Convergence of random walks on the circle generated by an irrational rotation

Transactions of the American Mathematical Society

Volumn 350, Issue 9, 1998, Pages 3717-3741

  Francis Edward, S U ^a  

^a NONE

Author keywords

Continued fractions; Discrepancy; Erd s Turan inequality; Irrational rotation; Random walk; Rate of convergence; Uniform distribution of sequences

Indexed keywords

EID: 22044436777     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-98-02152-7     Document Type: Article

References (21)

1. D. Asimov, The grand tour, SIAM Jour. Sei. Statist. Comp. 6(1983), 128-143. MR 8Gh:62087
2. P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes, Vol. 11, Hayward, CA, 1988. MR 90a:60001
3. P. Diaconis and L. Saloff-Coste, Comparison Techniques for Random Walks on Finite Groups, Ann. Probab. 21(1993), 2131-2156. MR 95a:60009
4. P. Erdös and P. Turân, On a problem in the theory of uniform distribution I, Indag. Math. 10(1948), 370-378. MR 10:372c
5. B.M. KIoss, Limiting distributions on bicompact topological groups, Th. Probab. Appl 4(1959), 237-270
6. A.Ya. Khinchin, Continued Fractions, Univ. of Chicago Press, 1964. MR 28:5037
7. L. Kuipers and H. Neiderreiter, Uniform Distribution of Sequences, Wiley, New York 1974. MR 54:7415
8. S. Lang, Introduction to Diophantine Approximations, Springer-Verlag, 1995. M 96h:11067
9. W.J. LeVeque, An inequality connected with Weyl's criterion for uniform distribution Proc. Symp. Pure Math. 8(1965), 22-30. MR 31:3401
10. A. Lubotsky, R. Phillips, and P. Sarnak, Hecke Operators and Distributing Points on th Sphere, I, Comm. Pure. Appl. Math. 39(1986), S149-S186. MR 88m:11025a
11. J. Marsden, Elementary Classical Analysis, W.H. Freeman and Co., 1974. MR 50:10161
12. Alathematica, version 2. Wolfram Research, Inc., 1991.
13. P. Matthews, Covering problems for random walks on spheres and finite groups, Ph.D Thesis, Dept. of Statistics, Stanford Univ., 1985.
14. H. Niederreiter and W. Philipp, Berry-Esseen bounds and a theorem of Erdös and Turd on uniform distribution mod 1, Duke Math. J. 40(1973), 633-649. MR 49:2642
15. U. Porod, The cut-off phenomenon for random reflections, Ann. Probab. 24(1996), 74-96 MR 97e:60012
16. C. Radin, The pinwheel tilings of the plane, Annals of Math. 139(1994), 661-702. M 95d:52021
17. E. Rains, personal communication, 1995.
18. J.S. Rosenthal, Random rotations: characters and random walks on SO(n), Ann. Probab 22(1994), 398-423. MR 95c:60008
19. W.M. Schmidt, Diophantine Approximation. Lecture Notes in Math. No. 785, SpringerVerlag, 1980. MR 81j:10038
20. N.J.A. Sloane, Encrypting by Random Rotations, in Cryptography, Lecture Notes in Computer Science, no. 149. T.Beth, editor. Berlin, 1983. MR 85i:94017
21. F.B. Su, Methods for Quantifying Rates of Convergence for Random Walks on Groups Ph.D. Thesis, Harvard University, 1995.