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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description | 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. |
doi_str_mv | 10.1134/S1063454120010124 |
format | Article |
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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.</description><identifier>ISSN: 1063-4541</identifier><identifier>EISSN: 1934-7855</identifier><identifier>DOI: 10.1134/S1063454120010124</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Analysis ; Approximation ; Continuity (mathematics) ; Covariance ; Decomposition ; Dependence ; Eigenvalues ; Eigenvectors ; Entropy ; Fields (mathematics) ; Integrals ; Intersections ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Operators (mathematics) ; Random processes</subject><ispartof>Vestnik, St. Petersburg University. Mathematics, 2020, Vol.53 (1), p.29-36</ispartof><rights>Pleiades Publishing, Ltd. 2020</rights><rights>2020© Pleiades Publishing, Ltd. 2020</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c311t-9168745185e25bf57d6f9cf144eb68cd4be6f6475f3b9482783b72ab6bdb93493</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1134/S1063454120010124$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S1063454120010124$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Zani, M.</creatorcontrib><creatorcontrib>Khartov, A. A.</creatorcontrib><title>Decomposition of Additive Random Fields</title><title>Vestnik, St. Petersburg University. Mathematics</title><addtitle>Vestnik St.Petersb. Univ.Math</addtitle><description>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.</description><subject>Analysis</subject><subject>Approximation</subject><subject>Continuity (mathematics)</subject><subject>Covariance</subject><subject>Decomposition</subject><subject>Dependence</subject><subject>Eigenvalues</subject><subject>Eigenvectors</subject><subject>Entropy</subject><subject>Fields (mathematics)</subject><subject>Integrals</subject><subject>Intersections</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operators (mathematics)</subject><subject>Random processes</subject><issn>1063-4541</issn><issn>1934-7855</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp1UMtKAzEUDaJgrX6AuwEXrkZz855lqVaFguBjHSaTRKZ0JmMyFfz7pozgQlzdA-dxDwehS8A3AJTdvgIWlHEGBGPAQNgRmkFFWSkV58cZZ7o88KfoLKUNxlwQTmfo-s41oRtCasc29EXwxcLajL9c8VL3NnTFqnVbm87Ria-3yV383Dl6X92_LR_L9fPD03KxLhsKMJYVCCUZB8Ud4cZzaYWvGg-MOSNUY5lxwgsmuaemYopIRY0ktRHGmty2onN0NeUOMXzuXBr1Juxin19qQhXljBCKswomVRNDStF5PcS2q-O3BqwPe-g_e2QPmTwpa_sPF3-T_zftAch0Xvk</recordid><startdate>2020</startdate><enddate>2020</enddate><creator>Zani, M.</creator><creator>Khartov, A. A.</creator><general>Pleiades Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2020</creationdate><title>Decomposition of Additive Random Fields</title><author>Zani, M. ; Khartov, A. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c311t-9168745185e25bf57d6f9cf144eb68cd4be6f6475f3b9482783b72ab6bdb93493</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Analysis</topic><topic>Approximation</topic><topic>Continuity (mathematics)</topic><topic>Covariance</topic><topic>Decomposition</topic><topic>Dependence</topic><topic>Eigenvalues</topic><topic>Eigenvectors</topic><topic>Entropy</topic><topic>Fields (mathematics)</topic><topic>Integrals</topic><topic>Intersections</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operators (mathematics)</topic><topic>Random processes</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Zani, M.</creatorcontrib><creatorcontrib>Khartov, A. A.</creatorcontrib><collection>CrossRef</collection><jtitle>Vestnik, St. Petersburg University. Mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Zani, M.</au><au>Khartov, A. A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Decomposition of Additive Random Fields</atitle><jtitle>Vestnik, St. Petersburg University. Mathematics</jtitle><stitle>Vestnik St.Petersb. Univ.Math</stitle><date>2020</date><risdate>2020</risdate><volume>53</volume><issue>1</issue><spage>29</spage><epage>36</epage><pages>29-36</pages><issn>1063-4541</issn><eissn>1934-7855</eissn><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.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S1063454120010124</doi><tpages>8</tpages><oa>free_for_read</oa></addata></record> |
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subjects | 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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