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Concentration of measure for Brownian particle systems interacting through their ranks. (English) Zbl 1297.82023

Summary: 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.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
91G10 Portfolio theory
60J65 Brownian motion
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References:

[1] Arguin, L.-P. and Aizenman, M. (2009). On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 1080-1113. · Zbl 1177.60050
[2] Banner, A. D., Fernholz, R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296-2330. · Zbl 1099.91056
[3] Banner, A. D. and Ghomrasni, R. (2008). Local times of ranked continuous semimartingales. Stochastic Process. Appl. 118 1244-1253. · Zbl 1147.60052
[4] Chatterjee, S. and Pal, S. (2010). A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 123-159. · Zbl 1188.60049
[5] Chatterjee, S. and Pal, S. (2011). A combinatorial analysis of interacting diffusions. J. Theoret. Probab. 24 939-968. · Zbl 1236.60093
[6] Dembo, A. (1997). Information inequalities and concentration of measure. Ann. Probab. 25 927-939. · Zbl 0880.60018
[7] Dembo, A. and Zeitouni, O. (1996). Transportation approach to some concentration inequalities in product spaces. Electron. Commun. Probab. 1 83-90 (electronic). · Zbl 0916.28003
[8] Djellout, H., Guillin, A. and Wu, L. (2004). Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 2702-2732. · Zbl 1061.60011
[9] Dupuis, P. and Ishii, H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stoch. Stoch. Rep. 35 31-62. · Zbl 0721.60062
[10] Dupuis, P. and Ramanan, K. (1999). Convex duality and the Skorokhod problem. I. Probab. Theory Related Fields 115 153-195. · Zbl 0944.60061
[11] Dupuis, P. and Ramanan, K. (1999). Convex duality and the Skorokhod problem. II. Probab. Theory Related Fields 115 197-236. · Zbl 0944.60062
[12] Fernholz, E. R. (2002). Stochastic Portfolio Theory . Springer, New York. · Zbl 1049.91067
[13] Fernholz, R. and Karatzas, I. (2009). Stochastic portfolio theory: A survey. In Handbook of Numerical Analysis : Mathematical Modeling and Numerical Methods in Finance 89-168. Elsevier, Amsterdam.
[14] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302-308. · Zbl 0462.60073
[15] Ichiba, T. and Karatzas, I. (2010). On collisions of Brownian particles. Ann. Appl. Probab. 20 951-977. · Zbl 1235.60111
[16] Ichiba, T., Papathanakos, V., Banner, A., Karatzas, I. and Fernholz, R. (2011). Hybrid atlas models. Ann. Appl. Probab. 21 609-644. · Zbl 1230.60046
[17] Jourdain, B. and Malrieu, F. (2008). Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF. Ann. Appl. Probab. 18 1706-1736. · Zbl 1185.65013
[18] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89 . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002
[19] Marton, K. (1996). Bounding \(\bar{d}\)-distance by informational divergence: A method to prove measure concentration. Ann. Probab. 24 857-866. · Zbl 0865.60017
[20] McKean, H. P. and Shepp, L. A. (2005). The advantage of capitalism vs. socialism depends on the criterion. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. ( POMI ) 328 160-168, 279-280. · Zbl 1121.91068
[21] Pal, S. (2012). Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Related Fields 154 225-254. · Zbl 1259.60091
[22] Pal, S. and Pitman, J. (2008). One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18 2179-2207. · Zbl 1166.60061
[23] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Springer, Berlin. · Zbl 0917.60006
[24] Ruzmaikina, A. and Aizenman, M. (2005). Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 82-113. · Zbl 1096.60042
[25] Shkolnikov, M. (2009). Competing particle systems evolving by i.i.d. increments. Electron. J. Probab. 14 728-751. · Zbl 1190.60039
[26] Shkolnikov, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21 1911-1932. · Zbl 1238.60113
[27] Shkolnikov, M. (2012). Large systems of diffusions interacting through their ranks. Stochastic Process. Appl. 122 1730-1747. · Zbl 1276.60087
[28] Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1-34. · Zbl 0858.60019
[29] Talagrand, M. (1996). Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 587-600. · Zbl 0859.46030
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