The effects of multifractality on the statistics of return intervals

• Published in: The European physical journal. ST, Special topics Special topics, 2008-07, Vol.161 (1), p.181-193
• Main Authors: Bogachev, M. I., Eichner, J. F., Bunde, A.
• Format: Article
• Language: English
• Subjects: Atomic; Classical and Continuum Physics; Condensed Matter Physics; Materials Science; Measurement Science and Instrumentation; Molecular; Optical and Plasma Physics; Physics; Physics and Astronomy
• ISSN: 1951-6355; 1951-6401
• DOI: 10.1140/epjst/e2008-00760-5

. We study the statistics of the return intervals in multifractal data sets with and without linear correlations. In the absence of linear correlations, we find that the nonlinear correlations inherent in multifractal data yield (i) a power-law decay of the autocorrelation function of the return intervals, (ii) a power-law increase of the conditional return period as function of the previous return interval, and (iii) a power-law decay of the probability density function of the return intervals. These features remain unchanged in the presence of linear long-term correlations. Deviations observed in the asymptotic behaviour are probably due to finite size effects. We compare our results with those obtained for uncorrelated and for monofractal long-term correlated data, and demonstrate significant differences. Applications can be found in studying the dynamics of several processes characterised by multifractality, such as turbulence, climate dynamics, heartbeat dynamics, stock market dynamics, and tele-traffic in large networks.