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401-5005-57L DR D-USYS , D-BAUG , D-MAVT , D-INFK , D-MTEC , D-MATH , D-BIOL , D-GESS , D-ITET , D-ARCH , D-CHAB

Non-Standard Asymptotics and Particle Systems

Lecturers: Prof. Dr. Piet Groeneboom
VVZ CR n/a

Last Updated: 2026-02-05 15:14:58

Abstract

"Nachdiplomvorlesung"

Content

Many nonparametric estimators which arise as solutions of optimization problems (maximum likelihood, least squares, etc.) have non-normal limit distributions, and converge at rates slower than n^(-1/2). This type of limit behavior is commonly called "non-standard asymptotics", and the local limit often has a characterization in terms of a stationary point process. The latter was shown, for example, in [4] for the Grenander estimator of a decreasing density. Results of this type have counterparts in the theory of interacting particle systems on the integers, such as TASEP (totally asymmetric simple exclusion processes) and in the theory of interacting particle systems on the real line, such as Hammersley's process. The connection between these seemingly unrelated fields is discussed in [3] for Hammersley's process, with a follow-up for TASEP in [1]. It is my intention to discuss recent results on these matters and to explore the rather fascinating connection between the fields further during the course. I will also discuss the (many) open problems.

Resources

Literature

[1] Balasz, M., Cator, E.A. and Seppalainen, T. (2006). Cube root fluctuations for the corner growth associated to the exclusion process. Electronic Journal of Probability, 11, 1094-1132. [2] Cator, E.A. and Groeneboom, P. (2005). Hammersley's process with sources and sinks. Annals of Probability, 33, 879-903. [3] Cator, E.A. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley's process. Annals of Probability, 34, 1273-1295. [4] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probability theory and related fields, 81, 79-109. [5] Groeneboom, P., Maathuis, M.H. and J.A. Wellner (2007). Current status data with competing risks: Consistency and rates of convergence of the MLE. To appear in the Annals of Statistics. [6] Groeneboom, P., Maathuis, M.H. and J.A. Wellner (2007). Current status data with competing risks: Limiting distribution of the MLE. To appear in the Annals of Statistics.

General Information

Language
English
Levels
DR

Examination

Type
no performance assessment

Course Components

Type Title Time & Place Hours
lecture Non-Standard Asymptotics and Particle Systems
Beginn: 09.10.2007
  • Tue 11:15-13:00 (HG G 19.2)
2 h weekly

Offered In