Uniform Relative Stability for Gaussian Arrays

The general phase-transition results for the exact support recovery investigated in Chap. 4 exploit a concentration of maxima phenomenon for dependent arrays known as the uniform relative stability (URS) property. This property holds for a wide range of models. The purpose of this chapter is to provide a complete characterization of URS for Gaussian triangular arrays. The main result shows that the Gaussian arrays are URS if and only if they are uniformly decreasingly dependent (UDD), which is a simple condition expressed in terms of the covariance structure of the array. In particular, classic results of Simeon Berman are recovered and extended. The proofs utilize tools such as Slepian’s lemma, the Sudakov–Fernique inequality, as well as, a curious result on the structure of correlation matrices derived using elementary Ramsey’s theory.