Ergodic theorem for nonstationary random walks on compact abelian groups

Monakov, Grigorii

doi:10.1090/proc/16848

Ergodic theorem for nonstationary random walks on compact abelian groups
HTML articles powered by AMS MathViewer

by Grigorii Monakov;

Proc. Amer. Math. Soc. 152 (2024), 3855-3866

DOI: https://doi.org/10.1090/proc/16848

Published electronically: July 1, 2024

HTML | PDF | Request permission

Abstract:

We consider a nonstationary random walk on a compact metrizable abelian group. Under a classical strict aperiodicity assumption we establish a weak-* convergence to the Haar measure, Ergodic Theorem and Large Deviation Type Estimate.

References

Alejandro D. de Acosta, Large deviations for Markov chains, Cambridge Tracts in Mathematics, vol. 229, Cambridge University Press, Cambridge, 2022. MR 4586459, DOI 10.1017/9781009053129
M. Anoussis and D. Gatzouras, A spectral radius formula for the Fourier transform on compact groups and applications to random walks, Adv. Math. 188 (2004), no. 2, 425–443. MR 2087233, DOI 10.1016/j.aim.2003.11.001
Yves Benoist and Jean-François Quint, Random walks on reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 62, Springer, Cham, 2016. MR 3560700, DOI 10.1007/978-3-319-47721-3
Daniel Berend and Vitaly Bergelson, Ergodic and mixing sequences of transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 353–366. MR 776873, DOI 10.1017/S0143385700002509
Arno Berger and Steven N. Evans, A limit theorem for occupation measures of Lévy processes in compact groups, Stoch. Dyn. 13 (2013), no. 1, 1250008, 16. MR 3007246, DOI 10.1142/S0219493712500086
R. N. Bhattacharya, Speed of convergence of the $n$-fold convolution of a probability measure on a compact group, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 1–10. MR 326795, DOI 10.1007/BF00533331
Garrett Birkhoff, A note on topological groups, Compositio Math. 3 (1936), 427–430. MR 1556955
Bence Borda, Equidistribution of random walks on compact groups, Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021), no. 1, 54–72 (English, with English and French summaries). MR 4255167, DOI 10.1214/20-aihp1070
Bence Borda, Equidistribution of random walks on compact groups II. The Wasserstein metric, Bernoulli 27 (2021), no. 4, 2598–2623. MR 4303897, DOI 10.3150/21-BEJ1324
Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674, DOI 10.1007/978-1-4684-9172-2
Ao Cai, Pedro Duarte, and Silvius Klein, Mixed random-quasiperiodic cocycles, Bull. Braz. Math. Soc. (N.S.) 53 (2022), no. 4, 1469–1497. MR 4502841, DOI 10.1007/s00574-022-00313-9
Ao Cai, Pedro Duarte, and Silvius Klein, Furstenberg theory of mixed random-quasiperiodic cocycles, Comm. Math. Phys. 402 (2023), no. 1, 447–487. MR 4616680, DOI 10.1007/s00220-023-04726-5
A. Cai, P. Duarte, and S. Klein, Statistical properties for mixing Markov chains with applications to dynamical systems, arXiv:2210.16908, 2022.
Jean-Pierre Conze and Albert Raugi, Limit theorems for sequential expanding dynamical systems on $[0,1]$, Ergodic theory and related fields, Contemp. Math., vol. 430, Amer. Math. Soc., Providence, RI, 2007, pp. 89–121. MR 2331327, DOI 10.1090/conm/430/08253
Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. MR 2571413, DOI 10.1007/978-3-642-03311-7
Zach Dietz and Sunder Sethuraman, Large deviations for a class of nonhomogeneous Markov chains, Ann. Appl. Probab. 15 (2005), no. 1A, 421–486. MR 2115048, DOI 10.1214/105051604000000990
Dmitry Dolgopyat and Bassam Fayad, Limit theorems for toral translations, Hyperbolic dynamics, fluctuations and large deviations, Proc. Sympos. Pure Math., vol. 89, Amer. Math. Soc., Providence, RI, 2015, pp. 227–277. MR 3309100, DOI 10.1090/pspum/089/01492
Dmitry Dolgopyat and Omri M. Sarig, Local limit theorems for inhomogeneous Markov chains, Lecture Notes in Mathematics, vol. 2331, Springer, Cham, [2023] ©2023. MR 4647462, DOI 10.1007/978-3-031-32601-1
Randal Douc, Eric Moulines, Pierre Priouret, and Philippe Soulier, Markov chains, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2018. MR 3889011, DOI 10.1007/978-3-319-97704-1
Pedro Duarte and Silvius Klein, Lyapunov exponents of linear cocycles, Atlantis Studies in Dynamical Systems, vol. 3, Atlantis Press, Paris, 2016. Continuity via large deviations. MR 3468528, DOI 10.2991/978-94-6239-124-6
Alex Furman, Random walks on groups and random transformations, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 931–1014. MR 1928529, DOI 10.1016/S1874-575X(02)80014-5
Basilis Gidas, Nonstationary Markov chains and convergence of the annealing algorithm, J. Statist. Phys. 39 (1985), no. 1-2, 73–131. MR 798248, DOI 10.1007/BF01007975
PawełGóra, Abraham Boyarsky, and Christopher Keefe, Absolutely continuous invariant measures for non-autonomous dynamical systems, J. Math. Anal. Appl. 470 (2019), no. 1, 159–168. MR 3865128, DOI 10.1016/j.jmaa.2018.09.060
Anton Gorodetski and Victor Kleptsyn, Non-stationary version of ergodic theorem for random dynamical systems, Mosc. Math. J. 23 (2023), no. 4, 515–532. MR 4676292, DOI 10.17323/1609-4514-2023-23-4-515-532
Donald Hackler, Compact groups of real power need not be metrizable, Proc. Amer. Math. Soc. 63 (1977), no. 1, 187. MR 442896, DOI 10.1090/S0002-9939-1977-0442896-0
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, 1950. MR 33869, DOI 10.1007/978-1-4684-9440-2
Shizuo Kakutani, Über die Metrisation der topologischen Gruppen, Proc. Imp. Acad. Tokyo 12 (1936), no. 4, 82–84 (German). MR 1568424
Yukiyosi Kawada, On the probability distribution on a compact group. II, Proc. Phys.-Math. Soc. Japan (3) 23 (1941), 669–686. MR 20086
Yukiyosi Kawada and Kiyosi Itô, On the probability distribution on a compact group. I, Proc. Phys.-Math. Soc. Japan (3) 22 (1940), 977–998. MR 3462
B. Kloss, \ifTeXMLÐ Ð²ÐµÑÐ¾ÑÑÐ½Ð¾ÑÑÐ½ÑÑ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸ÑÑ Ð½Ð° Ð±Ð¸ÐºÐ¾Ð¼Ð¿Ð°ÐºÑÐ½ÑÑ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ Ð³ÑÑÐ¿Ð¿Ð°Ñ \else\textcyr{O veroyatnostnykh raspredeleniyakh na bikompaktnykh topologicheskikh gruppakh} (Russian, English summary), Teor. Veroyatn. Primen. 4 (1959), 255–290 (Title translation), Probability distributions on bicompact topological groups.
Paul Lévy, L’addition des variables aléatoires définies sur un circonférence, Bull. Soc. Math. France 67 (1939), 1–41 (French). MR 372, DOI 10.24033/bsmf.1288
Guo Xin Liu and Wen Liu, On the strong law of large numbers for functionals of countable nonhomogeneous Markov chains, Stochastic Process. Appl. 50 (1994), no. 2, 375–391. MR 1273781, DOI 10.1016/0304-4149(94)90129-5
Wen Liu and Guo Xin Liu, A class of strong laws for functionals of countable nonhomogeneous Markov chains, Statist. Probab. Lett. 22 (1995), no. 2, 87–96. MR 1327733, DOI 10.1016/0167-7152(94)00053-B
Liu Wen and Yang Weiguo, An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains, Stochastic Process. Appl. 61 (1996), no. 1, 129–145. MR 1378852, DOI 10.1016/0304-4149(95)00068-2
A. Prékopa, A. Rényi, and K. Urbanik, \ifTeXMLÐ Ð¿ÑÐµÐ´ÐµÐ»ÑÐ½Ð¾Ð¼ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ð¸ Ð´Ð»Ñ ÑÑÐ¼Ð¼ Ð½ÐµÐ·Ð°Ð²Ð¸ÑÐ¸Ð¼ÑÑ ÑÐ»ÑÑÐ°Ð¹Ð½ÑÑ Ð²ÐµÐ»Ð¸ÑÐ¸Ð½ Ð½Ð° Ð±Ð¸ÐºÐ¾Ð¼Ð¿Ð°ÐºÑÐ½ÑÑ ÐºÐ¾Ð¼Ð¼ÑÑÐ°ÑÐ¸Ð²Ð½ÑÑ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ Ð³ÑÑÐ¿Ð¿Ð°Ñ \else\textcyr{O predel′nom raspedelenii dlya summ nezavisimykh slucha\char"1A nykh velichin na bikompaktnykh kommutativnykh topologicheskikh gruppakh} (Russian, English summary), Acta Math. Acad. Sci. Hungar. 7 (1956), 11–16 (Title translation), On the limiting distribution of sums of independent random variables in bicompact commutative topological groups.
Kenneth A. Ross and Daming Xu, Norm convergence of random walks on compact hypergroups, Math. Z. 214 (1993), no. 3, 415–423. MR 1245203, DOI 10.1007/BF02572414
Karl Stromberg, Probabilities on a compact group, Trans. Amer. Math. Soc. 94 (1960), 295–309. MR 114874, DOI 10.1090/S0002-9947-1960-0114874-4
K. Urbanik, On the limiting probability distribution on a compact topological group, Fund. Math. 44 (1957), 253–261. MR 92921, DOI 10.4064/fm-44-3-253-261
S. R. S. Varadhan, Large deviations and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 46, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. MR 758258, DOI 10.1137/1.9781611970241.bm
Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
G. Vitali, Sui gruppi di punti e sulle funzioni di variabili reali (Italian), Atti Accad. Sci. Torino 43 (1908), 75–92 (Title translation), On groups of points and functions of real variables.
Weiguo Yang, Convergence in the Cesàro sense and strong law of large numbers for nonhomogeneous Markov chains, Linear Algebra Appl. 354 (2002), 275–288. Ninth special issue on linear algebra and statistics. MR 1927662, DOI 10.1016/S0024-3795(02)00298-7
Weiguo Yang, Strong law of large numbers for countable nonhomogeneous Markov chains, Linear Algebra Appl. 430 (2009), no. 11-12, 3008–3018. MR 2517854, DOI 10.1016/j.laa.2009.01.016
Bei Wang, Zhiyan Shi, and Weiguo Yang, The strong law of large numbers for moving average of continuous state nonhomogeneous Markov chains, Stochastics 92 (2020), no. 5, 732–745. MR 4137062, DOI 10.1080/17442508.2019.1652608

AMS Website Down

Proceedings of the American Mathematical Society

Ergodic theorem for nonstationary random walks on compact abelian groups
HTML articles powered by AMS MathViewer

Abstract: