Measuring the roughness of random paths by increment ratios

A statistic based on increment ratios is defined and studied for measuring the roughness of random paths. Its asymptotic properties are related to an eventual tangent process. The case of rough Gaussian processes is studied in details. Under very general assumptions not requiring stationarity conditions, a strong law of large numbers and a central limit theorem are established. Proofs are obtained from a general moment bound for a product of Gaussian vector's functions extending the moment bound in Taqqu (1977, Lemma 4.5), and a central limit theorem for Gaussian multidimensional triangular arrays generalizing the result of Arcones (1994).