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Theory and Practice of Empirical Processes

  • Michel Talagrand
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics book series (MATHE3, volume 60)

Abstract

In Chapter 9 we investigate how to control the supremum of the empirical process over a class of functions. The fundamental theoretical question in this direction is whether there exists a “best possible” method to control this supremum at a given size of the random sample. We offer a natural candidate for such a “best possible” method, in the spirit of the Bednorz-Latała result of Chapter 5. Whether this natural method is actually optimal is a major open problem. To illustrate that meditating on these theoretical questions might help to solve practical problems, we present a somewhat streamlined proofs of two deep recent results.

Keywords

Empirical Process Invariant Probability Measure Admissible Sequence Spread Part Entropy Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Adamczak, R., Litvak, A.E., Pajor, A., Tomczak-Jaegermann, N.: Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Am. Math. Soc. 23(2), 535–561 (2010) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Adamczak, R., Litvak, A.E., Pajor, A., Tomczak-Jaegermann, N.: Sharp bounds on the rate of convergence of the empirical covariance matrix. C. R. Math. Acad. Sci. Paris 349(3–4), 195–200 (2011) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Adamczak, R., Latała, R., Litvak, A.E., Pajor, A., Tomczak-Jaegermann, N.: Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails. Ann. Inst. Henri Poincaré Probab. Stat. 48(4), 1103–1136 (2012) CrossRefMATHGoogle Scholar
  4. 4.
    Bourgain, J.: Random points in isotropic convex sets. In: Convex Geometric Analysis, Berkeley, CA, 1996. Math. Sci. Res. Inst. Publ., vol. 34, pp. 53–58. Cambridge Univ. Press, Cambridge (1999) Google Scholar
  5. 5.
    Dudley, R.M.: Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics, vol. 63. Cambridge University Press, Cambridge (1999) CrossRefMATHGoogle Scholar
  6. 6.
    Giné, E., Zinn, J.: Some limit theorems for empirical processes. Ann. Probab. 12, 929–989 (1984) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Klartag, B., Mendelson, S.: Empirical processes and random projections. J. Funct. Anal. 225(1), 229–245 (2005) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Latała, R.: Weak and strong moments of random vectors. In: Marcinkiewicz Centenary Volume. Banach Center Publ., vol. 95, pp. 115–121. Polish Acad. Sci. Inst. Math, Warsaw (2011) Google Scholar
  9. 9.
    Mendelson, S.: Empirical processes with a bounded ψ 1-diameter. Geom. Funct. Anal. 20(4), 988–1027 (2010) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Mendelson, S., Pajor, A., Tomczak-Jaegermann, N.: Reconstruction and subgaussian operators in asymptotic geometric analysis. Geom. Funct. Anal. 17(4), 1248–1282 (2007) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Mendelson, S., Paouris, G.: On generic chaining and the smallest singular value of random matrices with heavy tails. J. Funct. Anal. 262(9), 3775–3811 (2012) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Talagrand, M.: The Glivenko-Cantelli problem. Ann. Probab. 15, 837–870 (1987) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Talagrand, M.: Regularity of infinitely divisible processes. Ann. Probab. 21, 362–432 (1993) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Michel Talagrand
    • 1
  1. 1.Institut de MathématiquesUniversité Paris VIParis Cedex 05France

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