  Linear fractional stable motion, denoted by $\{X_{H,\al}(t)\}_{t\in \R}$, is
one of the most classical stable processes; it depends on two parameters $H\in
(0,1)$ and $\al\in (0,2)$. The parameter $H$ characterizes the self-similarity
property of $\{X_{H,\al}(t)\}_{t\in \R}$ while the parameter $\al$ governs the
tail heaviness of its finite dimensional distributions; throughout our article
we assume that the latter distributions are symmetric, that $H>1/\al$ and that
$H$ is known. We show that, on the interval $[0,1]$, the asymptotic behaviour
of the maximum, at a given scale $j$, of absolute values of the wavelet
coefficients of $\{X_{H,\al}(t)\}_{t\in \R}$, is of the same order as
$2^{-j(H-1/\al)}$; then we derive from this result a strongly consistent (i.e.
almost surely convergent) statistical estimator for the parameter $\al$.
