Michael Woodroofe
University of Michigan
Title:The conditional central limit question for stationary processes

Let ...,X(-1),X(0),X(1),X(2),... denote a stationary ergodic process for which the X(k) have mean 0 and a finite positive variance. Further, let S(n) = X(1)+...+X(n) and sigma(n)^2 = E(S(n)^2) and suppose that sigma(n) tends to infinity as n tends to infinity. The conditional central limit question is whether the conditional distribution of S(n) /\ sigma(n) given ... X(-1),X(0) converges to the standard normal distribution. There has been substantial recent progress on this question, leading conditions that are necessary and nearly sufficient. On one hand the conditions can be phrased in terms of growth restrictions on E(S(n) | ... X(-1),X(0)). On the other, they may be phrased in terms of approximate solutions to Poisson's Equation for the Markov Chain W(n) = (... X(n-1),X(n)) and solutions to a fractional version of Poisson's Equation. This progress will be reviewed, and some current work described.
©2004, Department of Mathematical Sciences
Last Modified: December 6, 2004