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

Statistical mechanics and time-series analysis by L\'evy-parameters with the possibility of real-time application