Statistics 608a
Approximation of probability distributions

Syllabus

1. Some exact distributions
2. Some bounds
1. Markov, Chernoff's bounds (iid case and Markov).
2. binomial inequalities
3. concentrations inequalities + applications to qualitative estimates on the running time of rejection algorithm for simulating r.v.'s
4. bounds using coupling techniques + applications for testing unimodality or monotone failure rate
5. folklore bounds (Dvoretsky-Kiefer-Wolfowitz) + applications
3. The CLT and linear form of the empirical measure. Mainly some examples, the idea of differentiable statistics.
4. Berry-Esseen with the proof via Stein-Chen method.
5. Edgeworth expansions
1. Basics on characteristic functions.
2. Edgeworth expansion for the mean, and for U-statistics.
3. Application to improved confidence intervals for the mean.
6. Laplace method
1. in R: application to Gaussian and gamma integrals
2. in R^d: some elementary differential geometry of hypersurfaces (normal vector, second fundamental form)
• Laplace approximation;
• application to the power of tests in LAN families under directions not so close to the null hypothesis.
7. Saddle point approximations: application to the distribution of the determinant, the norm of random matrices; application to the distribution of the empirical covariance of some time series models.
8. Weyl tube formula type estimates : application to correlation coefficients.
9. If time permits (but this is already plenty):
1. some more Stein-Chen method;
2. some integral over asymptotic sets and small cones (plenty of applications to approximate distribution of correlation coefficient, percentage of inertia explained by the first principal components, etc.)