Statistics 608a
Approximation of probability distributions


  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 Rd: some elementary differential geometry of hypersurfaces (normal vector, second fundamental form)
  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.)