On the Maximum of Dependent Gaussian Random Variables: A Sharp Bound for the Lower Tail
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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,...,X_n) be a centered Gaussian vector, with standardized entries, whose correlation matrix R satisfies _i≠ j R_ij≤ρ_0 for some constant ρ_0∈ (0,1). Then, for any ϵ_0∈ (0,√(1-ρ_0)), we establish an upper bound on the probability P(_1≤ i≤ n X_i≤ϵ_0√(2(n))) that is a function of ρ_0, ϵ_0, and n. Furthermore, we show the bound is sharp, in the sense that it is attained up to a constant, for each ρ_0 and ϵ_0.
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