Richard Bradley
Indiana University
Title: A strictly stationary, ``causal,'' 5-tuplewise independent counterexample to the central limit theorem
A strictly stationary sequence of random variables is constructed with the following properties: (i) the random variables each take the values -1 and +1 with probability 1/2 each, (ii) every five of the random variables are independent of each other, (iii) the sequence is ``causal'' in a certain sense (and hence ``isomorphic to a Bernoulli shift''), (iv) the sequence has a trivial double tail sigma-field, and (v) regardless of the normalization used, the partial sums do not converge to a (nondegenerate) normal law (even along a subsequence). The example has appeared in [1], and it has some features in common with an earlier construction (for an arbitrary fixed positive integer N), by Alexander Pruss and the author [2], of a strictly stationary (ergodic) N-tuplewise independent counterexample to the central limit theorem.
[1] R.C. Bradley. A strictly stationary, ``causal,'' 5-tuplewise independent counterexample to the central limit theorem. Latin American Journal of Probability and Mathematical Statistics (ALEA) 7 (2010) 377-450.
[2] R.C. Bradley and A.R. Pruss. A strictly stationary, N-tuplewise independent counterexample to the central limit theorem. Stochastic Processes and their Applications 119 (2009) 3300-3318.
©2010, Department of Mathematical Sciences
Last Modified: February 26, 2009