Stochastic processes where we know that ancient history is still relevant for the future predictions, even if we know the recent history; how do we analyse such things?
I haven't said anything about the types of generating process here. Maybe there is an explicit long-time dependency in your process which is not mediated through a hidden state; can we statistically distinguish these cases? Sometimes, e.g. [#Küns86], but generally you probably want a model, or how will you do it?
There are some obvious process model which have a long-memory in this sense, such as stack automata, or Hawkes processes with non-exponential kernels.
Note that “long memory” can work with not only time series but any random field, e.g. spatial, or random fields indexed by any number of dimensions, over whatever topology, causal or non-causal. I guess you'd need a notion of distance to make this meaningful, so let's presume we are on a metric space at least.
For now, see learning theory for dependent data, which has some concrete results for predictive time series. Also see ergodic theory, for one perspective on the difficulty of sampling dependent series.
- GrJo80: C. W. J. Granger, Roselyne Joyeux (1980) An Introduction to Long-Memory Time Series Models and Fractional Differencing. Journal of Time Series Analysis, 1(1), 15–29. DOI
- GiSu99: L Giraitis, 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.
- Küns86: Hans Rudolf Künsch (1986) Discrimination between monotonic trends and long-range dependence. Journal of Applied Probability, 23(4), 1025–1030.
- SaSo10: A. I. Saichev, D. Sornette (2010) Generation-by-generation dissection of the response function in long memory epidemic processes. The European Physical Journal B, 75(3), 343–355. DOI
- Mcle98: A. Ian McLeod (1998) Hyperbolic decay time series. Journal of Time Series Analysis, 19(4), 473–483. DOI
- Bera10: Jan Beran (2010) Long-range dependence. Wiley Interdisciplinary Reviews: Computational Statistics, 2(1), 26–35. DOI
- PiTa17: Vladas Pipiras, Murad S. Taqqu (2017) Long-range dependence and self-similarity. Cambridge, United Kingdom ; New York, NY, USA: Cambridge University Press
- Hurv02: Clifford M. Hurvich (2002) Multistep forecasting of long memory series using fractional exponential models. International Journal of Forecasting, 18(2), 167–179. DOI
- BHKS06: István Berkes, Lajos Horváth, Piotr Kokoszka, Qi-Man Shao (2006) On discriminating between long-range dependence and changes in mean. The Annals of Statistics, 34(3), 1140–1165. DOI
- Lahi93: S N Lahiri (1993) On the moving block bootstrap under long range dependence. Statistics & Probability Letters, 18(5), 405–413. DOI
- CsMi99: S Csörgö, J Mielniczuk (1999) Random-design regression under long-range dependent errors. Bernoulli, 5(2), 209–224. DOI
- Bera92: Jan Beran (1992) Statistical Methods for Data with Long-Range Dependence. Statistical Science, 7(4), 404–416.
- Bera94: Jan Beran (1994) Statistics for Long-Memory Processes. CRC Press
- ScHu16: Francois G. Schmitt, Yongxiang Huang (2016) Stochastic analysis of scaling time series: from turbulence theory to applications. Cambridge: Cambridge University Press
- BeTe96: Jan Beran, Norma Terrin (1996) Testing for a change of the long-memory parameter. Biometrika, 83(3), 627–638. DOI
- DoOT03: P Doukhan, G Oppenheim, M S Taqqu (2003) Theory and applications of long-range dependence. Birkhauser