In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. We focus on the case when the processes have finite number of points attaining their maximal variance, but, unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maximal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.

在本文中，我们考虑了非平稳高斯过程的极值分布，并针对这种分布的渐近行为提出了新的理论结果。我们关注的情况是过程具有达到其最大方差的有限数量的点，但是与该领域以前已知的事实不同，我们的主定理产生了带有指数衰减余项的相应分布函数的渐近表示。例如，基于勒让德多项式，该结果可以有效地用于研究投影密度估计。更准确地说，我们构造了伴随定律的序列，该定律用多项式率近似了所考虑估计的最大偏差的分布。此外，我们构造了密度的置信带，