Linear fractional stable motion: a wavelet estimator of the $\text{al}$ parameter

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