Expected Supremum of a Random Linear Combination of Shifted Kernels

TL;DR

This paper investigates the expected maximum value of a linear combination of shifted sinc kernels with random coefficients, revealing different growth rates depending on the coefficient distribution.

Contribution

It provides new bounds on the expected supremum for Gaussian and bounded coefficients, highlighting differences from orthonormal function cases.

Findings

Expected supremum with Gaussian coefficients is of order √log n.

Expected supremum with bounded coefficients is of order log log n.

Distinct growth behaviors compared to orthonormal functions.

Abstract

We address the expected supremum of a linear combination of shifts of the sinc kernel with random coefficients. When the coefficients are Gaussian, the expected supremum is of order \sqrt{\log n}, where n is the number of shifts. When the coefficients are uniformly bounded, the expected supremum is of order \log\log n. This is a noteworthy difference to orthonormal functions on the unit interval, where the expected supremum is of order \sqrt{n\log n} for all reasonable coefficient statistics.