eprintid: 1876 rev_number: 7 eprint_status: archive userid: 6 dir: disk0/00/00/18/76 datestamp: 2013-11-06 11:50:13 lastmod: 2016-04-06 09:55:08 status_changed: 2013-11-06 11:50:13 type: article metadata_visibility: show creators_name: Podobnik, Boris creators_name: Matia, Kaushik creators_name: Chessa, Alessandro creators_name: Ivanov, Plamen Ch. creators_name: Lee, Youngki creators_name: Stanley, H. Eugene creators_id: creators_id: creators_id: alessandro.chessa@imtlucca.it creators_id: creators_id: creators_id: title: Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions ispublished: pub subjects: QC divisions: EIC full_text_status: none keywords: Random walks; Stochastic processes; Fluctuation phenomena; Central limit theory 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 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. date: 2001 date_type: published publication: Physica A: Statistical Mechanics and its Applications volume: 300 number: 1–2 publisher: Elsevier pagerange: 300 - 309 id_number: 10.1016/S0378-4371(01)00390-9 refereed: TRUE issn: 0378-4371 official_url: http://www.sciencedirect.com/science/article/pii/S0378437101003909 citation: Podobnik, Boris and Matia, Kaushik and Chessa, Alessandro and Ivanov, Plamen Ch. and Lee, Youngki and Stanley, H. Eugene Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions. Physica A: Statistical Mechanics and its Applications, 300 (1–2). 300 - 309. ISSN 0378-4371 (2001)