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The following anticoncentration property is proved. The probability that the$k$-order statistic of an arbitrarily correlated jointly Gaussian random vector$X$ with unit variance components lies within an interval of length$\varepsilon$ is bounded above by $2{\varepsilon}k ({ 1+\mathrm{E}[\|X\|_\infty]}) $. This bound has implications for generalized error rate control instatistical high-dimensional multiple hypothesis testing problems, which arediscussed subsequently.

Dimension-Free Anticoncentration Bounds for Gaussian Order Statistics with Discussion of Applications to Multiple Testing