Concentration Inequalities

Stephane Boucheron, Gabor Lugosi, and Pascal Massart 
Concentration inequalities for functions of independent random variables is an area of probability theory that has witnessed a great revolution in the last few decades, and has applications in a wide variety of areas such as machine learning, statistics, discrete mathematics, and high-dimensional geometry. Roughly speaking, if a function of many independent random variables does not depend too much on any of the variables then it is concentrated in the sense that with high probability, it is close to its expected value. This book offers a host of inequalities to illustrate this rich theory in an accessible way by covering the key developments and applications in the field.The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities to transportation arguments to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are also presented.A self-contained introduction to concentration inequalities, it includes a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.Publisher: Oxford University Press.

Publication Date: February 7, 2013 (January 28, 2016, Paperback) | ISBN-10: 0199535256  or 0198767657  | ISBN-13: 978-0199535255 or 978-0198767657 
  1. Introduction
  2. Basic inequalities
  3. Bounding the variance
  4. Basic information inequalities
  5. Logarithmic Sobolev inequalities
  6. The entropy method
  7. Concentration and isoperimetry
  8. The transportation method
  9. Influences and threshold phenomena
  10. Isoperimetry on the hypercube and Gaussian spaces
  11. The variance of suprema of empirical processes
  12. Suprema of empirical processes: exponential inequalities
  13. The expected value of suprema of empirical processes
  14. \(\Phi\)-entropies
  15. Moment inequalities

Oxford Scholarship Online : DOI:10.1093/acprof:oso/9780199535255.001.0001