Statistics 608a
Approximation of probability distributions
Syllabus

Some exact distributions

Some bounds

Markov, Chernoff's bounds (iid case and Markov).

binomial inequalities

concentrations inequalities + applications to qualitative estimates on
the running time of rejection algorithm for simulating r.v.'s

bounds using coupling techniques + applications for testing unimodality
or monotone failure rate

folklore bounds (DvoretskyKieferWolfowitz) + applications

The CLT and linear form of the empirical measure. Mainly some examples,
the idea of differentiable statistics.

BerryEsseen with the proof via SteinChen method.

Edgeworth expansions

Basics on characteristic functions.

Edgeworth expansion for the mean, and for Ustatistics.

Application to improved confidence intervals for the mean.

Laplace method

in R: application to Gaussian and gamma integrals

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.

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.

Weyl tube formula type estimates : application to correlation coefficients.

If time permits (but this is already plenty):

some more SteinChen method;

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.)