Anomalous diffusion index for Levy motions

In modelling complex systems as real diffusion processes it is common to analyse its diffusive regime through the study of approximating sequences of random walks. For the partial sums S-n = xi(1) + xi(2) + ... + xi(n) one considers the approximating sequence of processes X-(n)(t) = a(n) (S-[knt] - b(n)). Then, under sufficient smoothness requirements we have the convergence to the desired diffusion, X(" (t) X(t). A key assumption usually presumed is the finiteness of the second moment, and, hence the validity of the Central Limit Theorem. Under anomalous diffusive regime the asymptotic behavior of S-n may well be non-Gaussian and n (-1) E(S-n(2)) --> infinity. Such random walks have been referred by physicists as Levy motions or Levy flights. In this work, we introduce an alternative notion to classify these regimes, the diffusion index gamma chi. For some gamma(0)(chi) properly chosen let gamma chi = inf{gamma : 0 < gamma <= gamma(0)(chi), lim sup (t -->infinity) t(-1) E\ X(t)\ (1/gamma) < infinity). Relationship between gamma chi, the infinitesimal diffusion coefficients and the diffusion constant will be explored. Illustrative examples as well as estimates, based on extreme order statistics, for gamma chi will also be presented.