Karhunen-Loève expansion of Random Measures

2022/03/27 by Ricardo Carrizo Vergara, Vergara, Ricardo Carrizo
Computer Science · Decision Sciences · Mathematics · #60G12 #60G57 #60H05 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Probability and Risk Models #Rough Sets and Fuzzy Logic #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2203.14202

openalex publication_date 2022/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an orthogonal expansion for real, function-regulated, second-order random measures over ℝ^d with measure covariance. Such a expansion, which can be seen as a Karhunen-Loève decomposition, consists in a series of deterministic real measures weighted by uncorrelated real random variables with the variances forming a convergent series. The convergence of the series is in a mean-square sense stochastically and against measurable bounded test functions (with compact support if the random measure is not finite) in the measure sense, which implies set-wise convergence. This is proven taking advantage of the extra requirement of having a covariance measure over ℝ^d×ℝ^d describing the covariance structure of the random measure, for which we also provide a series expansion. These results cover for instance the cases of Gaussian White Noise, Poisson and Cox point processes, and can be used to obtain expansions for trawl processes.

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