Systems with correlations in the variance: Generating power law tails in probability distributions

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Abstract
We study how the presence of correlations in physical variables contributes to the form of probability distributions. We investigate a process with correlations in the variance generated by i) a Gaussian or ii) a truncated Lévy distribution. For both i) and ii), we find that due to the correlations in the variance, the process "dynamically" generates power law tails in the distributions, whose exponents can be controlled through the way the correlations in the variance are introduced. For ii), we find that the process can extend a truncated distribution beyond the truncation cutoff, which leads to a crossover between a Lévy stable power law and the present "dynamically generated" power law. We show that the process can explain the crossover behavior recently observed in the S&P500 stock index.