Small sample spaces for Gaussian processes

IF 1.5 2区 数学 Q2 STATISTICS & PROBABILITY
Bernoulli Pub Date : 2021-03-04 DOI:10.3150/22-bej1483
T. Karvonen
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引用次数: 10

Abstract

It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process $X$ is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a"small"set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of $X$ in the $\sigma$-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of $X$ and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen-Lo\`{e}ve theorem.
高斯过程的小样本空间
已知高斯过程$X$的样本在给定的再生核Hilbert空间(RKHS)中的隶属度受一定的核优势条件控制。然而,如何识别包含样本的“小”函数集(不一定是向量空间)还不太清楚。本文提出了一种识别此类集合的通用方法。我们使用可被视为希尔伯特标度的推广的标度RKHS,将样本支持集定义为最大集,该最大集包含在由标度RKHS-集合诱导的$\sigma$-代数中的$X$定律下的全测度的每个元素中。然后,这个潜在的不可测量集合被证明由那些函数组成,这些函数可以根据$X$的协方差核的RKHS的正交基进行扩展,并且它们的平方基系数有界于零和无穷大,这是Karhunen-Lo提出的结果\`{e}ve定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Bernoulli
Bernoulli 数学-统计学与概率论
CiteScore
3.40
自引率
0.00%
发文量
116
审稿时长
6-12 weeks
期刊介绍: BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.
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