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Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics
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  • Published: 31 January 2009

Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics

  • Ester Gabetta1 &
  • Eugenio Regazzini1,2 

Probability Theory and Related Fields volume 146, pages 451–480 (2010)Cite this article

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Abstract

This paper is part of our efforts to show how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the convergence to equilibrium of the solutions of the Kac equation. Here, we consider convergence with respect to the following metrics: Kolmogorov’s uniform metric; 1 and 2 Gini’s dissimilarity indices (widely known as 1 and 2 Wasserstein metrics); χ-weighted metrics. Our main results provide new bounds, or improvements on already well-known ones, for the corresponding distances between the solution of the Kac equation and the limiting Gaussian (Maxwellian) distribution. The study is conducted both under the necessary assumption that initial data have finite energy, without assuming existence of moments of order greater than 2, and under the condition that the (2 + δ)-moment of the initial distribution is finite for some δ > 0.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Ester Gabetta & Eugenio Regazzini

  2. IMATI-CNR, Milan, Italy

    Eugenio Regazzini

Authors
  1. Ester Gabetta
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  2. Eugenio Regazzini
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Corresponding author

Correspondence to Ester Gabetta.

Additional information

E. Gabetta has been supported by MIUR, grant 2006/015821 and E. Regazzini by MIUR, grant 2006/134525.

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Gabetta, E., Regazzini, E. Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics. Probab. Theory Relat. Fields 146, 451–480 (2010). https://doi.org/10.1007/s00440-008-0196-0

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  • Received: 23 March 2007

  • Revised: 25 July 2008

  • Published: 31 January 2009

  • Issue Date: March 2010

  • DOI: https://doi.org/10.1007/s00440-008-0196-0

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Keywords

  • Boltzmann (Kac) equation
  • Berry–Esseen inequality
  • Central limit theorem
  • Gini’s (Tanaka, Wasserstein) metrics
  • Kolmogorov’s metric
  • χ-Weighted metrics

Mathematics Subject Classification (2000)

  • 60F05
  • 82C40
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