Mixing Conditions, Central Limit Theorems, and Invariance Principles: A Survey of the Literature with Some New Results on Heteroscedastic Sequences

This article is a survey of the main results on the central limit theorem (CLT) and its invariance principle (IP) for mixing sequences that have been obtained in the probabilistic literature in the last fifty years or so with a view towards econometric applications. Each of these theorems specifies a set of moment, dependence, and heterogeneity conditions on the underlying sequence that ensures the validity of CLT and IP. Special emphasis is paid to the case in which the underlying sequence has just barely infinite variance, since this case is relevant to econometrics applications that involve high-frequency financial data. Moreover, two new results on IPs that apply to heteroscedastic sequences are obtained. The first IP applies to sequences whose variances evolve over time in a polynomial-like fashion, whereas the second IP concerns sequences that experience a single variance break at some point within the sample.