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Slepian and Sudakov-Fernique type inequalities, which compare expectations ofmaxima of Gaussian random vectors under certain restrictions on the covariancematrices, play an important role in probability theory, especially in empiricalprocess and extreme value theories. Here we give explicit comparisons ofexpectations of smooth functions and distribution functions of maxima ofGaussian random vectors without any restriction on the covariance matrices. Wealso establish an anti-concentration inequality for the maximum of a Gaussianrandom vector, which derives a useful upper bound on the L\'{e}vy concentrationfunction for the Gaussian maximum. The bound is dimension-free and applies tovectors with arbitrary covariance matrices. This anti-concentration inequalityplays a crucial role in establishing bounds on the Kolmogorov distance betweenmaxima of Gaussian random vectors. These results have immediate applications inmathematical statistics. As an example of application, we establish aconditional multiplier central limit theorem for maxima of sums of independentrandom vectors where the dimension of the vectors is possibly much larger thanthe sample size.

Comparison and anti-concentration bounds for maxima of Gaussian random vectors