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,.
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