Long memory time series

Hurst exponents, non-stationarity etc. I used to do a lot of work in this area, but have not now for so long that I no longer claim any authority.


Many interesting things, but for now, note that many natural generic models of long-memory in time series turn out ot be fractal models, so note power laws, \(1/f\) noise, fractional brownian motion etc. Link to branching processes.


Beran, Jan. 1992. “Statistical Methods for Data with Long-Range Dependence.” Statistical Science 7 (4): 404–16.
———. 1994. Statistics for Long-Memory Processes. CRC Press.
———. 2010. Long-Range Dependence.” Wiley Interdisciplinary Reviews: Computational Statistics 2 (1): 26–35.
Beran, Jan, and Norma Terrin. 1996. Testing for a Change of the Long-Memory Parameter.” Biometrika 83 (3): 627–38.
Berkes, István, Lajos Horváth, Piotr Kokoszka, and Qi-Man Shao. 2006. On Discriminating Between Long-Range Dependence and Changes in Mean.” The Annals of Statistics 34 (3): 1140–65.
Brouste, Alexandre, Jacques Istas, and Sophie Lambert-Lacroix. 2016. Conditional Fractional Gaussian Fields with the Package FieldSim.” R JOURNAL 8 (1): 38–47.
Chen, Yanguang. 2011. Zipf’s Law, 1/f Noise, and Fractal Hierarchy.” Chaos, Solitons & Fractals, November, S0960077911001901.
Csörgö, S, and J Mielniczuk. 1999. Random-Design Regression Under Long-Range Dependent Errors.” Bernoulli 5 (2): 209–24.
Dieker, Ton. 2004. Simulation of Fractional Brownian Motion.” MSc Theses, University of Twente, Amsterdam, The Netherlands.
Doukhan, P, G Oppenheim, and M S Taqqu. 2003. Theory and Applications of Long-Range Dependence. Birkhauser.
Giraitis, L, and D Surgailis. 1999. “Central Limit Theorem for the Empirical Process of a Linear Sequence with Long Memory.” Journal of Statistical Planning and Inference 80 (1-2): 81–93.
Granger, C. W. J., and Roselyne Joyeux. 1980. An Introduction to Long-Memory Time Series Models and Fractional Differencing.” Journal of Time Series Analysis 1 (1): 15–29.
Greaves-Tunnell, Alexander, and Zaid Harchaoui. 2019. A Statistical Investigation of Long Memory in Language and Music.” arXiv:1904.03834 [Cs, Eess, Stat], June.
Hurvich, Clifford M. 2002. Multistep Forecasting of Long Memory Series Using Fractional Exponential Models.” International Journal of Forecasting, Forecasting Long Memory Processes, 18 (2): 167–79.
Künsch, Hans Rudolf. 1986. “Discrimination Between Monotonic Trends and Long-Range Dependence.” Journal of Applied Probability 23 (4): 1025–30.
Lahiri, S N. 1993. On the Moving Block Bootstrap Under Long Range Dependence.” Statistics & Probability Letters 18 (5): 405–13.
Mandelbrot, Benoit B. 1999. Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976). New York: Springer-Verlag.
McCauley, Joseph L. 2008. Time Vs. Ensemble Averages for Nonstationary Time Series.” Physica A: Statistical Mechanics and Its Applications 387 (22): 5518–22.
McLeod, A. Ian. 1998. Hyperbolic Decay Time Series.” Journal of Time Series Analysis 19 (4): 473–83.
Papavasiliou, Anastasia, and Kasia B. Taylor. 2016. Approximate Likelihood Construction for Rough Differential Equations.” arXiv:1612.02536 [Math, Stat], December.
Pipiras, Vladas, and Murad S. Taqqu. 2017. Long-Range Dependence and Self-Similarity. Cambridge Series in Statistical and Probabilistic Mathematics 45. Cambridge, United Kingdom ; New York, NY, USA: Cambridge University Press.
Saichev, A. I., and D. Sornette. 2010. Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes.” The European Physical Journal B 75 (3): 343–55.
Schmitt, Francois G., and Yongxiang Huang. 2016. Stochastic Analysis of Scaling Time Series: From Turbulence Theory to Applications. Cambridge: Cambridge University Press.

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