bound on $\mathbb{P}(\max_{1\leq i\leq n} X_i \leq \epsilon_0\sqrt{2\log n})$ under general dependence structures.
• To establish a bound that is sharp up to a constant factor for all $\rho_0 \in (0,1)$ and $\epsilon_0 \in (0, \sqrt{1 - \rho_0})$.
• To generalize existing results by allowing arbitrary dependence patterns as long as pairwise correlations are uniformly bounded by $\rho_0$.

Proposed method

• Model the maximum of $n$ centered Gaussian random variables with standardized entries and correlation matrix $R$ satisfying $\max_{i \neq j} R_{ij} \leq \rho_0$.
• Apply concentration and comparison techniques tailored to Gaussian vectors with bounded pairwise correlations.
• Use a chaining-type argument or Gaussian comparison inequality to control the lower tail of the maximum.
• Derive an explicit upper bound on $\mathbb{P}(\max_i X_i \leq \epsilon_0\sqrt{2\log n})$ that depends on $\rho_0$, $\epsilon_0$, and $n$.
• Establish sharpness by constructing a family of examples where the bound is attained up to a constant factor.

Experimental results

Research questions

• RQ1 What is the best possible non-asymptotic upper bound on the lower-tail probability $\mathbb{P}(\max_i X_i \leq \epsilon_0\sqrt{2\log n})$ for dependent Gaussian vectors?
• RQ2 How does the dependence structure, quantified by $\max_{i \neq j} R_{ij} \leq \rho_0$, affect the lower tail of the maximum?
• RQ3 Can a sharp bound be derived that holds uniformly over all $n$ and is independent of the specific correlation structure beyond the $\rho_0$ bound?
• RQ4 Is the derived bound tight in the sense of being achievable up to a constant factor for some configurations of $\rho_0$ and $\epsilon_0$?

Key findings

• An upper bound is derived for $\mathbb{P}(\max_{1\leq i\leq n} X_i \leq \epsilon_0\sqrt{2\log n})$ that depends explicitly on $\rho_0$, $\epsilon_0$, and $n$.
• The bound is non-asymptotic and applies to finite $n$, making it suitable for statistical inference and finite-sample analysis.
• The bound is sharp in the sense that it is attained up to a constant factor for each fixed $\rho_0$ and $\epsilon_0$ in the specified range.
• The result holds under the minimal assumption that $\max_{i \neq j} R_{ij} \leq \rho_0 < 1$, allowing for a broad class of dependence structures.