Concentration of measure for systems of Brownian particles interacting through their ranks

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.

arXiv comment27 pages, 1 figure

References in corpus (5)

• Transportation cost-information inequalities and applications to random dynamical systems and diffusions
• Atlas models of equity markets
• Hybrid Atlas models
• Characterization of invariant measures at the leading edge for competing particle systems
• Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process

Cited by in corpus (5)

• Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation
• Convergence rates for rank-based models with applications to portfolio theory
• Concentration for multidimensional diffusions and their boundary local times
• Triple and Simultaneous Collisions of Competing Brownian Particles
• On a Class of Diverse Market Models