A study of the ordered Chinese Restaurant Process and random dissections
| Author(s) | Rivera-Lopez, Kelvin | |
| Date Accessioned | 2022-02-22T12:58:50Z | |
| Date Available | 2022-02-22T12:58:50Z | |
| Publication Date | 2021 | |
| SWORD Update | 2021-09-30T19:14:41Z | |
| Abstract | The ordered Chinese Restaurant Process and polygonal dissections are modern variants of classical objects – namely, the Chinese Restaurant Process and polygonal triangulations. While these classical objects are well-studied in the literature, much less is known about their modern counterparts. The general aim of this thesis is to close this gap by presenting the modern analogues of some classical results. ☐ In the case of the ordered Chinese Restaurant Process, our primary interest lies in the associated up-down chains, a family of Markov processes on integer compositions. We show that, in some scaling limit, these chains converge to a diffusion on the open subsets of (0, 1). This is the analogue of a result of Petrov, in which the limit of the up-down chains associated with the classical Chinese Restaurant Process is identified. Consequently, we construct an ordered analogue to Petrov’s diffusion, and by extension, the infinitely-many-neutral-alleles diffusion model of Ethier and Kurtz. We also study the process obtained by projecting the up-down chain to its first coordinate. We state a condition on the initial distribution of the up-down chain that leads to this process having the Markov property and being intertwined with the up-down chain. In particular, these properties hold when the up-down chain is running in stationarity. ☐ Our study of polygonal dissections is focused on describing the maximum vertex degree of a random dissection. We present a concentration inequality for this random variable that is analogous to a result of Gao and Wormald concerning triangulations. As a result, we resolve a conjecture posed in 2012 by Curien and Kortchemski. | en_US |
| Advisor | Rizzolo, Douglas | |
| Degree | Ph.D. | |
| Department | University of Delaware, Department of Mathematical Sciences | |
| DOI | https://doi.org/10.58088/d4tc-6y62 | |
| Unique Identifier | 1298707841 | |
| URL | https://udspace.udel.edu/handle/19716/30500 | |
| Language | en | |
| Publisher | University of Delaware | en_US |
| URI | https://login.udel.idm.oclc.org/login?url=https://www.proquest.com/dissertations-theses/study-ordered-chinese-restaurant-process-random/docview/2591344481/se-2?accountid=10457 | |
| Keywords | Chinese restaurant process | |
| Keywords | Concentration inequality | |
| Keywords | Dissection | |
| Keywords | Intertwining | |
| Keywords | Trees | |
| Keywords | Up-down Markov chain | |
| Title | A study of the ordered Chinese Restaurant Process and random dissections | en_US |
| Type | Thesis | en_US |