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The rate of escape for anisotropic random walks in a tree
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  • Published: October 1987

The rate of escape for anisotropic random walks in a tree

  • Stanley Sawyer1 &
  • Tim Steger2 

Probability Theory and Related Fields volume 76, pages 207–230 (1987)Cite this article

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  • 39 Citations

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Summary

Let G be the group generated by L free involutions, whose Cayley graph T is the infinite homogeneous tree with L edges at every node. A general central limit theorem and law of the iterated logarithm is proven for left-invariant random walks Z n on G or T which applies to the distance of Z n from a fixed point, as well as to the distribution of the last R letters in Z n . For nearest neighbor random walks, we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point. A generalization for Z n with a finitely supported step distribution is derived and discussed.

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References

  • Aomoto, K.: Spectral theory on a free group an algebraic curves. J. Fac. Sci. Univ. Tokyo, Sect. IA 31, 297–317 (1984)

    Google Scholar 

  • Arzberger, P.: A probabilistic and algebraic treatment of regular inbreeding systems. J. Math. Biol. 22, 175–197 (1985)

    Google Scholar 

  • Cartier, P.: Harmonic analysis on trees. Proc. Symp. Pure Math. 26, 419–424 (1973)

    Google Scholar 

  • Cartwright D., Soardi, P.: Random walks on free products, quotients, and amalgams. Nagoya Math. J. 102, 163–180 (1986a)

    Google Scholar 

  • Cartwright, D., Soardi, P.: A local limit theorem for random walks on the Cartesian product of discrete groups Manuscript (1986b)

  • Derriennic, Y.: Marche aléatoire sur le groupe libre et frontière de Martin. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 261–276 (1975)

    Google Scholar 

  • Derriennic, Y.: Quelques applications du théorème ergodique sousadditif. Astérisque 74, 183–201 (1980)

    Google Scholar 

  • Derriennic, Y., Guivarc'h Y.: Théorème de renouvellement pour les groupes non moyennables. C.R. Acad. Sci. Paris 277A, 613–615 (1973)

    Google Scholar 

  • Dixmier, J.: Les moyennes invariantes dans les sémi-groupes et leurs applications. Acta Sci. Math. Szeged 12A, 213–227 (1950)

    Google Scholar 

  • Dynkin, E., Malyutov, M.: Random walks on groups with a finite number of generators. Sov. Math. 2, 399–402 (1961)

    Google Scholar 

  • Faraut, J., Picardello, M.: The Plancherel measure for symmetric graphs. Ann. Mat. Pura Appl. IV. Ser. 138, 151–155 (1984)

    Google Scholar 

  • Figà-Talamanca, A., Steger, T.: Harmonic analysis on trees. Symp. Math. (1986)

  • Figà-Talamanca, A., Picardello, M.: Harmonic analysis on free groups. Lect. Notes Pure Appl. Math. 87 (1983)

  • Furstenberg, H.: Random walks and discrete subgroups of Lie groups. Adv. Probab. Rel. Topics 1, 1–63 (1971)

    Google Scholar 

  • Gerl, P.: Irrfahrten auf F2. Monatsh. Math. 84, 29–35 (1977)

    Google Scholar 

  • Gerl, P., Woess, W.: Local limits and harmonic functions for nonisotropic random walks on free groups. Probab. Th. Rel. Fields 71, 341–355 (1986)

    Google Scholar 

  • Greenleaf, F.: Invariant means on topological groups and their applications. New York: Van Nostrand 1969

    Google Scholar 

  • Guivarc'h Y.: Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire. Astérisque 74, 47–98 (1980)

    Google Scholar 

  • Hall, P., Heyde, C.: Martingale limit theory and its applications. New York: Academic Press 1980

    Google Scholar 

  • Iozzi, A., Picardello, M.: Spherical functions on symmetric graphs. Lect Notes Math. 992, 344–386 (1983)

    Google Scholar 

  • Ishitani, J.: A central limit theorem for the subadditive process and its application to products of random matrices. Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 565–575 (1977)

    Google Scholar 

  • Kaimanovich, K., Vershik, A.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457–490 (1983)

    Google Scholar 

  • Kesten, H.: Full Banach mean values on countable groups. Math. Scand. 7 146–156 (1959)

    Google Scholar 

  • Kingman, J.F.C.: The ergodic theory of subadditive processes. J. R. Stat. Soc. Ser. B. 30, 499–510 (1968)

    Google Scholar 

  • Kingman, J.F.C.: Subadditive processes. Lect. Notes Math. 539 (1976)

  • Koranyi, A., Picardello, M., Taibleson, M.: Hardy spaces on non-homogeneous trees. Symp. Math., in press (1987)

  • Kuhn, G., Soardi, P.: The Plancherel measure for polygonal graphs. Ann. Mat. Pura Appl. 34, 393–401 (1983)

    Google Scholar 

  • Levit, B., Molchanov, S.: Invariant chains on free groups with a finite number of generators (in Russian). Vest. Moscow Univ. 6, 80–88 (1971)

    Google Scholar 

  • Picardello, M., Woess, W.: Martin boundaries of random walks: Ends of trees and groups. Manuscript (1986)

  • Sawyer, S.: Isotropic random walks on a tree. Z. Wahrscheinlichkeitstheor. Verw. Geb. 42, 279–292 (1978)

    Google Scholar 

  • Steger, T.: Anisotropic harmonic analysis for homogeneous trees., Ph.D. thesis, Washington University, St. Louis (1986)

  • Taibleson, M.: Hardy spaces of harmonic functions on homogeneous isotropic trees. Math. Nachr., in press (1987)

  • Widder, D.: The Laplace transform. Princeton: Princeton University Press 1946

    Google Scholar 

  • Woess, W.: Random walks and periodic continued fractions. Adv. Appl. Probab. 17, 67–84 (1985)

    Google Scholar 

  • Woess, W.: Nearest neighbor random walks on free products of discrete groups. Bollettino U.M.I. (6) 5-B, 962–982 (1986)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Washington University, 63130, St. Louis, MO, USA

    Stanley Sawyer

  2. Department of Mathematics, Yale University, 06520, New Haven, CT, USA

    Tim Steger

Authors
  1. Stanley Sawyer
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  2. Tim Steger
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Additional information

Partially supported by grant NSF MCS85-04315

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Sawyer, S., Steger, T. The rate of escape for anisotropic random walks in a tree. Probab. Th. Rel. Fields 76, 207–230 (1987). https://doi.org/10.1007/BF00319984

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  • Received: 18 January 1986

  • Revised: 01 April 1987

  • Issue Date: October 1987

  • DOI: https://doi.org/10.1007/BF00319984

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Keywords

  • Generate Function
  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Statistical Theory
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