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Random walks on amalgams. (English) Zbl 0564.60069
Let \(\Gamma\) be a locally compact amalgam of compact groups. The action of \(\Gamma\) on a suitable tree is used to study all random walks on \(\Gamma\) which can be described as nearest neighbour random walks on the tree. In particular, local limit theorems are derived, i.e. the asymptotic behaviour of n-step transition probabilities is determined.

60G50 Sums of independent random variables; random walks
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A05 Measures on groups and semigroups, etc.
05C05 Trees
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[1] Akemann, Ch. A., Ostrand, Ph. A.: Computing norms in groupC *-algebras. Am. J. Math.98, 1015-1047 (1976). · Zbl 0342.22008 · doi:10.2307/2374039
[2] Alperin, R.: Locally compact groups acting on trees. Pacific J. Math.100, 23-32 (1982). · Zbl 0513.22003
[3] Avez, A.: Limite de quotients pour les marches aléatoires sur des groupes. C. R. Acad. Sci. Paris (Sér. A)276, 317-320 (1973). · Zbl 0273.60006
[4] Baldi, P., Bourgerol, P., Crepel, P.: Théorème central limite local sur les extensions compactes de ? d . Ann. Inst. H. Poincaré Sect. B14, 99-112 (1978). · Zbl 0382.60013
[5] Bender, E. A.: Asymptotic methods in enumeration. SIAM Review16, 485-515 (1974). · Zbl 0294.05002 · doi:10.1137/1016082
[6] Berg, Ch., Christensen, J. P. R.: Sur la norme des opérateurs de convolution. Invent. Math.23, 173-178 (1974). · Zbl 0267.22007 · doi:10.1007/BF01405169
[7] Bourgerol, P.: Théorème central limit local sur certains groupes de Lie. Ann. Sci. Ec. Norm. Sup.14, 403-432 (1981).
[8] Dixmier, J.: Les moyennes invariantes dans les sémigroupes et leurs applications. Acta Sci. Math. Szeged12 A, 213-227 (1950). · Zbl 0037.15501
[9] Gerl, P.: Irrfahrten aufF 2. Mh. Math.84, 29-35 (1977). · Zbl 0372.60096 · doi:10.1007/BF01637023
[10] Gerl, P., Woess, W.: Local limit theorems and harmonic functions for nonisotropic random walks on free groups. Z. Wahrscheinlichkeitsth. To appear. · Zbl 0562.60011
[11] Gnedenko, B. V., Kolmogorov, A. N.: Limit Distributions for Sums of Independent Random Variables. Reading, Mass.: Addison Wesley. 1954. · Zbl 0056.36001
[12] Guivarc’h, Y., Keane, M., andRoynette, B.: Marches Aléatoires sur les Groupes de Lie. Lect. Notes Math. 624. Berlin-Heidelberg-New York: Springer. 1977.
[13] Guivarc’h, Y.: Théorèmes quotients pour les marches aléatoires. Astérisque74, 15-28 (1980).
[14] Guivarc’h, Y.: Sur le loi des grands nombres et le rayon spectral d’une marche aléatoire. Astérisque74, 47-98 (1980).
[15] Ito, K., Kawada, Y.: On the probability distribution on a compact group. Proc. Phys. Math. Soc. Japan22, 977-999 (1940). · Zbl 0026.13801
[16] Picardello, M. A.: Spherical functions and local limit theorems on free groups. Ann. Mat. Pur. Appl. (IV)33, 177-191 (1983). · Zbl 0527.60011 · doi:10.1007/BF01766017
[17] Sawyer, S.: Isotropic random walks in a tree. Z. Wahrscheinlichkeitsth.42, 279-292 (1978). · Zbl 0362.60075 · doi:10.1007/BF00533464
[18] Serre, J. P.: Arbres, Amalgames, SL2. Astérisque46, 1977.
[19] Szegö, G.: Orthogonal Polynomials. Providence, R. I.: Amer. Math. Soc. 1939. · JFM 65.0278.03
[20] Woess, W.: Aperiodische Wahrscheinlichkeitsmaße auf topologischen Gruppen. Mh. Math.90, 339-345 (1980). · Zbl 0435.60013 · doi:10.1007/BF01540853
[21] Woess, W.: A random walk on free products of finite groups. In: Probability Measures on Groups. Lect. Notes Math. 1064. Berlin-Heidelberg-New York: Springer. 1984. · Zbl 0542.60073
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