Open Access
August 2014 Concentration of measure for Brownian particle systems interacting through their ranks
Soumik Pal, Mykhaylo Shkolnikov
Ann. Appl. Probab. 24(4): 1482-1508 (August 2014). DOI: 10.1214/13-AAP954


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.


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Soumik Pal. Mykhaylo Shkolnikov. "Concentration of measure for Brownian particle systems interacting through their ranks." Ann. Appl. Probab. 24 (4) 1482 - 1508, August 2014.


Published: August 2014
First available in Project Euclid: 14 May 2014

zbMATH: 1297.82023
MathSciNet: MR3211002
Digital Object Identifier: 10.1214/13-AAP954

Primary: 60H10 , 82C22 , 91G10

Keywords: atlas model , Brownian particle systems , concentration of measure , Skorokhod maps , Stochastic Portfolio Theory , transportation cost inequalities

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.24 • No. 4 • August 2014
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