Statistics 607b, Spring 2005
Inequalities for probability and statistics

Instructor:   Mr. D. Pollard.
Time: MW 1:00-2:15
Office hours:   TBA
Place:   24 HH Rm. 107
A guided tour of some inequalities useful in statistical and probabilistic problems. The course will be broken into independent segments, each treating a specific method and an illustrative application. Acquaintance with probability at the 600 level helpful for some segments. Possible topics: convexity arguments; tail bounds for martingales and independent summands; metric entropy and maximal inequalities; VC dimension and combinatorial methods; distances between probability measures; majorizing measures and generic chaining; isoperimetric inequalities; concentration inequalities; Gaussian processes. Applications to: statistical inference; asymptotic theory; minimax rates of convergence; machine learning; complexity.


Homework plus one more substantial project.


See the reference list for some books and original papers. I will also prepare handouts, with other material cited, for some topics.

Intended topics

I cannot possibly cover everything in the following list properly in one semester. Topics near the end will probably be covered only briefly or omitted altogether.

  1. Tails of the normal distribution: classical bounds; Mill's ratio; conditional distributions; LIL.

  2. Convexity: log moment generating functions; Hoeffding's inequality; Bennett's inequality; large deviation bounds; Orlicz norms; subgaussian distributions.

  3. Metric entropy: chaining inequality for processes via Orlicz norms; Gaussian processes.

  4. Distances between measures: total variation; Hellinger; Kullback-Leibler; Fano's lemma; Assouad's inequality; minimax rates of convergence; maximum likelihood in one dimension via Hellinger distance.

  5. Covariance inequalities: FKG; application to 2-dimensional Ising model.

  6. Combinatorial bounds: Vapnik-Cervonenkis and beyond.

  7. Concentration inequalities: martingale method; Talagrand's method for product measures; tensorization methods.

  8. Poisson and Binomial: coupling bound; Le Cam-Hodges method; Chen-Stein method; unimodality; majorization.

  9. Berry-Esseen bound: via Fourier inversion method; Stein's method.

  10. Tusnady's inequality: coupling of empirical and Gaussian processes.

  11. Majorizing measures: the "generic chaining".

  12. Sudakov and Fernique inequalities for Gaussian processes; Borell's isopermetric inequality.