Linear Multifractional Stable Motion: an a.s. uniformly convergent estimator for Hurst function

Since the middle of the 90's, multifractional processes have been introduced for overcoming some limitations of the well-known Fractional Brownian Motion model due to the constancy in time of its Hurst parameter $H$; in their context, this parameter becomes a H\"older continuous function $H(\cdot)$ depending on the time variable~$t$. Global and local sample path roughness of a multifractional process are determined by values of its parameter $H(\cdot)$; therefore, since about two decades, several authors have been interested in their statistical estimation, starting from discrete variations of the process. Because of complex dependence structures of variations, in order to show consistency of estimators one has to face challenging problems. The main goal of our article, is to introduce in the setting of the symmetric $\alpha$-stable non-anticipative moving average Linear Multifractional Stable Motion (LMSM), where $\alpha\in (1,2)$, a new strategy for dealing with such kind of problems. In contrast with previously developed strategies, this new one, does not require to look for sharp estimates of covariances related to functionals of variations; roughly speaking, it consists in expressing variations in such a way that they become independent random variables up to negligible remainders. Thanks to it, we obtain: $(i)$ a strongly consistent estimator of $\min_{t\in I} H(t)$ for any compact interval $I$; $(ii)$ more importantly, a strongly consistent estimator of the whole function $H(\cdot)$, which converges almost surely in the sense of the uniform norm. Such kind of almost sure result in $L^\infty$-norm is rather unusual in the literature on statistical estimation of functions.