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

## TL;DR

This paper introduces a real-time method for analyzing Lévy processes using cumulant functions, avoiding complex procedures, and explores their thermodynamic properties with applications to financial time-series like DAX and S&P-500.

## Contribution

It provides explicit formulas for Lévy-parameters from cumulant functions, enabling real-time analysis without maximum-likelihood or least-squares methods.

## Key findings

- Successful application to DAX and S&P-500 time-series
- Explicit formulas for Lévy-parameters derived
- Thermodynamic properties of Lévy systems discussed

## Abstract

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\,.