Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T01:51:37.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Group representations and applied probability

Published online by Cambridge University Press:  14 July 2016

E. J. Hannan
E. J. Hannan*
Affiliation:
Australian National University
Get access Rights & Permissions [Opens in a new window]

Extract

This account is meant to serve as an introduction to certain mathematical ideas which can be of considerable importance in applied probability and in statistical theory. Since a wide field is to be covered in a short space, the account has to depend on heuristic arguments and a sympathetic understanding from those familiar with the technical details of the subject is called for. Constant reference to the sources of the underlying mathematics is not called for, as it will be clear that the author is not claiming these results as his own. The references at the end of the paper, which are arranged so as to aid those readers who wish to proceed further, will also indicate the sources of the mathematical and statistical ideas used. We assume some knowledge of the groups, though most concepts are defined below. By beginning from the finite dimensional case and motivating later concepts in terms of this we hope to make the later, rather technical sections, more accessible.

Type
Review Paper
Information
Journal of Applied Probability , Volume 2 , Issue 1 , June 1965 , pp. 1 - 68
Copyright
Copyright © Sheffield: Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, T. W. (1958) Introduction to Mutivariate Statistical Analysis. John Wiley, New York.Google Scholar
Boerner, H. (1963) Representations of Groups. North-Holland, Amsterdam.Google Scholar
Chevalley, D. (1946) Theory of Lie Groups. Princeton U. P. Google Scholar
Curtis, C. W. and Reiner, T. (1962) Representation Theory of Finite Groups and Associative Algebras. Interscience, New York.Google Scholar
Fisher, R.A. (1947) The theory of linkage in polysomic inheritance. Phil. Trans. Roy. Soc. London B 233, 55–87.Google Scholar
Gangolli, R. (1964) Isotropic infinitely divisible distributions on symmetric spaces. Acta Math. 111, 213–246.CrossRefGoogle Scholar
Gelfand, I. M. and Pyatecki-Sapiro, I. (1959) Theory of representations and theory of automorphic functions. Uspehi Mat. Nauk. 14, 171–194.Google Scholar
Gelfand, I. M. and Sapiro, Z. YA. (1952) Representations of the group of rotations in three dimensional space. Uspehi Mat. Nauk 7, (A.M.S. Translations (1956) 2, 207–316.).Google Scholar
Godement, R. (1957) Introduction aux travaux de A. Selberg. Séminaire Bourbaki 144, 1–16.Google Scholar
Godement, R. (1962) La formule des traces de Selberg. Séminaire Bourbaki 244, 1–10.Google Scholar
Grenander, U. (1963) Probabilities on Algebraic Structures. John Wiley, New York.Google Scholar
Hall, M. (1959) The Theory of Groups. Macmillan, New York.Google Scholar
Halmos, P. R. (1947) Finite Dimensional Vector Spaces. (2nd Edition) Princeton U.P. Google Scholar
Hannan, E. J. (1960) Time Series Analysis. Methuen & Co., London.Google Scholar
Hannan, E. J. (1961) Applications of the representation theory of groups and algebras to some statistical problems. Research Reports (Part 1), Summer Research Institute, Australian Mathematical Society.Google Scholar
Helgason, S. (1962) Differential Geometry and Symmetric Spaces. Academic Press, New York.Google Scholar
Helson, H. and Lowdenslager, D. (1958) Prediction theory and Fourier series in several variables. Acta. Math. 99. 167–202.CrossRefGoogle Scholar
Hunt, G. A. (1956) Semigroups of measures on Lie groups. Trans. Amer. Math. Soc. 81, 264–293.Google Scholar
James, A. T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25, 40–75.Google Scholar
James, A. T. (1957) The relationship algebra of an experimental design. Ann. Math. Statist. 28, 993–1002.Google Scholar
James, A. T. (1960) The distribution of the latent roots of the covariance matrix. Ann Math. Statist. 31, 151–148.Google Scholar
James, A. T. (1961) Zonal polynomials of the real positive definite symmetric matrices. Ann. Math. 74, 456–469.Google Scholar
James, A. T. (1964) Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35, 475–501.Google Scholar
James, A. T., and Papazian, H. P. (1961) Enumeration of quad types in diploids and tetra-ploids. Genetics 46, 817–829.Google Scholar
Kampé De Fériet, J. (1948) Analyse harmonique des fonctions aléatoires stationnaires d'ordre 2 définies sur un group abélien localement compact. C.P. Acad. Sci., Paris 226, 868–870.Google Scholar
Kawada, Y. and Ito, K. (1940) On the probability distribution on a compact group. Proc. Phys.-Mat. Soc. Japan 22,.Google Scholar
Kloss, B. M. (1959) Probability distributions on bicompact topological groups. Theor. Prob. Appl. 4, 237–270.Google Scholar
Kloss, B. M. (1961) Limiting distributions on bicompact Abelian groups. Theor. Prob. Appl. 6, 361–389.Google Scholar
Kloss, B.M. (1962) Stable distributions on locally bicompact groups. Theor. Prob. Appl. 7, 237–257.Google Scholar
Littlewood, D. E. (1950) The Theory of Group Characters. Oxford.Google Scholar
Loève, M. (1955) Probability Theory. Van Nostrand, New York.Google Scholar
Loomis, L.H. (1953) An Introduction to Abstract Harmonic Analysis. Van Nostrand, New York.Google Scholar
Mackey, G.W. (1963) Infinite dimensional group representations. Bull. Amer. Math. Soc. 69, 628–686.Google Scholar
Mclaren, A.D. (1963) On group representations and invariant stochastic processes. Proc. Camb. Phil. Soc. 59, 431–450.Google Scholar
Mann, H. A. (1960) The algebra of a linear hypothesis. Ann. Math. Statist. 31, 1–15.Google Scholar
Naimark, M. A. (1959) Normed Rings. Noordhoff, Groningen.Google Scholar
Naimark, M. A. (1964) Linear Representations of the Lorentz Group. Macmillan, New York.Google Scholar
Pontrjagin, L. (1939) Topological Groups. Princeton U.P. Google Scholar
Selberg, A. (1956) Harmonic analysis and discontinuous groups in weakly symmetric Riemannian manifolds with applications to Dirichlet series. J. Ind. Math. Soc. 20, 47–57.Google Scholar
Tamagawa, T. (1960) On Selberg's trace formula. J. Fac. Sci. Tokyo. Sect. I. 8, 363.Google Scholar
Wehn, O.F. (1962) Probabilities on Lie Groups. Proc. Nat. Acad. Sci., U.S.A. 48, 791–795.Google Scholar
Weil, A. (1951) L'intégration dans les Groupes Topologiques et ses Applications. Hermann, Paris.Google Scholar
Weyl, H. (1946) The Clasical Groups. Princeton U.P. Google Scholar
Yaglom, A. M. (1961) Second order homogeneous random fields. Fourth Berkeley Symposium on Mathematical Statistics and Probability. 2, 593–622. University of California Press, Berkeley.Google Scholar