Central Limit Theory under the Strong Mixing Condition

R. Bradley

ABSTRACT: This talk is intended for a general audience; its main starting point will be just a rudimentary notion of ``probability.'' There will be a brief review of the ``central limit theorem'' (the law of averages involving the standard bell curve) for independent observations from the same ``population.'' Then the talk will get to central limit theorems for ``weakly dependent'' observations from a given ``population,'' with the ``weak dependence'' being the strong mixing condition. The Bernstein ``blocking'' technique will be discussed. Also, a theorem of Helson and Sarason will be given, showing a connection between the strong mixing condition and Fourier analysis.

Central Limit Theory under Other Mixing Conditions, and a look at Ibragimov's Conjecture

R. Bradley

ABSTRACT: Again, this talk is intended for a general audience, though it will build on the first talk. It will give a comparison of results in central limit theory under three mixing conditions that are stronger than strong mixing --- namely, ``phi-mixing,'' ``rho-mixing,'' and ``interlaced rho-mixing.'' There will be a discussion of Ibragimov's conjecture, which involves the central limit question under the phi-mixing condition. That conjecture was posed in the 1960's by I.A. Ibragimov and remains unsolved today. We shall look briefly at a central limit theorem of M. Peligrad which confirms the general spirit of that conjecture.

Every ``lower psi-mixing'' Markov chain is ``interlaced rho-mixing''; a possible but uncertain approach to Ibragimov's Conjecture

R. Bradley

ABSTRACT: This talk will start out at a not-too-technical level, but will then develop into a more technical form than in the first two talks. It will deal with a possible but uncertain, off-beat approach to Ibragimov's Conjecture. That approach is motivated by the following theorem: Every ``lower psi-mixing'' Markov chain is ``interlaced rho-mixing.'' The ``lower psi-mixing'' condition will be defined (it is stronger than phi-mixing), and a sketch of a proof of that theorem will be given.