SMOLUCHOWSKI PROCESSES AND NONPARAMETRIC ESTIMATION OF FUNCTIONALS OF PARTICLE DISPLACEMENT DISTRIBUTIONS FROM COUNT DATA

Alexander Goldenshluger, Royi Jacobovic

Research output: Contribution to journalArticlepeer-review

Abstract

Suppose that particles are randomly distributed in Rd, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. This paper studies properties of the Smoluchowski processes and considers related statistical problems. In the first part of the paper we revisit probabilistic properties of the Smoluchowski process in a unified and principled way: explicit formulas for generating functionals and moments are derived, conditions for stationarity and Gaussian approximation are discussed, and relations to other stochastic models are highlighted. The second part deals with statistics of the Smoluchowski processes. We consider two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In the setting of the undeviated uniform motion we study the problems of estimating the mean speed and the speed distribution, while for the Brownian displacement model the problem of estimating the diffusion coefficient is considered. In all these settings we develop estimators with provable accuracy guarantees.

Original languageEnglish
Pages (from-to)1224-1270
Number of pages47
JournalAnnals of Applied Probability
Volume34
Issue number1
DOIs
StatePublished - Feb 2024

Bibliographical note

Publisher Copyright:
© 2024 Institute of Mathematical Statistics. All rights reserved.

Keywords

  • covariance function
  • generating functions
  • kernel estimators
  • nonparametric estimation
  • Smoluchowski processes
  • stationary processes

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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