On the rate of concentration of maxima in Gaussian arrays

TL;DR

This paper investigates how quickly the maximum values in Gaussian arrays concentrate, providing bounds and conditions that inform support recovery and phase transitions in high-dimensional statistical models.

Contribution

It derives bounds on the rate of maximum concentration in Gaussian arrays and establishes conditions for support recovery phase transitions, extending classical assumptions.

Findings

Bounds on concentration rates are established.

Conditions for support recovery phase transitions are identified.

Optimal concentration rates are analyzed under general assumptions.

Abstract

Recently in Gao and Stoev (2018) it was established that the concentration of maxima phenomenon is the key to solving the exact sparse support recovery problem in high dimensions. This phenomenon, known also as relative stability, has been little studied in the context of dependence. Here, we obtain bounds on the rate of concentration of maxima in Gaussian triangular arrays. These results are used to establish sufficient conditions for the uniform relative stability of functions of Gaussian arrays, leading to new models that exhibit phase transitions in the exact support recovery problem. Finally, the optimal rate of concentration for Gaussian arrays is studied under more general assumptions than the ones implied by the classic condition of Berman (1964).