Invariance Principles for Dependent Processes Indexed by Besov Classes with an Application to a Hausman Test for Linearity

Abstract: This paper considers functional central limit theorems for stationary absolutely regular mixing processes. Bounds for the entropy with bracketing are derived using recent results in Nickl and P\"otscher (2007). More specifically, their bracketing metric entropy bounds are extended to a norm defined in Doukhan, Massart and Rio (1995, henceforth DMR) that depends both on the marginal distribution of the process and on the mixing coefficients. Using these bounds, and based on a result in DMR, it is shown that for the class of weighted Besov spaces polynomially decaying tail behavior of the function class is sufficient to obtain a functional central limit theorem under minimal conditions. A second class of functions that allow for a functional central limit theorem under minimal conditions are smooth functions defined on bounded sets. Similarly, a functional CLT for polynomially explosive tail behavior is obtained under additional moment conditions that are easy to check. An application to a Hausman specification test illustrates the theory.