New GP theory predicts minima overshoot and minimizer locations

A new mathematical paper by Enkelejd Hashorva and Svyatoslav Novikov (arXiv:2607.20714) explores the asymptotic behavior of the minima of Gaussian processes. For a centered Gaussian process X(t) on a compact metric space K with continuous sample paths, let M be the minimum. The authors show that, conditional on M being above a high threshold u, the scaled overshoot u(M-u) converges in distribution to an exponential random variable with mean σ²_*, the minimum covariance energy of the process.

This result generalizes earlier findings for smooth Gaussian processes to a broader class. The paper also addresses the location of the minimizer: any weak subsequential limit of the conditional law of a measurable minimizer is an optimal covariance-energy measure. If that measure is unique, the conditional law converges weakly to it. The theory is illustrated with stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet. The work has implications for Bayesian optimization, uncertainty quantification, and the study of random landscapes in machine learning.