On Levy stable and semistable distributions

This work emphasizes the special role played by semistable and log-semistab le distributions as relevant statistical models of various observable and " internal" variables in physics. Besides of their representation, some of th eir remarkable properties (chiefly semi-self-similarity) are displayed in s ome detail. One of their characteristic features is a log-periodic variatio n of the scale parameter which appears in the standard Levy alpha -stable d istributions whose Fourier representations are re-derived in a self-contain ed way.