Estimation of the local regularity index of a sample path (Q1598478)

We consider processes that satisfy a local Hölder condition with unknown coefficient \(\gamma_0\). We study two families of estimators for \(\gamma_0\): the first one based upon the whole sample path over \([0,T_N]\), \(T_N\nearrow\infty\), and the second one constructed with \(n\) observations at sampling rate \(\delta_n\), \(\delta_n\to 0\) and \(n\delta_n\to \infty\) as \(n\to \infty\). For the almost sure convergence, we give the rates for these two families.