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

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 in statistical high-dimensional multiple hypothesis testing problems, which are discussed subsequently.