Linear fractional stable motion: A wavelet estimator of the αα parameter

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