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- W4302760520 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. The aim of this paper is to develop such a bound, while also allowing for many types of dependence. Let $(xi_1,dots,xi_N)$ be a centered Gaussian vector with standardized entries, whose correlation matrix $R$ satisfies $max_{ineq j} R_{ij}leq rho_0$ for some constant $rho_0in (0,1)$. Then, for any $epsilon_0in(0,sqrt{1-rho_0})$, we establish an upper bound on the probability $mathbb{P}(max_{1leq jleq N} xi_jleq epsilon_0sqrt{2log(N)})$ in terms of $(rho_0,epsilon_0,N)$. The bound is also sharp, in the sense that it is attained up to a constant, independent of $N$. Next, we apply this result in the context of high-dimensional statistics, where we simplify and weaken conditions that have recently been used to establish near-parametric rates of bootstrap approximation. Lastly, an interesting aspect of this application is that it makes use of recent refinements of Bourgain and Tzafriri's restricted invertibility principle." @default.
- W4302760520 created "2022-10-07" @default.
- W4302760520 creator A5037982747 @default.
- W4302760520 creator A5056911642 @default.
- W4302760520 date "2018-09-23" @default.
- W4302760520 modified "2023-09-28" @default.
- W4302760520 title "A Sharp Lower-tail Bound for Gaussian Maxima with Application to Bootstrap Methods in High Dimensions" @default.
- W4302760520 doi "https://doi.org/10.48550/arxiv.1809.08539" @default.
- W4302760520 hasPublicationYear "2018" @default.
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