Universal and non-universal properties of recurrence intervals of rare events

This paper is devoted to the statistical analysis on the recurrence intervals of rare events that are defined above a given threshold. The memory property of original records is found to have significant effects on the distribution and the correlation structure of recurrence intervals, so that some universal and non-universal properties arise. (i) For long-range persistent records by the ARFIMA processes, where large values are likely to follow large values, the recurrence intervals yield stretched exponential distributions and further show long-range persistence. (ii) For long-range anti-persistent records by the ARFIMA processes, the recurrence intervals obey exponential distributions, also absence of autocorrelation. (iii) For short-range autocorrelated records, the distribution and the correlation structure of recurrence intervals both depend on the parameters of the model and the threshold of rare events.

[1]  H. Kantz,et al.  Recurrence time analysis, long-term correlations, and extreme events. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Enrico Scalas,et al.  Waiting-times and returns in high-frequency financial data: an empirical study , 2002, cond-mat/0203596.

[3]  Luciano Telesca,et al.  Are global terrorist attacks time-correlated? , 2006 .

[4]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[5]  Wei-Xing Zhou,et al.  Nonuniversal distributions of stock returns in an emerging market , 2010, 1003.5984.

[6]  Wei-Xing Zhou,et al.  Tests of nonuniversality of the stock return distributions in an emerging market. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  T. Ouarda,et al.  Regional flood frequency estimation with canonical correlation analysis , 2001 .

[8]  Kazuko Yamasaki,et al.  Scaling and memory of intraday volatility return intervals in stock markets. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  H. Stanley,et al.  Scaling of seismic memory with earthquake size. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  G. F. Newell,et al.  Zero Crossing Probabilities for Gaussian Stationary Processes , 1962 .

[11]  Shlomo Havlin,et al.  Statistics of return intervals in long-term correlated records. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Ian F. Blake,et al.  Level-crossing problems for random processes , 1973, IEEE Trans. Inf. Theory.

[13]  Kazuko Yamasaki,et al.  Indication of multiscaling in the volatility return intervals of stock markets. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  D Delignières,et al.  Degeneracy and long-range correlations. , 2013, Chaos.

[15]  P. Ivanov,et al.  Common scaling patterns in intertrade times of U. S. stocks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  E. Casson,et al.  Simulation and extremal analysis of hurricane events , 2000 .

[17]  Kazuko Yamasaki,et al.  Scaling and memory in volatility return intervals in financial markets. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[18]  R. Bartsch,et al.  Statistical analysis of fluctuations in the ECG morphology , 2005 .

[19]  M. Ginovyan,et al.  Prediction Error for Continuous-Time Stationary Processes with Singular Spectral Densities , 2010 .

[20]  Luciano Telesca,et al.  Investigating the temporal variation of the scaling behavior in rainfall data measured in central Argentina by means of detrended fluctuation analysis , 2012 .

[21]  Fotios Siokis.  Multifractal analysis of stock exchange crashes , 2013 .

[22]  C. Peng,et al.  Mosaic organization of DNA nucleotides. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  V. Keilis-Borok,et al.  Variations of trends of indicators describing complex systems: change of scaling precursory to extreme events. , 2010, Chaos.

[24]  Luciano Telesca,et al.  Analysis of seismic sequences by using the method of visibility graph , 2012 .

[25]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1971 .

[26]  Tian Qiu,et al.  Scaling and memory effect in volatility return interval of the Chinese stock market , 2008, 0805.2194.

[27]  Woo-Sung Jung,et al.  Volatility return intervals analysis of the Japanese market , 2007, 0709.1725.

[28]  L. Telesca,et al.  On the scaling behavior of rain event sequence recorded in Basilicata region (Southern Italy) , 2004 .

[29]  Shlomo Havlin,et al.  Long-term persistence and multifractality of river runoff records: Detrended fluctuation studies , 2003 .

[30]  Jae Woo Lee,et al.  Power law of quiet time distribution in the Korean stock-market , 2007 .

[31]  Armin Bunde,et al.  Effect of nonlinear correlations on the statistics of return intervals in multifractal data sets. , 2007, Physical review letters.

[32]  C. Granger.  Long memory relationships and the aggregation of dynamic models , 1980 .

[33]  Shlomo Havlin,et al.  Return intervals of rare events in records with long-term persistence , 2004 .

[34]  John B. Rundle,et al.  Simulation-Based Distributions of Earthquake Recurrence Times on the San Andreas Fault System , 2006 .

[35]  Tamás Bódai,et al.  Annual variability in a conceptual climate model: snapshot attractors, hysteresis in extreme events, and climate sensitivity. , 2012, Chaos.

[36]  Holger Kantz,et al.  Return interval distribution of extreme events and long-term memory. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  H. Stanley,et al.  Cross-correlations between volume change and price change , 2009, Proceedings of the National Academy of Sciences.

[38]  S. Havlin,et al.  Detecting long-range correlations with detrended fluctuation analysis , 2001, cond-mat/0102214.

[39]  Shlomo Havlin,et al.  Long-term memory: a natural mechanism for the clustering of extreme events and anomalous residual times in climate records. , 2005, Physical review letters.

[40]  Pengjian Shang,et al.  Scaling analysis of stock markets. , 2014, Chaos.

[41]  L. Telesca,et al.  Long-range correlations in two-dimensional spatio-temporal seismic fluctuations , 2007 .

[42]  Ivo Grosse,et al.  Fractionally integrated process with power-law correlations in variables and magnitudes. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Adilson E. Motter,et al.  Beyond Word Frequency: Bursts, Lulls, and Scaling in the Temporal Distributions of Words , 2009, PloS one.

[44]  L. Telesca,et al.  Detrended fluctuation analysis of the spatial variability of the temporal distribution of Southern California seismicity , 2004 .

[45]  Wei‐Xing Zhou,et al.  Scaling and memory in the return intervals of realized volatility , 2009, 0904.1107.

[46]  Luciano Telesca,et al.  Analysis of the temporal properties in car accident time series , 2008 .

[47]  H. Stanley,et al.  Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. , 1995, Chaos.

[48]  Shlomo Havlin,et al.  The effect of long-term correlations on the return periods of rare events , 2003 .