Skip to main content

Applications of non-commutative fourier analysis to probability problems

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1362)

Keywords

  • Random Walk
  • Irreducible Representation
  • Finite Group
  • Symmetric Group
  • Simple Random Walk

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aldous, D. (1982). Markov chains with almost exponential hitting times. Stochastic Proc. Appl. 13, 305–310.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Aldous, D. (1983a). Random walk on finite groups and rapidly mixing Markov chains. In Seminaire de Probabilities XVII, 243–297. Lecture Notes in Mathematics 986.

    MathSciNet  MATH  Google Scholar 

  • Aldous, D. (1983b). Minimization algorithms and random walk on the d-cube. Ann. Prob. 11, 403–413.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Aldous, D. and Diaconis, P. (1986). Shuffling cards and stopping times. American Mathematical Monthly 93, 333–348.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Advances in Appl. Math. 8, 69–97.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Beran, R. (1979). Exponential models for directional data. Ann. Statist. 7, 1162–1178.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Bovey, J. D. (1980). An approximate probability distribution for the order of elements of the symmetric group. Bull. London Math. Soc. 12, 41–46.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Chung, F., Diaconis, P. and Graham, R. L. (1987). A random walk problem involving random number generation. To appear in Ann. Prob.

    Google Scholar 

  • Cohen, P. J. (1959). Factorization in group algebras. Duke Math. J. 26, 199–205.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Critchlow, D. (1985). Metric Methods for Analyzing Partially Ranked Data. Lecture Notes in Statistics No. 34. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Curtis, C. W. and Reiner, I. (1982). Representation of Finite Groups and Associative Algebra, 2nd edition. Wiley, New York.

    Google Scholar 

  • Diaconis, P. (1982). Lectures on the use of group representations in probability and statistics. Typed Lecture Notes, Department of Statistics, Harvard University.

    Google Scholar 

  • Diaconis, P. (1987). Group Representations in Probability and Statistics. Institute of Mathematical Statistics, Hayward.

    MATH  Google Scholar 

  • Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 57, 159–179.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Diaconis, P. and Shahshahani, M. (1986a). Products of random matrices as they arise in the study of random walks on groups. Contemporary Mathematics 50, 183–195.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Diaconis, P. and Shahshahani, M. (1986b). On square roots of the uniform distribution on compact groups. Proc. American Math'l Soc. 98, 341–348.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Diaconis, P. and Shahshahani, M. (1987a). The subgroup algorithm for generating uniform random variables. Prob. in Engineering and Info. Sciences 1, 15–32.

    CrossRef  MATH  Google Scholar 

  • Diaconis, P. and Shahshahani, M. (1987b). Time to reach stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math'l Analysis 18, 208–218.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Diaconis, P. and Stanley, R. (1986). Mathematical aspects of cooking potatoes. Unpublished manuscript.

    Google Scholar 

  • Dies, J. E. (1983). ChaÎnes de Markov sur les Permutations. Lecture Notes in Math. 1010. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Flatto, L., Odlyzko, A. M. and Wales, D. B. (1985). Random shuffles and group representations. Ann. Prob. 13, 154–178.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Furst, M., Hopcroft, J. and Luka, E. (1980). Polynomial time algorithms for permutation groups. Proc. 21st FOCS I, 36–41.

    Google Scholar 

  • Gleason, A. (1952). Groups without small subgroups. Amer. Math. 56, 193–212.

    MathSciNet  MATH  Google Scholar 

  • Greenhalgh, A. (1987). Ph.D. disseration, Department of Mathematics, Stanford University.

    Google Scholar 

  • Griffeath, D. (1975). A maximal coupling for Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 95–100.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Griffeath, D. (1978). Coupling methods for Markov chains. In G. C. Rota (ed.) STUDIES IN PROBABILITY AND ERGODIC THEORY, 1–43.

    Google Scholar 

  • Hall, M. (1959). The Theory of Groups. MacMillan, New York.

    MATH  Google Scholar 

  • Herstein, I. N. (1975). Topics in Algebra, 2nd edition. Wiley, New York.

    MATH  Google Scholar 

  • Hewitt, E. and Zukerman, H. (1966). Singular measures with absolutely continuous convolution squares. Proc. Camb. Phil. Soc. 62, 399–420.

    CrossRef  MathSciNet  Google Scholar 

  • Ingram, R. E. (1950). Some characters of the symmetric group. Proc. Amer. Math. Soc. 1, 358–369.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • James, G. D. (1978). The Representation Theory of the Symmetric Groups. Lecture Notes in Mathematics 682. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • James, G. and Kerber, A. (1981). The Representation Theory of the Symmetric Group. Addison-Wesley, Reading, Massachusetts.

    MATH  Google Scholar 

  • Knuth, D. (1981). The Art of Computer Programming. Vol. II, 2nd edition. Addison-Wesley, Menlo Park, California.

    MATH  Google Scholar 

  • Levy, P. (1953). Premiers Elements de l'Arithmetique des Substitutions Aleatoires. C.R. Acad. Sci. 237, 1488–1489.

    MathSciNet  MATH  Google Scholar 

  • Lewis, T. (1967). The factorization of the rectangular distribution. J. Appl. Prob. 4, 529–542.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Lukacs, E. (1970). Characteristic Functions, 2nd edition. Griffin, London.

    MATH  Google Scholar 

  • Mackey, G. (1978). UNITARY GROUP REPRESENTATIONS IN PHYSICS, PROBABILITY, AND NUMBER THEORY. Benjamin/Cummings.

    Google Scholar 

  • Mackey, G. (1980). Harmonic analysis as the exploitation of symmetry. Bull. Amer. Math. Soc. 3, 543–697.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York.

    MATH  Google Scholar 

  • Matthews, P. (1985). Covering problems for random walks on spheres and finite groups. Ph.D. dissertation, Department of Statistics, Stanford University.

    Google Scholar 

  • Matthews, P. (1987). Covering problems for Brownian motion on a sphere. To appear Ann. Prob.

    Google Scholar 

  • Naimark, M. and Stern, A. (1982). Theory of Group Representations. Springer-Verlag, New York.

    CrossRef  MATH  Google Scholar 

  • Pascaud, J. (1973). Anneaux de groups réduits. C.R. Acad. Sci. Paris, Sér. A 277, 719–722.

    MathSciNet  MATH  Google Scholar 

  • Pemantle, R. (1987). An analysis of overhand shuffles. To appear Ann. Probl.

    Google Scholar 

  • Pitman, J. W. (1976). On the coupling of Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 35, 315–322.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Pontrijagin, L. S. (1966). TOPOLOGICAL GROUPS. Gordon and Breach.

    Google Scholar 

  • Rotman, J. (1973). The Theory of Groups: An Introduction, 2nd edition. Allyn and Bacon, Boston.

    MATH  Google Scholar 

  • Rubin, H. (1967). Supports of convolutions of identical distributions. Proc. Fifth Berkeley Symp. on Mathematics, Statistics, and Probability, Vol. 2 415–422.

    MathSciNet  MATH  Google Scholar 

  • Sehgal, S. K. (1975). Nilpotent elements in group rings. Manuscripta Math. 15, 65–80.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Serre, J. P. (1977). Linear Representations of Finite Groups. Springer-Verlag, New York.

    CrossRef  MATH  Google Scholar 

  • Suzuki, M. (1982). Group Theory, I, II. Spring-Verlag, New York.

    CrossRef  Google Scholar 

  • Verducci, J. (1982). Discriminating between two probabilities on the basis of ranked preferences. Ph.D. dissertation, Department of Statistics, Stanford University.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1988 Springer-Verlag

About this paper

Cite this paper

Diaconis, P. (1988). Applications of non-commutative fourier analysis to probability problems. In: Hennequin, PL. (eds) École d'Été de Probabilités de Saint-Flour XV–XVII, 1985–87. Lecture Notes in Mathematics, vol 1362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086179

Download citation

  • DOI: https://doi.org/10.1007/BFb0086179

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50549-5

  • Online ISBN: 978-3-540-46042-8

  • eBook Packages: Springer Book Archive