Detecting the Complexity of a Functional Time Series

Consider a time series that takes values in a general topological space, and suppose that its Small-ball probability is factorized into two terms that play the role of a surrogate density and a volume term. The latter allows us to study the complexity of the underlying process. In some cases, the volume term can be analytically specified in a parametric form as a function of a complexity index. This work presents the study of an estimator for such an index whenever the volume term is monomial. Weak consistency and asymptotic Gaussianity are shown under an appropriate dependence structure, providing theoretical support for the construction of confidence intervals. A Monte Carlo simulation is performed to evaluate the performance of the approach under various conditions. Finally, the method is applied to identify the complexity of two real data sets.