Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon
{"title":"论超扩散大象随机漫步的极限规律","authors":"Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon","doi":"arxiv-2409.06836","DOIUrl":null,"url":null,"abstract":"When the memory parameter of the elephant random walk is above a critical\nthreshold, the process becomes superdiffusive and, once suitably normalised,\nconverges to a non-Gaussian random variable. In a recent paper by the three\nfirst authors, it was shown that this limit variable has a density and that the\nassociated moments satisfy a nonlinear recurrence relation. In this work, we\nexploit this recurrence to derive an asymptotic expansion of the moments and\nthe asymptotic behaviour of the density at infinity. In particular, we show\nthat an asymmetry in the distribution of the first step of the random walk\nleads to an asymmetry of the tails of the limit variable. These results follow\nfrom a new, explicit expression of the Stieltjes transformation of the moments\nin terms of special functions such as hypergeometric series and incomplete beta\nintegrals. We also obtain other results about the random variable, such as\nunimodality and, for certain values of the memory parameter, log-concavity.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the limit law of the superdiffusive elephant random walk\",\"authors\":\"Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon\",\"doi\":\"arxiv-2409.06836\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"When the memory parameter of the elephant random walk is above a critical\\nthreshold, the process becomes superdiffusive and, once suitably normalised,\\nconverges to a non-Gaussian random variable. In a recent paper by the three\\nfirst authors, it was shown that this limit variable has a density and that the\\nassociated moments satisfy a nonlinear recurrence relation. In this work, we\\nexploit this recurrence to derive an asymptotic expansion of the moments and\\nthe asymptotic behaviour of the density at infinity. In particular, we show\\nthat an asymmetry in the distribution of the first step of the random walk\\nleads to an asymmetry of the tails of the limit variable. These results follow\\nfrom a new, explicit expression of the Stieltjes transformation of the moments\\nin terms of special functions such as hypergeometric series and incomplete beta\\nintegrals. We also obtain other results about the random variable, such as\\nunimodality and, for certain values of the memory parameter, log-concavity.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"28 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06836\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06836","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the limit law of the superdiffusive elephant random walk
When the memory parameter of the elephant random walk is above a critical
threshold, the process becomes superdiffusive and, once suitably normalised,
converges to a non-Gaussian random variable. In a recent paper by the three
first authors, it was shown that this limit variable has a density and that the
associated moments satisfy a nonlinear recurrence relation. In this work, we
exploit this recurrence to derive an asymptotic expansion of the moments and
the asymptotic behaviour of the density at infinity. In particular, we show
that an asymmetry in the distribution of the first step of the random walk
leads to an asymmetry of the tails of the limit variable. These results follow
from a new, explicit expression of the Stieltjes transformation of the moments
in terms of special functions such as hypergeometric series and incomplete beta
integrals. We also obtain other results about the random variable, such as
unimodality and, for certain values of the memory parameter, log-concavity.