The Annals of Applied Probability

Concentration of measure for Brownian particle systems interacting through their ranks

Soumik Pal and Mykhaylo Shkolnikov

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Abstract

We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using transportation cost inequalities for stochastic processes we provide uniform fluctuation bounds for the ordered particles, their local time of collisions and various associated statistics over intervals of time. For example, such processes, when exponentiated and rescaled, exhibit power law decay under stationarity; we derive concentration bounds for the empirical estimates of the index of the power law over large intervals of time. A key ingredient in our proofs is a novel upper bound on the Lipschitz constant of the Skorokhod map that transforms a multidimensional Brownian path to a path which is constrained not to leave the positive orthant.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 4 (2014), 1482-1508.

Dates
First available in Project Euclid: 14 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1400073655

Digital Object Identifier
doi:10.1214/13-AAP954

Mathematical Reviews number (MathSciNet)
MR3211002

Zentralblatt MATH identifier
1297.82023

Subjects
Primary: 82C22: Interacting particle systems [See also 60K35] 60H10: Stochastic ordinary differential equations [See also 34F05] 91G10: Portfolio theory

Keywords
Brownian particle systems concentration of measure transportation cost inequalities Skorokhod maps stochastic portfolio theory Atlas model

Citation

Pal, Soumik; Shkolnikov, Mykhaylo. Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24 (2014), no. 4, 1482--1508. doi:10.1214/13-AAP954. https://projecteuclid.org/euclid.aoap/1400073655


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