We consider a probabilistic approach to compute the Wiener-Young Φ-variation of fractal functions in the Takagi class. Here, the Φ-variation is understood as a generalization of the quadratic variation or, more generally, the p th variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions Φ we consider form a very wide class of increasing functions that are regularly varying at zero. Moreover, for each such function Φ, our results provide in a straightforward manner functions in the Takagi class that have nontrivial and linear Φ-variation. The proof of our main result relies on a limit theorem for certain sums of Bernoulli random variables that converge to an infinite Bernoulli convolution.
We say that a continuous real-valued function x admits the Hurst roughness exponent H if the p th variation of x converges to zero if p > 1/H and to infinity if p < 1/H. For the sample paths of many stochastic processes, such as fractional Brownian motion, the Hurst roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber-Schauder coefficients of x under which the Hurst roughness exponent exists and is given as the limit of the classical Gladyshev estimates H n (x). This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because no assumption whatsoever is made on the possible dynamics of the function x. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber-Schauder expansion of x. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence ( H n ) n∈N . We also discuss how a dynamic change in the Hurst roughness parameter of a time series can be detected. Our results are illustrated by means of high-frequency financial time series.
We prove limit theorems for the weighted quadratic variation of trifractional Brownian motion and n-th order fractional Brownian motion. Furthermore, a sufficient condition for the L Pconvergence of the weighted quadratic variation for Gaussian processes is obtained as a byproduct. As an application, we give a statistical estimator for the self-similarity index of trifractional Brownian motion. These theorems extend results of Baxter, Gladyshev, and Norvaiša.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.