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

– 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. Many natural phenomena are described by distributions with scale-invariant behavior in the central part and power law tails. To explain such a behavior the Lévy process [1] has been employed in finance [2], fluid dynamics [3], polymers [4], city growth [5], geophysical [6] and biological [7] systems. An intense activity has been developed in order to understand the origin of these ubiquitous power law distributions [8]. The Lévy process, however, is characterized by a distribution with infinite moments and in applications this might be a problem, e.g., analysis of autocorrelations in time series requires a finite second moment. To address this problem, probability distributions of the Lévy type with both abrupt [9] and exponential cutoffs [10] have been proposed. A second problem is that the Lévy process has been introduced for independent and identically distributed stochastic variables, while for some systems there is a clear evidence of correlations in the variance (e.g., for many important market indices [11]). Moreover, a crossover between a Lévy stable power law and a power law with an exponent out of the Lévy regime, was recently found in the analysis of price changes [12]. We investigate how a stochastic process with no correlations in the variables but rather in their variance can be introduced to account for the empirical observations of a Lévy stable form of the probability distribution in the central part and a crossover to a power law behavior different than the Lévy in the far tails. (∗) E-mail: [email protected]