{"title":"Hammond-Sheffield瓮中通过合并概率的渐近高斯性","authors":"Jan Lukas Igelbrink, A. Wakolbinger","doi":"10.30757/ALEA.v20-04","DOIUrl":null,"url":null,"abstract":"For the renormalised sums of the random $\\pm 1$-colouring of the connected components of $\\mathbb Z$ generated by the coalescing renewal processes in the\"power law P\\'olya's urn\"of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, Gonz\\'alez Casanova, Kurt, and Span\\`o.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn\",\"authors\":\"Jan Lukas Igelbrink, A. Wakolbinger\",\"doi\":\"10.30757/ALEA.v20-04\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For the renormalised sums of the random $\\\\pm 1$-colouring of the connected components of $\\\\mathbb Z$ generated by the coalescing renewal processes in the\\\"power law P\\\\'olya's urn\\\"of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, Gonz\\\\'alez Casanova, Kurt, and Span\\\\`o.\",\"PeriodicalId\":49244,\"journal\":{\"name\":\"Alea-Latin American Journal of Probability and Mathematical Statistics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-01-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Alea-Latin American Journal of Probability and Mathematical Statistics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.30757/ALEA.v20-04\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/ALEA.v20-04","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn
For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the"power law P\'olya's urn"of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, Gonz\'alez Casanova, Kurt, and Span\`o.
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.