On local maxima of smooth Gaussian nonstationary processes and stationary planar fields with trends

Abstract: We present exact formulas for both the expected number and the height distribution of local maxima (peaks) in two distinct categories of smooth, non-centered Gaussian fields: (i) nonstationary Gaussian processes and (ii) stationary planar Gaussian fields. For case (i), we introduce a novel parameter related to conditional correlation that significantly simplifies the computation of these formulas. Notably, the peak height distribution is solely dependent on this single parameter. In case (ii), traditional methods involving GOE random matrices are ineffective for non-isotropic fields with mean functions. To address this, we apply specific transformations that enable the derivation of formulas using generalized chi-squared density functions. These derived results provide essential tools for calculating p-values and power in applications of signal and change point detection within environments characterized by non-isotropic Gaussian noise.