On the rate of concentration of maxima in Gaussian arrays

Recently in Gao and Stoev (2018) it was established that the concentration of maxima phenomenon is the key to solving the exact sparse support recovery problem in high dimensions. This phenomenon, known also as relative stability, has been little studied in the context of dependence. Here, we obtain bounds on the rate of concentration of maxima in Gaussian triangular arrays. These results are used to establish sufficient conditions for the uniform relative stability of functions of Gaussian arrays, leading to new models that exhibit phase transitions in the exact support recovery problem. Finally, the optimal rate of concentration for Gaussian arrays is studied under general assumptions implied by the classic condition of Berman (1964).