A sharp lower-tail bound for Gaussian maxima with application to bootstrap methods in high dimensions

Although there is an extensive literature on the maxima of Gaussian processes, are relatively few non-asymptotic bounds their lower-tail probabilities. The aim this paper to develop such a bound, while also allowing for many types dependence. Let (ξ1,…,ξN) be centered vector with standardized entries, whose correlation matrix R satisfies maxi≠jRij≤ρ0 some constant ρ0∈(0,1). Then, any ϵ0∈(0,1−ρ0), we establish upper bound probability P(max1≤j≤Nξj≤ϵ0 2log(N)) in terms (ρ0,ϵ0,N). sharp, sense that it attained up constant, independent N. Next, apply result context high-dimensional statistics, where simplify and weaken conditions have recently been used near-parametric rates bootstrap approximation. Lastly, interesting aspect application makes use recent refinements Bourgain Tzafriri’s “restricted invertibility principle”.