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

2010/11/10 by Soumik Pal, Pal, Soumik, Mykhaylo Shkolnikov +1 · 1 citation
Mathematics · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Point processes and geometric inequalities #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1011.2443

Abstract

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

Citations

• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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