Statistics 607a, Fall 2000
Inequalities in probability and statistics

References

Asymptopia: drafts of a manuscript on asymptotic theory

Handout on k-means

Intended topics

A study of a variety of useful inequalities.
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.

1. Convex functions: epigraphs; separating hyperplanes; conjugate convex functions; special convex functions (\Delta and \psi); log moment generating functions; Hoeffding's inequality; Bennett's inequality; large deviation bounds; argmax of a Gaussian process.

2. Orlicz norms: maximal inequalities; subgaussian distributions.

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

4. LIL via Bennett's inequality, for bounded summands; Kolmogorov's condition for LIL.

5. Metric entropy: maximal inquality for processes via Orlicz norms; maximum likelihood in one dimension; VC lemma; Glivenko-Cantelli theorem and rates; k-means.

6. Distances between measures: total variation; Hellinger; Kullback-Leibler; Fano's lemma; minimax rates of convergence.

7. Poisson approximation to Binomial: coupling bound; Le Cam-Hodges method; Stein's method.

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

9. Majorizing measures (if time permits).

10. Sudakov and Fernique inequalities for Gaussian processes.

11. Concentration inequalities: Borell's isopermetric inequality; Talagrand method for product measures.