Abstract
Subadditivity of a sequence of positive real numbers x 1, … refers to the property x m+n ≤ x m + x n, n ≥ 1. For such sequences, it is a calculus exercise to verify that \(\lim _{n\to \infty }{x_n\over n} = \inf _{m\ge 1}{x_m\over m}\). The extension of this notion to almost sure convergence of a corresponding class of stochastic processes is the objective of this chapter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Fekete (1923).
- 2.
Kingman (1976) provided the initial breakthrough in exploiting subadditivity for an ergodic theory of stationary processes. Liggett (1985) provided the strengthening given here and finds applications for which the hypothesis of Kingman is too strong. The original version of Kingman contains assumption (d), but he required the conditions that Z m,k + Z k,n ≥ Z m,n, m = 2, …, n − 1, and that the distribution of {Z m+k,n+k : m = 0, 1, …, n − 1} be independent of k. These prove to be too strong for some applications.
- 3.
- 4.
BCPT, p.17.
- 5.
BCPT, pp. 142–145.
- 6.
- 7.
See Dascaliuc et al. (2022a) for related calculations.
- 8.
Furstenberg and Kesten (1960).
- 9.
This purely mathematical result has important consequences in physics where it is used to quantify important notions of disorder and localization. Comtet et al. (2013) provide a readable review from this perspective.
- 10.
- 11.
See Key (1987) and the references therein for examples and illustrative applications of the maximal Lyapunov exponent.
References
Auffinger A, Damron M, Hanson J (2017) 50 years of first-passage percolation, vol 68. Am Math Soc
Bhattacharya R, Waymire E (2021) Random walk, Brownian motion, and martingales. Graduate text in mathematics. Springer, New York
Comtet A, Texier C, Touriguy Y (2013) Lyapunov exponents, one-dimensional Anderson localization and products of random matrices. J Phys A: Math Theor 46:254003
Dascaliuc R, Pham T, Thomann E, Waymire E (2022a) Doubly stochastic yule cascades (part I): the explosion problem in the time-reversible case. J Funct Anal. in press
Durrett R (1991) Probability theory and examples, 2nd edn. Wadsworth, Brooks & Cole, Pacific, Grove
Fekete M (1923) Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematische Zeitschrift 17(1):228–249
Furstenberg H, Kesten H (1960) Products of random matrices. Ann Math Statist 31:451–469
Hammersley JM, Welsh DJA (1965) First passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In: Bernoulli–Bayes–Laplace Anniversary Volume. Springer, Berlin, pp 61–110
Key E (1987) Computable examples of the maximal Lyapunov exponent. Probab Theory Rel. Fields 75(1):97–107
Kingman JFC (1976) Subadditive ergodic theory. Ann Probab 883–909
Liggett TM (1985) Interacting particle systems. Springer, New York
Liggett TM (1985) An improved subadditive ergodic theorem. Ann Probab 13(5):1279–1285
Kallenberg O (2002) Foundations of modern probability, 2nd edn. Springer, NY
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bhattacharya, R., Waymire, E. (2022). Subadditive Ergodic Theory. In: Stationary Processes and Discrete Parameter Markov Processes. Graduate Texts in Mathematics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-031-00943-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-031-00943-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-00941-9
Online ISBN: 978-3-031-00943-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)