@article{eprints1876,
          author = {Boris Podobnik and Kaushik Matia and Alessandro Chessa and Plamen Ch. Ivanov and Youngki Lee and H. Eugene Stanley},
         journal = {Physica A: Statistical Mechanics and its Applications},
           pages = {300 -- 309},
          volume = {300},
       publisher = {Elsevier},
           title = {Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions },
            year = {2001},
          number = {1?2},
        abstract = {We model the time series of the S\&P500 index by a combined process, the AR+GARCH process, where \{AR\} denotes the autoregressive process which we use to account for the short-range correlations in the index changes and \{GARCH\} denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long-range correlations in the variance. We study the AR+GARCH process with an initial distribution of truncated L{\'e}vy form. We find that this process generates a new probability distribution with a crossover from a L{\'e}vy stable power law to a power law with an exponent outside the L{\'e}vy range, beyond the truncation cutoff. We analyze the sum of n variables of the AR+GARCH process, and find that due to the correlations the AR+GARCH process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n{--}a feature which is observed in the probability distribution of the S\&P500 index. We find that this power-law stability depends on the characteristic scale in the correlations. We also find that inclusion of short-range correlations through the \{AR\} process is needed to obtain convergence to a limiting Gaussian distribution for large n as observed in the data. },
        keywords = {Random walks; Stochastic processes; Fluctuation phenomena; Central limit theory},
             url = {http://eprints.imtlucca.it/1876/}
}