  We develop a method that relates the truncated cumulant-function of the
fourth order with the L\'evian cumulant-function. This gives us explicit
formulas for the L\'evy-parameters, which allow a real-time analysis of the
state of a random-motion. Cumbersome procedures like maximum-likelihood or
least-square methods are unnecessary. Furthermore, we treat the L\'evy-system
in terms of statistical mechanics and work out it's thermodynamic properties.
This also includes a discussion of the fractal nature of relativistic
corrections. As examples for a time-series analysis, we apply our results on
the time-series of the German DAX and the American S\&P-500\,.
