Header

UZH-Logo

Maintenance Infos

The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes


Bauer, Erich; Bertoin, Jean (2015). The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes. Electronic Journal of Probability, 20(98):online.

Abstract

We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time $\boldsymbol{t}$ is encoded by a partition $\Pi$($\mathit{t}$) of $\mathbb{N}$ into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $\Pi$ is Markovian with transitions determined by a splitting rates measure $\mathbf{r}$. However, somewhat surprisingly, $\mathbf{r}$ fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of $\Pi$($\mathit{t}$) We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.

Abstract

We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time $\boldsymbol{t}$ is encoded by a partition $\Pi$($\mathit{t}$) of $\mathbb{N}$ into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $\Pi$ is Markovian with transitions determined by a splitting rates measure $\mathbf{r}$. However, somewhat surprisingly, $\mathbf{r}$ fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of $\Pi$($\mathit{t}$) We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.

Statistics

Citations

Dimensions.ai Metrics
4 citations in Web of Science®
4 citations in Scopus®
4 citations in Microsoft Academic
Google Scholar™

Altmetrics

Downloads

5 downloads since deposited on 27 Jan 2016
2 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:16 September 2015
Deposited On:27 Jan 2016 10:03
Last Modified:18 Apr 2018 11:46
Publisher:Institute of Mathematical Statistics
ISSN:1083-6489
Funders:Swiss National Science Foundation grant
OA Status:Hybrid
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/EJP.v20-3866
Project Information:
  • : FunderSNSF
  • : Grant ID
  • : Project TitleSwiss National Science Foundation grant

Download

Download PDF  'The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes'.
Preview
Content: Published Version
Language: English
Filetype: PDF
Size: 316kB
View at publisher