{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T04:26:49Z","timestamp":1685593609925},"reference-count":15,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,6,1]]},"abstract":"Abstract<\/jats:title>\n Finite-dimensional (FD) models \n \n \n \n \n X<\/m:mi>\n d<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n X_{d}(t)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, i.e., deterministic functions of time and finite sets of \ud835\udc51 random variables, are constructed for stationary and nonstationary Gaussian processes \n \n \n \n X<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n X(t)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with continuous samples defined on a bounded time interval \n \n \n \n [<\/m:mo>\n 0<\/m:mn>\n ,<\/m:mo>\n \u03c4<\/m:mi>\n ]<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n [0,\\tau]<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThe basis functions of these FD models are finite sets of eigenfunctions of the correlation functions of \n \n \n \n X<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n X(t)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and of trigonometric functions.\nNumerical illustrations are presented for a stationary Gaussian process \n \n \n \n X<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n X(t)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with exponential correlation function and a nonstationary version of this process obtained by time distortion.\nIt was found that the FD models are consistent with the theoretical results in the sense that their samples approach the target samples as the stochastic dimension is increased.<\/jats:p>","DOI":"10.1515\/mcma-2023-2006","type":"journal-article","created":{"date-parts":[[2023,5,4]],"date-time":"2023-05-04T16:44:35Z","timestamp":1683218675000},"page":"127-142","source":"Crossref","is-referenced-by-count":0,"title":["Monte Carlo estimates of extremes of stationary\/nonstationary Gaussian processes"],"prefix":"10.1515","volume":"29","author":[{"given":"Mircea Dan","family":"Grigoriu","sequence":"first","affiliation":[{"name":"School of Civil and Environmental Engineering and Center for Applied Mathamatics , Cornell University , Ithaca , NY 14853\u20133501 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,5,25]]},"reference":[{"key":"2023053118460993316_j_mcma-2023-2006_ref_001","unstructured":"P. Billingsley,\nConvergence of Probability Measures,\nJohn Wiley & Sons, New York, 1968."},{"key":"2023053118460993316_j_mcma-2023-2006_ref_002","unstructured":"H. Cram\u00e9r and M. R. Leadbetter,\nStationary and Related Stochastic Processes,\nJohn Wiley & Sons, New York, 1967."},{"key":"2023053118460993316_j_mcma-2023-2006_ref_003","doi-asserted-by":"crossref","unstructured":"M. Grigoriu,\nSimulation of nonstationary Gaussian processes by random trigonometric polynomials,\nJ. Engrg. Mech. 119 (1993), 328\u2013343.","DOI":"10.1061\/(ASCE)0733-9399(1993)119:2(328)"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_004","doi-asserted-by":"crossref","unstructured":"M. Grigoriu,\nStochastic Calculus. Applications in Science and Engineering,\nBirkh\u00e4user, Boston, 2002.","DOI":"10.1007\/978-0-8176-8228-6"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_005","doi-asserted-by":"crossref","unstructured":"M. Grigoriu,\nEvaluation of Karhunen\u2013Lo\u00e8ve, spectral, and sampling representations for stochastic processes,\nJ. Engrg. Mech. 132 (2006), no. 2, 179\u2013189.","DOI":"10.1061\/(ASCE)0733-9399(2006)132:2(179)"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_006","doi-asserted-by":"crossref","unstructured":"D. B. Hern\u00e1ndez,\nLectures on Probability and Second Order Random Fields,\nSer. Adv. Math. Appl. Sci. 30,\nWorld Scientific, River Edge, 1995.","DOI":"10.1142\/2491"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_007","unstructured":"K. It\u00f4 and M. Nisio,\nOn the convergence of sums of independent Banach space valued random variables,\nOsaka Math. J. 5 (1968), 35\u201348."},{"key":"2023053118460993316_j_mcma-2023-2006_ref_008","doi-asserted-by":"crossref","unstructured":"T. T. Kadota,\nTerm-by-term differentiability of Mercer\u2019s expansion,\nProc. Amer. Math. Soc. 18 (1967), 69\u201372.","DOI":"10.1090\/S0002-9939-1967-0203397-1"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_009","doi-asserted-by":"crossref","unstructured":"V. A. Ogorodnikov and S. M. Prigarin,\nNumerical Modelling of Random Processes and Fields: Algorithms and Applications,\nVSP, Utrecht, 1996.","DOI":"10.1515\/9783110941999"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_010","doi-asserted-by":"crossref","unstructured":"S. M. Prigarin,\nSpectral Models of Random Fields in Monte Carlo Simulation,\nVSP, Boston, 2001.","DOI":"10.1515\/9783110941982"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_011","unstructured":"W. Rudin,\nReal and Complex Analysis,\nMcGraw-Hill, New York, 1974."},{"key":"2023053118460993316_j_mcma-2023-2006_ref_012","doi-asserted-by":"crossref","unstructured":"K. K. Sabelfeld,\nRandom Fields and Stochastic Lagrangian Models. Analysis and Applications in Turbulence and Porous Media,\nWalter de Gruyter, Berlin, 2012.","DOI":"10.1515\/9783110296815"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_013","doi-asserted-by":"crossref","unstructured":"G. Samorodnitsky,\nProbability tails of Gaussian extrema,\nStochastic Process. Appl. 38 (1991), no. 1, 55\u201384.","DOI":"10.1016\/0304-4149(91)90072-K"},{"key":"2023053118460993316_j_mcma-2023-2006_ref_014","unstructured":"G. P. Tolstov,\nFourier Series,\nDover, New York, 1962."},{"key":"2023053118460993316_j_mcma-2023-2006_ref_015","doi-asserted-by":"crossref","unstructured":"H. Xu and M. Grigoriu,\nFinite dimensional models for extremes of Gaussian and non-Gaussian processes,\nProbab. Engrg. 68 (2022), 10.1016\/j.probengmech.2022.103199.","DOI":"10.1016\/j.probengmech.2022.103199"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2023-2006\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2023-2006\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T18:46:43Z","timestamp":1685558803000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2023-2006\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,25]]},"references-count":15,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,5,10]]},"published-print":{"date-parts":[[2023,6,1]]}},"alternative-id":["10.1515\/mcma-2023-2006"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2023-2006","relation":{},"ISSN":["0929-9629","1569-3961"],"issn-type":[{"value":"0929-9629","type":"print"},{"value":"1569-3961","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,25]]}}}