The Living Thing / Notebooks :

Ergodic theory / mixing

TBD.

Relevance to actual stochastic processes and dynamical systems, especially linear. and nonlinear system identification, and long-memory systems.

Keywords to look up:

Not much material here, but please see learning theory for dependent data for some interesting categorisations of mixing and transcendence of stupid mixing conditions.

Mixing zoo

\(\beta\)-mixing

\(\phi\)-mixing

Sequential Rademacher complexity

Refs

AlBo05
Aly, E.-E. A. A., & Bouzar, N. (2005) Stationary solutions for integer-valued autoregressive processes. International Journal of Mathematics and Mathematical Sciences, 2005(1), 1–18. DOI.
BrMa01
Brémaud, P., & Massoulié, L. (2001) Hawkes branching point processes without ancestors. Journal of Applied Probability, 38(1), 122–135. DOI.
GaLe14
Gallo, S., & Leonardi, F. G.(2014) Nonparametric statistical inference for the context tree of a stationary ergodic process. arXiv:1411.7650 [Math, Stat].
GöKü96
Götze, F., & Künsch, H. R.(1996) Second-order correctness of the blockwise bootstrap for stationary observations. The Annals of Statistics, 24(5), 1914–1933. DOI.
Gray09
Gray, R. M.(2009) Probability, random processes, and ergodic properties. . Springer Verlag
IsFu01
Ishioka, S., & Fuchikami, N. (2001) Thermodynamics of computing: Entropy of nonergodic systems. Chaos, 11(3), 734–746. DOI.
KePe06
Keane, M., & Petersen, K. (2006) Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem. IMS Lecture Notes-Monograph Series Dynamics & Stochastics, 48. DOI.
Lair78
Laird, N. (1978) Nonparametric Maximum Likelihood Estimation of a Mixing Distribution. Journal of the American Statistical Association, 73(364), 805–811. DOI.
LiIS12
Livan, G., Inoue, J., & Scalas, E. (2012) On the non-stationarity of financial time series: impact on optimal portfolio selection. Journal of Statistical Mechanics: Theory and Experiment, 2012(7), P07025. DOI.
McSS11
McDonald, D. J., Shalizi, C. R., & Schervish, M. (2011) Risk bounds for time series without strong mixing. arXiv:1106.0730 [Cs, Stat].
MoYG96
Morvai, G., Yakowitz, S., & Györfi, L. (1996) Nonparametric inference for ergodic, stationary time series. The Annals of Statistics, 24(1), 370–379. DOI.
Palm82
Palmer, R. G.(1982) Broken ergodicity. Advances in Physics, 31(6), 669–735. DOI.
Rose84
Rosenblatt, M. (1984) Asymptotic Normality, Strong Mixing and Spectral Density Estimates. The Annals of Probability, 12(4), 1167–1180. DOI.
RyRy10
Ryabko, D., & Ryabko, B. (2010) Nonparametric Statistical Inference for Ergodic Processes. IEEE Transactions on Information Theory, 56(3), 1430–1435. DOI.
ShWu07
Shao, X., & Wu, W. B.(2007) Asymptotic spectral theory for nonlinear time series. The Annals of Statistics, 35(4), 1773–1801. DOI.
Shie98
Shields, P. C.(1998) The interactions between ergodic theory and information theory. Information Theory, IEEE Transactions on, 44(6), 2079–2093. DOI.
Stei97
Steif, J. E.(1997) Consistent estimation of joint distributions for sufficiently mixing random fields. The Annals of Statistics, 25(1), 293–304.
StNe95
Stein, D. L., & Newman, C. M.(1995) Broken ergodicity and the geometry of rugged landscapes. Physical Review E, 51(6), 5228–5238. DOI.
ThWe12
THOUVENOT, J.-P., & Weiss, B. (2012) Limit Laws for Ergodic Processes. Stochastics and Dynamics, 12(1), 1150012. DOI.
Whit54
Whittle, P. (1954) On stationary processes in the plane. Biometrika, 41(3/4), 434–449.