Invariance Principles for Dependent Processes Indexed by Besov Classes with an Application to a Hausman Test for Linearity
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 Pötscher (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 dependence 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 (1978) specification test for linearity of the conditional
mean illustrates the theory.