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Random walks in Euclidean space. (English) Zbl 1315.60034
Let \(X_1, X_2,\dots\) be independent identically distributed random isometries of the Euclidean space \(R^d\). Let \(x_0\in R^d\) be any point, and consider the sequence of points \[ Y=x_0, \dots, Y_e(X_{l-1}(\dots (x_0))), \dots. \]
The author’s main objective is to obtain a local limit theorem in the following sense. Under some conditions on \(Y_0\) and \(X\) there are \(v_0\in \mathbb{R}^d\) and \(c>0\) depending only on the distribution of \(X_j\) such that \[ \lim_{l\to0}l^{d/2}E[f(Y_0-lv_0)]=c\int f(r)\,dx, \] where \(f\) is any continuous and compactly supported function.

MSC:
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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References:
[1] M. Ådahl, I. Melbourne, and M. Nicol, ”Random iteration of Euclidean isometries,” Nonlinearity, vol. 16, iss. 3, pp. 977-987, 2003. · Zbl 1037.37006
[2] V. I. Arnolcprimed and A. L. Krylov, ”Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain,” Dokl. Akad. Nauk SSSR, vol. 148, pp. 9-12, 1963. · Zbl 0237.34008
[3] P. Baldi, P. Bougerol, and P. Crépel, ”Théorème central limite local sur les extensions compactes de \({\mathbf R}^d\),” Ann. Inst. H. Poincaré Sect. B, vol. 14, iss. 1, pp. 99-111, 1978. · Zbl 0382.60013
[4] H. Bass, ”Theorems of Jordan and Burnside for algebraic groups,” J. Algebra, vol. 82, iss. 1, pp. 245-254, 1983. · Zbl 0506.20016
[5] J. Bourgain and A. Gamburd, ”On the spectral gap for finitely-generated subgroups of \( SU(2)\),” Invent. Math., vol. 171, iss. 1, pp. 83-121, 2008. · Zbl 1135.22010
[6] J. Bourgain and A. Gamburd, ”A spectral gap theorem in \({ SU}(d)\),” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 14, iss. 5, pp. 1455-1511, 2012. · Zbl 1254.43010
[7] E. Breuillard, Random walks on Lie groups. · Zbl 1083.60008
[8] D. L. Burkholder, ”Distribution function inequalities for martingales,” Ann. Probability, vol. 1, pp. 19-42, 1973. · Zbl 0301.60035
[9] J. -P. Conze and Y. Guivarc’h, ”Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts,” Discrete Contin. Dyn. Syst., vol. 33, iss. 9, pp. 4239-4269, 2013. · Zbl 1329.37004
[10] C. M. Dawson and M. A. Nielsen, ”The Solovay-Kitaev algorithm,” Quantum Inf. Comput., vol. 6, iss. 1, pp. 81-95, 2006. · Zbl 1152.81706
[11] D. Dolgopyat, ”On mixing properties of compact group extensions of hyperbolic systems,” Israel J. Math., vol. 130, pp. 157-205, 2002. · Zbl 1005.37005
[12] L. G. Gorostiza, ”The central limit theorem for random motions of \(d\)-dimensional Euclidean space,” Ann. Probability, vol. 1, pp. 603-612, 1973. · Zbl 0263.60010
[13] M. Gotô, ”A Theorem on compact semi-simple groups,” J. Math. Soc. Japan, vol. 1, pp. 270-272, 1949. · Zbl 0041.36208
[14] A. Grintsyavichyus, ”The domain of normal attraction of a stable law for the group of motions of a Euclidean space,” Litovsk. Mat. Sb., vol. 25, iss. 3, pp. 39-52, 1985.
[15] Y. Guivarc’h, ”Equirépartition dans les espaces homogènes,” in Théorie Ergodique, New York: Springer-Verlag, 1976, vol. 532, pp. 131-142. · Zbl 0368.28024
[16] D. A. Kavzdan, ”Uniform distribution on a plane,” Trudy Moskov. Mat. Ob\vs\vc., vol. 14, pp. 299-305, 1965. · Zbl 0225.26018
[17] . Y. S. Khokhlov, ”A local limit theorem for the composition of random motions of Euclidean space,” Dokl. Akad. Nauk SSSR, vol. 260, iss. 2, pp. 295-299, 1981. · Zbl 0494.60026
[18] . Y. S. Khokhlov, ”The domain of normal attraction of a semistable distribution on a semidirect product compact group and \(\mathbb R^d\),” J. Math. Sci., vol. 76, iss. 1, pp. 2147-2152, 1995.
[19] Y. A. Kitaev, ”Quantum computations: algorithms and error correction,” Uspekhi Mat. Nauk, vol. 52, iss. 6(318), pp. 53-112, 1997. · Zbl 0917.68063
[20] V. M. Maximov, ”Local theorems for Euclidean motions. I,” Z. Wahrsch. Verw. Gebiete, vol. 51, iss. 1, pp. 27-38, 1980. · Zbl 0427.60010
[21] S. T. Rachev and J. E. Yukich, ”Rates of convergence of \(\alpha\)-stable random motions,” J. Theoret. Probab., vol. 4, iss. 2, pp. 333-352, 1991. · Zbl 0724.60019
[22] B. Roynette, ”Théorème central-limite pour le groupe des déplacements de \({\mathbf R}^d\),” Ann. Inst. H. Poincaré Sect. B, vol. 10, pp. 391-398 (1975), 1974. · Zbl 0324.60026
[23] V. N. Tutubalin, ”The central limit theorem for random motions of Euclidean space,” Vestnik Moskov. Univ. Ser. I Mat. Meh., vol. 22, iss. 6, pp. 100-108, 1967.
[24] P. Pál. Varjú, ”Random walks in compact groups,” Doc. Math., vol. 18, pp. 1137-1175, 2013. · Zbl 1278.60011
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