Abstract
Long-range dependent (LRD) phenomena were first exhibited by Hurst for hydrology purposes. This phenomenon occurs from the superposition of independent sources, e.g. confluent rivers provide this behaviour (see Fig. 4.2). Such aggregation procedures provide this new phenomenon. Hurst (Trans Am Soc Civ Eng 116:770–799, 1951) originally determined the optimum dam sizing for the Nile river’s volatile rain and drought conditions observed over a long period of time. LRD is characterized by slow decorrelation properties and the behaviour of partial sums’s variances. This phenomenon is discussed above, see Sects. 4.3 and 4.4. Asymptotic properties of instantaneous functions of Gaussian processes are provided in Remark 5.2.4. Infinite moving averages models with LRD properties are provided in Sects. 6.4 and 6.6. We refer the reader to Doukhan et al. (Theory and applications of long-range dependence. Birkhaüser, Boston, 2002b) for much more.
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Notes
- 1.
Note that the same relation \(a_n=o(A_n)\) does not hold when \(a_k\) admits a geometric decay rate since in this case tails \(A_n\) and the first term \(a_n\) of a series admit the same order of magnitude.
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Doukhan, P. (2018). Long-Range Dependence. In: Stochastic Models for Time Series. Mathématiques et Applications, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-76938-7_10
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DOI: https://doi.org/10.1007/978-3-319-76938-7_10
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