Appendix D - Other Notes and Topics

Summary

A number of new books on long-range dependence and self-similarity have been published in the last ten years or so. On the statistics and modeling side, these include (in alphabetical order):

1. • Beran et al. [127], Cohen and Istas [253], Giraitis et al. [406], Leonenko [612], Palma [789].

On the probability side, these include:

1. • Berzin, Latour, and León [138], Biagni et al. [146], Embrechts and Maejima [350], Major [673], Mishura [722], Nourdin [760], Nourdin and Peccati [763], Nualart [769], Peccati and Taqqu [797], Rao [842], Samorodnitsky [879, 880], Tudor [963].

On the application side, these include:

1. • Dmowska and Saltzman [319] regarding applications in geophysics, Park and Willinger [794] and Sheluhin et al. [903] in connection to telecommunications, Robinson [854] and Teyssière and Kirman [958] in connection to economics and finance.

Collections of articles include:

1. • Doukhan, Oppenheim, and Taqqu [328], Rangarajan and Ding [841].

See also the following books of related interest:

1. • Houdré and Pérez-Abreu [488], Janson [529], Marinucci and Peccati [689], Meerschaert and Sikorskii [712], Terdik [954].

We have not covered in this monograph a number of other interesting topics related to long-range dependence and/or self-similarity, including:

1. • Long-range dependence for point processes: In a number of applications, data represent events recorded in time (e.g., spike trains in brain science, packet arrivals in computer networks, and so on). Such data can be modeled through point processes (e.g., Daley and Vere-Jones [282, 283]). Long-range dependence for point processes was defined and studied in Daley [281], Daley and Vesilo [284], Daley, Rolski, and Vesilo [285]. See also the monograph of Lowen and Teich [648].

2. • Other nonlinear time series with long-range dependence: A large portion of this monograph (in particular, Chapter 2) focuses on linear time series exhibiting long-range dependence, with the exception of the non-linear time series defined as functions of linear series in Chapter 5. A number of other nonlinear time series with long-range dependence features have been considered in the literature, especially in the context of financial time series, e.g., FIGARCH and related models (Ding, Granger, and Engle [316], Bail-lie, Bollerslev, and Mikkelsen [89], Andersen and Bollerslev [29], Andersen, Bollerslev, Diebold, and Ebens [30], Comte and Renault [257], Tayefi and Ramanathan [953], Deo et al. [302]). See also Shao and Wu [902], Wu and Shao [1013], Baillie and Kapetanios [87].

In the context of financial time series, long-range dependence has been associated to their volatility process.