TY  - JOUR
UR  - http://www.sciencedirect.com/science/article/pii/S0378437101003909
AV  - none
TI  - Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions 
KW  - Random walks; Stochastic processes; Fluctuation phenomena; Central limit theory
N2  - 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 Lévy form. We find that this process generates a new probability distribution with a crossover from a Lévy stable power law to a power law with an exponent outside the Lévy 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. 
SN  - 0378-4371
EP  -  309
ID  - eprints1876
JF  - Physica A: Statistical Mechanics and its Applications
IS  - 1?2
Y1  - 2001///
SP  - 300 
A1  - Podobnik, Boris
A1  - Matia, Kaushik
A1  - Chessa, Alessandro
A1  - Ivanov, Plamen Ch.
A1  - Lee, Youngki
A1  - Stanley, H. Eugene
PB  - Elsevier
VL  - 300
ER  -