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It was known that extreme recurrence intervals can better characterize the significant influence of long\u2010term memory than using the GEV model. Our statistical analyses based on time series datasets following the L\u00e9vy stable distribution confirm that the stretched exponential distribution can describe a wide spectrum of memory behavior transition from exponentially distributed intervals (without memory) to power\u2010law distributed ones (with strong memory or fractal scaling property), extending the previous evaluation of the stretched exponential function using Gaussian\/exponential distributed random data. Further deviation and discussion of a historical paradox (i.e., the residual waiting time tends to increase with an increasing elapsed time under long\u2010term memory) are also provided, based on the theoretical analysis of the Bayesian law and the stretched exponential distribution.<\/jats:p>","DOI":"10.1155\/2018\/5913976","type":"journal-article","created":{"date-parts":[[2018,7,3]],"date-time":"2018-07-03T23:48:32Z","timestamp":1530661712000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Investigation of Stretched Exponential Function in Quantifying Long\u2010Term Memory of Extreme Events Based on Artificial Data following L\u00e9vy Stable Distribution"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8422-3871","authenticated-orcid":false,"given":"HongGuang","family":"Sun","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Yuan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0121-0950","authenticated-orcid":false,"given":"Yong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicholas","family":"Privitera","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2018,7,3]]},"reference":[{"key":"e_1_2_8_1_2","doi-asserted-by":"publisher","DOI":"10.1029\/2003GL019119"},{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.quascirev.2009.02.021"},{"key":"e_1_2_8_3_2","doi-asserted-by":"publisher","DOI":"10.1029\/2012gl051431"},{"key":"e_1_2_8_4_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.97.078501"},{"key":"e_1_2_8_5_2","volume-title":"Klimageschichte Mitteleuropas: 1000 Jahre Wetter, Klima, Katastrophen","author":"Glaser R.","year":"2001"},{"key":"e_1_2_8_6_2","doi-asserted-by":"publisher","DOI":"10.3354\/cr023089"},{"key":"e_1_2_8_7_2","volume-title":"Fractal River Basins: Chance and Selforganization","author":"Rodriguez-Iturbe I.","year":"1997"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4371(02)01662-X"},{"key":"e_1_2_8_9_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature04745"},{"key":"e_1_2_8_10_2","volume-title":"Extremes and Related Properties of Random Sequences and Processes","author":"Leadbetter M. 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