Decomposition of Additive Random Fields

We consider in this work an additive random field on [0, 1] d , which is a sum of d uncorrelated random processes. We assume that the processes have zero mean and the same continuous covariance function. There is a significant interest in the study of random fields of this type. For example, they ar...

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Vydané v:	Vestnik, St. Petersburg University. Mathematics Ročník 53; číslo 1; s. 29 - 36
Hlavní autori:	Zani, M., Khartov, A. A.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Moscow Pleiades Publishing 01.01.2020 Springer Nature B.V
Predmet:	Analysis Approximation Continuity (mathematics) Covariance Decomposition Dependence Eigenvalues Eigenvectors Entropy Fields (mathematics) Integrals Intersections Mathematical analysis Mathematics Mathematics and Statistics Operators (mathematics) Random processes decomposition additive random fields eigenpairs covariance function covariance operator average-case approximation complexity
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Abstract	We consider in this work an additive random field on [0, 1] d , which is a sum of d uncorrelated random processes. We assume that the processes have zero mean and the same continuous covariance function. There is a significant interest in the study of random fields of this type. For example, they arise in the theory of intersections and self-intersections of Brownian processes, in the problems concerning small ball probabilities, and in the finite-rank approximation problems with arbitrarily large parametric dimension d . In problems of the last kind, the spectral characteristics of the covariance operator play a key role. For a given additive random field, the dependence of eigenvalues of its covariance operator on eigenvalues of the covariance operator of the marginal processes is quite simple, provided that the identical 1 is an eigenvector of the latter operator. In the opposite case, the dependence is complex, and, therefore, it is hard to study these random fields. Here, summands of the decomposition of the random field into the sum of its integral and its centered version are orthogonal in L 2 ([0, 1] d ), but, in general, they are correlated. In the present paper, we propose another interesting decomposition for random fields (it was discovered by the authors while resolving finite-rank approximation problems in the average-case setting). In the obtained decomposition, the summands are orthogonal in L 2 ([0, 1] d ) and are uncorrelated. Moreover, for large d , they are respectively close to the integral and to the centered version of the random field with small relative mean square errors.
AbstractList	We consider in this work an additive random field on [0, 1]d, which is a sum of d uncorrelated random processes. We assume that the processes have zero mean and the same continuous covariance function. There is a significant interest in the study of random fields of this type. For example, they arise in the theory of intersections and self-intersections of Brownian processes, in the problems concerning small ball probabilities, and in the finite-rank approximation problems with arbitrarily large parametric dimension d. In problems of the last kind, the spectral characteristics of the covariance operator play a key role. For a given additive random field, the dependence of eigenvalues of its covariance operator on eigenvalues of the covariance operator of the marginal processes is quite simple, provided that the identical 1 is an eigenvector of the latter operator. In the opposite case, the dependence is complex, and, therefore, it is hard to study these random fields. Here, summands of the decomposition of the random field into the sum of its integral and its centered version are orthogonal in L2([0, 1]d), but, in general, they are correlated. In the present paper, we propose another interesting decomposition for random fields (it was discovered by the authors while resolving finite-rank approximation problems in the average-case setting). In the obtained decomposition, the summands are orthogonal in L2([0, 1]d) and are uncorrelated. Moreover, for large d, they are respectively close to the integral and to the centered version of the random field with small relative mean square errors. We consider in this work an additive random field on [0, 1] d , which is a sum of d uncorrelated random processes. We assume that the processes have zero mean and the same continuous covariance function. There is a significant interest in the study of random fields of this type. For example, they arise in the theory of intersections and self-intersections of Brownian processes, in the problems concerning small ball probabilities, and in the finite-rank approximation problems with arbitrarily large parametric dimension d . In problems of the last kind, the spectral characteristics of the covariance operator play a key role. For a given additive random field, the dependence of eigenvalues of its covariance operator on eigenvalues of the covariance operator of the marginal processes is quite simple, provided that the identical 1 is an eigenvector of the latter operator. In the opposite case, the dependence is complex, and, therefore, it is hard to study these random fields. Here, summands of the decomposition of the random field into the sum of its integral and its centered version are orthogonal in L 2 ([0, 1] d ), but, in general, they are correlated. In the present paper, we propose another interesting decomposition for random fields (it was discovered by the authors while resolving finite-rank approximation problems in the average-case setting). In the obtained decomposition, the summands are orthogonal in L 2 ([0, 1] d ) and are uncorrelated. Moreover, for large d , they are respectively close to the integral and to the centered version of the random field with small relative mean square errors.
Author	Zani, M. Khartov, A. A.
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Cites_doi	10.1090/S0002-9947-07-04233-X 10.1016/j.jco.2007.11.002 10.1016/j.jco.2015.05.002 10.1007/978-3-0348-8059-6_14 10.1137/0108003 10.1007/BFb0103934 10.1016/j.jco.2018.04.001 10.1016/j.jco.2019.02.002
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References	HickernellF. J.WasilkowskiG. W.WoźniakowskiH.Monte Carlo and Quasi-Monte Carlo Methods 20062008BerlinSpringer-Verlag KarolA.NazarovA.NikitinYa.Small ball probabilities for Gaussian random fields and tensor products of compact operatorsTrans. Am. Math. Soc.200836014431474235770210.1090/S0002-9947-07-04233-X KhartovA. A.ZaniM.Approximation complexity of sums of random processesJ. Complexity201954101399398321110.1016/j.jco.2019.02.002 BrownJ. L.Mean Square truncation error in series expansions of random functionsJ. Soc. Ind. Appl. Math.19608283215199910.1137/0108003 KhartovA. A.ZaniM.Asymptotic analysis of average case approximation complexity of additive random fieldsJ. Complexity2019522444394507010.1016/j.jco.2018.04.001 K. Ritter, Average-Case Analysis of Numerical Problems (Springer-Verlag, Berlin, 2000), in Ser.: Lecture Notes in Mathematics, Vol. 1733. X. Chen and W. V. Li, “Small deviation estimates for some additive processes,” in Proc. 3rd Conf. High Dimensional Probability, Sandjberg, Denmark, June 24–28,2002 (Birkhäuser, Basel, 2003), in Ser.: Progress in Probability, Vol. 55, 225–238. LifshitsM. A.ZaniM.Approximation of additive random fields based on standard information: Average case and probabilistic settingsJ. Complexity201531659674337750610.1016/j.jco.2015.05.002 LifshitsM. A.ZaniM.Approximation complexity of additive random fieldsJ. Complexity200824362379242675810.1016/j.jco.2007.11.002 5030_CR9 M. A. Lifshits (5030_CR5) 2015; 31 5030_CR1 A. Karol (5030_CR2) 2008; 360 A. A. Khartov (5030_CR7) 2019; 54 F. J. Hickernell (5030_CR3) 2008 A. A. Khartov (5030_CR6) 2019; 52 M. A. Lifshits (5030_CR4) 2008; 24 J. L. Brown (5030_CR8) 1960; 8
References_xml	– reference: KarolA.NazarovA.NikitinYa.Small ball probabilities for Gaussian random fields and tensor products of compact operatorsTrans. Am. Math. Soc.200836014431474235770210.1090/S0002-9947-07-04233-X – reference: K. Ritter, Average-Case Analysis of Numerical Problems (Springer-Verlag, Berlin, 2000), in Ser.: Lecture Notes in Mathematics, Vol. 1733. – reference: HickernellF. J.WasilkowskiG. W.WoźniakowskiH.Monte Carlo and Quasi-Monte Carlo Methods 20062008BerlinSpringer-Verlag – reference: KhartovA. A.ZaniM.Approximation complexity of sums of random processesJ. Complexity201954101399398321110.1016/j.jco.2019.02.002 – reference: BrownJ. L.Mean Square truncation error in series expansions of random functionsJ. Soc. Ind. Appl. Math.19608283215199910.1137/0108003 – reference: LifshitsM. A.ZaniM.Approximation complexity of additive random fieldsJ. Complexity200824362379242675810.1016/j.jco.2007.11.002 – reference: KhartovA. A.ZaniM.Asymptotic analysis of average case approximation complexity of additive random fieldsJ. Complexity2019522444394507010.1016/j.jco.2018.04.001 – reference: X. Chen and W. V. Li, “Small deviation estimates for some additive processes,” in Proc. 3rd Conf. High Dimensional Probability, Sandjberg, Denmark, June 24–28,2002 (Birkhäuser, Basel, 2003), in Ser.: Progress in Probability, Vol. 55, 225–238. – reference: LifshitsM. A.ZaniM.Approximation of additive random fields based on standard information: Average case and probabilistic settingsJ. Complexity201531659674337750610.1016/j.jco.2015.05.002 – volume: 360 start-page: 1443 year: 2008 ident: 5030_CR2 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-07-04233-X – volume: 24 start-page: 362 year: 2008 ident: 5030_CR4 publication-title: J. Complexity doi: 10.1016/j.jco.2007.11.002 – volume-title: Monte Carlo and Quasi-Monte Carlo Methods 2006 year: 2008 ident: 5030_CR3 – volume: 31 start-page: 659 year: 2015 ident: 5030_CR5 publication-title: J. Complexity doi: 10.1016/j.jco.2015.05.002 – ident: 5030_CR1 doi: 10.1007/978-3-0348-8059-6_14 – volume: 8 start-page: 28 year: 1960 ident: 5030_CR8 publication-title: J. Soc. Ind. Appl. Math. doi: 10.1137/0108003 – ident: 5030_CR9 doi: 10.1007/BFb0103934 – volume: 52 start-page: 24 year: 2019 ident: 5030_CR6 publication-title: J. Complexity doi: 10.1016/j.jco.2018.04.001 – volume: 54 start-page: 101399 year: 2019 ident: 5030_CR7 publication-title: J. Complexity doi: 10.1016/j.jco.2019.02.002
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SubjectTerms	Analysis Approximation Continuity (mathematics) Covariance Decomposition Dependence Eigenvalues Eigenvectors Entropy Fields (mathematics) Integrals Intersections Mathematical analysis Mathematics Mathematics and Statistics Operators (mathematics) Random processes
Title	Decomposition of Additive Random Fields
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