HOROWITZ SEMINAR

PROBABILITY, ERGODIC THEORY and DYNAMICAL SYSTEMS

  • Monday, 10/01/2005, at 16:00 in: room 210, Schreiber Bldg.

  • Vladimir Rotar (UCSD and Moscow U.)

    will speak on

  • Stein's Method, Edgeworth's Expansions, a Formula of Barbour

  • Abstract:

    Let W be the sum of dependent random variables, and h(x) be a function. We discuss an approach based on Stein's method, to providing an Edgeworth expansion for E{h(W)} in terms of certain characteristics of dependency, and of the smoothness of h and/or the distribution of W. The core of the class of dependency structures for which these characteristics are meaningful is the local dependency, but in fact, the class is essentially wider. The approach mentioned uses in particular the following nice representation:

    E{Wf(W)}=\sum_{m=0}^{r} [v_{m+1}E{f^{(m)}(W)}/m!] +R

    where f is an arbitrary r times differentiable function, f^{(m)} is its mth derivative, W is a r.v. with finite first r+1 moments, v _{m} is the mth cumulant of W or a characteristic close to it, and R is a remainder which may be small under suitable conditions. If W is a sum of r.v.'s, a good bound for the remainder should reflect the dependency structure between the summands. We review and discuss here known results on such bounds, and provide a new result that essentially widens possibilities of applications.

    To suggest a talk (yours, or someone elses), contact Jon. Aaronson:
    email

    Back to: Jon Aaronson's homepage.

      Last update on 20/10/02.