  Since the middle of the 90's, multifractional processes have been introduced
for overcoming some limitations of the classical Fractional Brownian Motion
model. In their context, the Hurst parameter becomes a Holder continuous
function H(?) of the time variable t. Linear Multifractional Stable Motion
(LMSM) is the most known one of them with heavy-tailed distributions. Generally
speaking, 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 LMSM, where $\alpha \in (1; 2)$, a new strategy for dealing with
such kind of problems. It can also be useful in other contexts. 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 of expressing variations in such a way that they become
independent random variables up to negligible remainders. Thanks to it, we
obtain, an almost surely and $L^p(\Omega), p\in(0; 4]$, consistent estimator of
the whole function $H(\cdot)$, which converges, uniformly in t, and even for
some Holder norms. Also, we obtain estimates for the rates of convergence.
Such kind of strong consistency results in uniform and Holder norms are rather
unusual in the literature on statistical estimation of functions.
