User:Boris Tsirelson/sandbox

Strong Mixing Conditions

Richard C. Bradley

Department of Mathematics, Indiana University, Bloomington, Indiana, USA

There has been much research on stochastic models that have a well defined, specific structure — for example, Markov chains, Gaussian processes, or linear models, including ARMA (autoregressive – moving average) models. However, it became clear in the middle of the last century that there was a need for a theory of statistical inference (e.g. central limit theory) that could be used in the analysis of time series that did not seem to "fit" any such specific structure but which did seem to have some "asymptotic independence" properties. That motivated the development of a broad theory of "strong mixing conditions" to handle such situations. This note is a brief description of that theory.

The field of strong mixing conditions is a vast area, and a short note such as this cannot even begin to do justice to it. Journal articles (with one exception) will not be cited; and many researchers who made important contributions to this field will not be mentioned here. All that can be done here is to give a narrow snapshot of part of the field.