  This paper is devoted to the introduction of a new class of consistent
estimators of the fractal dimension of locally self-similar Gaussian processes.
These estimators are based on convex combinations of sample quantiles of
discrete variations of a sample path over a discrete grid of the interval
$[0,1]$. We derive the almost sure convergence and the asymptotic normality for
these estimators. The key-ingredient is a Bahadur representation for sample
quantiles of non-linear functions of Gaussians sequences with correlation
function decreasing as $k^{-\alpha}L(k)$ for some $\alpha>0$ and some slowly
varying function $L(\cdot)$.
