×

Central limit theorems for Gromov hyperbolic groups. (English) Zbl 1217.60019

This paper proves a central limit theorem and a law of iterated logarithm for symmetric and nondegenerate random walks on transient hyperbolic groups, extending previous results for certain CAT\((-1)\)-groups. If \((X,d)\) is a Gromov hyperbolic space satisfying certain conditions, and \(\mu\) is a symmetric probability measure on \(\Gamma\), also satisfying certain conditions, then there is a positive constant \(\sigma\) such that
\[ {1 \over \sqrt{n}}\left(d(Z_n x_0,x_0)-nA(\mu)\right) \]
converges weakly to a nondegenerate Gaussian distribution and
\[ \limsup_{n \to \infty} {d(Z_nx_0,x_0)-nA(\mu)) \over \sqrt{n \log n \log n}}=\sigma>0 \text{ a.s. } \]

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ancona, A.: Positive Harmonic Functions and Hyperbolicity. Springer Lecture Notes, vol. 1344. Springer, Berlin (1987) · Zbl 0677.31006
[2] Ancona, A.: Théorie du Potentiel sur les Graphes et les Variétés. Springer Lecture Notes in Math., vol. 1427, pp. 4–112. Springer, Berlin (1990)
[3] Bear, H.S.: A geometric characterization of Gleason parts. Proc. Am. Math. Soc. 16, 407–412 (1965) · Zbl 0139.07402 · doi:10.1090/S0002-9939-1965-0181910-9
[4] Bellman, R.: Limit theorems for non-commutative operations. I. Duke Math. J. 21, 491–500 (1954) · Zbl 0057.11202 · doi:10.1215/S0012-7094-54-02148-1
[5] Blachère, S., Brofferio, S.: Internal diffusion limited aggregation on discrete groups having exponential growth. Probab. Theory Relat. Fields 137(3–4), 323–343 (2007) · Zbl 1106.60078 · doi:10.1007/s00440-006-0009-2
[6] Blachère, S., Haïssinsky, P., Mathieu, P.: Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab. 36(3), 1134–1152 (2008) · Zbl 1146.60008 · doi:10.1214/07-AOP356
[7] Blachère, S., Haïssinsky, P., Mathieu, P.: Harmonic measures versus quasiconformal measures for hyperbolic groups. Preprint · Zbl 1243.60005
[8] Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schrödinger Operators. Progress in Probability and Statistics, vol. 8. Birkhäuser Boston, Cambridge (1985). xii+283 pp., ISBN: 0-8176-3324-3 · Zbl 0572.60001
[9] Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999). xxii+643 pp., ISBN: 3-540-64324-9 · Zbl 0988.53001
[10] Dyubina, A.: An example of the rate of departure to infinity for a random walk on a group. Russ. Math. Surv. 54, 1023–1024 (1999) · Zbl 0964.60506 · doi:10.1070/RM1999v054n05ABEH000208
[11] Erschler, A.: Asymptotics of drift and entropy for a random walk on groups. Russ. Math. Surv. 56(3), 580–581 (2001) · Zbl 1026.60004 · doi:10.1070/RM2001v056n03ABEH000411
[12] Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31, 457–469 (1960) · Zbl 0137.35501 · doi:10.1214/aoms/1177705909
[13] Gilch, L.: Rate of escape of random walks. Ph.D. thesis, Graz (2007) · Zbl 1147.60030
[14] Gordin, M.I.: The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741 (1969) · Zbl 0212.50005
[15] Gordin, M.I., Holzmann, H.: The central limit theorem for stationary Markov chains under invariant splittings. Stoch. Dyn. 4(1), 15–30 (2004) · Zbl 1077.60023 · doi:10.1142/S0219493704000985
[16] Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. MSRI Publ., vol. 8, pp. 75–263. Springer, New York (1987)
[17] Guivarc’h, Y.: Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. Astérisque 74, 47–98 (1980)
[18] Guivarc’h, Y., Le Page, É.: Simplicité de spectres de Lyapunov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. In: Random Walks and Geometry, pp. 181–259. Walter de Gruyter GmbH & Co. KG, Berlin (2004)
[19] Hall, P., Heyde, C.C.: Martingale Limit Theory and its Applications. Academic Press, San Diego (1980) · Zbl 0462.60045
[20] Hennion, H., Hervé, L.: Central limit theorems for iterated random Lipschitz mappings. Ann. Probab. 32(3A), 1934–1984 (2004) · Zbl 1062.60017 · doi:10.1214/009117904000000469
[21] Kaimanovich, V.: The Poisson formula for groups with hyperbolic properties. Ann. Math. (2) 152(3), 659–692 (2000) · Zbl 0984.60088 · doi:10.2307/2661351
[22] Karlsson, A., Ledrappier, F.: On laws of large numbers for random walks. Ann. Probab. 34(5), 1693–1706 (2006) · Zbl 1111.60005 · doi:10.1214/009117906000000296
[23] Karlsson, A., Ledrappier, F.: Linear drift and Poisson boundary for random walks. Pure Appl. Math. Q. 3, 1027–1036 (2007) · Zbl 1142.60035
[24] Karlsson, A., Ledrappier, F.: Propriété de Liouville et vitesse de fuite du mouvement Brownien. C. R. Acad. Sci. Paris, Ser. I 344, 685–690 (2007) · Zbl 1122.60071
[25] Karlsson, A., Ledrappier, F.: Noncommutative ergodic theorems. Preprint · Zbl 1339.37005
[26] Kingman, J.F.C.: Subadditive ergodic theory. Ann. Probab. 1, 883–909 (1973) · Zbl 0311.60018 · doi:10.1214/aop/1176996798
[27] Ledrappier, F.: Some asymptotic properties of random walks on free groups. CRM Proc. Lect. Notes 28, 117–152 (2001) · Zbl 0994.60073
[28] Le Page, É.: Théorèmes de la limite centrale pour certains produits de matrices aléatoires. C. R. Acad. Sci. Paris Sér. I Math. 292(6), 379–382 (1981) · Zbl 0461.60023
[29] Liverani, C.: Decay of correlations. Ann. Math. (2) 142(2), 239–301 (1995) · Zbl 0871.58059 · doi:10.2307/2118636
[30] Nagnibeda, T., Woess, W.: Random walks on trees with finitely many cone types. J. Theor. Probab. 15(2), 383–422 (2002) · Zbl 1008.60061 · doi:10.1023/A:1014810827031
[31] Ol’shanskii, A.Yu.: Almost every group is hyperbolic. Int. J. Algebra Comput. 2(1), 1–17 (1992) · Zbl 0779.20016 · doi:10.1142/S0218196792000025
[32] Revelle, D.: Rate of escape of random walks on wreath products and related groups. Ann. Probab. 31(4), 1917–1934 (2003) · Zbl 1051.60047 · doi:10.1214/aop/1068646371
[33] Sawyer, S., Steger, T.: The rate of escape for anisoptopic random walks in a tree. Probab. Theory Relat. Fields 76(2), 207–230 (1987) · Zbl 0608.60064 · doi:10.1007/BF00319984
[34] Storm, P.A.: The barycenter method on singular spaces. Comment. Math. Helv. 82(1), 133–173 (2007) · Zbl 1118.53046 · doi:10.4171/CMH/87
[35] Varopoulos, N.T.: Théorie du potentiel sur les groupes et des varietés. C. R. Acad. Sci. Paris Sér. A–B 302, 203–205 (1986)
[36] Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge University Press, Cambridge (2000) · Zbl 0951.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.