Random Matrices

  • Göran Högnäs
  • Arunava Mukherjea
Part of the The University Series in Mathematics book series (USMA)

Abstract

In Chapter 4, we apply methods and results from Chapters 2 and 3 to random matrices. To include only results that are reasonably complete, we restrict our attention to the class of nonnegative matrices (that is, matrices whose entries are all nonnegative). Though there have been a great deal of results in this area in different directions [see Bougerol and Lacroix (1985) or Cohen et al. (1986)], here we restrict ourselves only to problems involving recurrence, tightness, invariant measures and laws of large numbers for products of random matrices. The reason is of course our own bias and also the desire to avoid duplicating work already available in books or well-known papers.

Keywords

Probability Measure Invariant Measure Compact Group Random Matrice Random Matrix 
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.

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References

  1. Barnsley, M. F., S. G. Demko, J. H. Elton, J. S. Geronimo, “Invariant measures for Markov processes arising from iterated function systems with place dependent probabilities,” Ann. Inst. Henri Poincaré 24, No. 3, 367–394 (1988).MathSciNetMATHGoogle Scholar
  2. Bougerol, P., “Tightness of products of random matrices and stability of linear stochastic systems,” Ann. Prob. 15, 40–74 (1987).MathSciNetMATHCrossRefGoogle Scholar
  3. Bougerol, P., J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, Boston—Basel—Stuttgart (1985).Google Scholar
  4. Breiman, L., Probability, Addison-Wesley, Reading Massachusetts (1968).MATHGoogle Scholar
  5. Cohen, J. E., H. Kesten, C. M. Newman, Editors, Random Matrices and their Applications, Contemporary Mathematics, Vol. 50, AMS, Providence, R. I. (1986).Google Scholar
  6. Darling, R. W. R., A. Mukherjea, “Stochastic flows on a countable set,” J. Th. Prob. 1, No. 2, 121–147 (1988).MathSciNetMATHCrossRefGoogle Scholar
  7. Darling, R. W. R., A. Mukherjea, “Discrete time voter models: a class of stochastic automata,” in: Probability Measures on Groups X, (H. Beyer, editor), pp. 83–94, Plenum Press, New York (1991).Google Scholar
  8. Darling, R. W. R., A. Mukherjea, “Probability measures on semigroups of nonnegative matrices,” in: The Analytical and Topological Theory of Semigroups, (K. H. Hofmann, J. D. Lawson, and J. S. Pym, editors), pp. 361–377, Walter de Gruyter, Berlin—New York (1990).Google Scholar
  9. Elton, J. H., “An ergodic theorem for iterated maps,” Ergodic Th. and Dynam. Systems 7, 481 (1987).MathSciNetMATHGoogle Scholar
  10. Furstenberg, H., “Noncommuting random products,” Trans. Amer. Math. Soc. 108, 377–428 (1963).MathSciNetMATHCrossRefGoogle Scholar
  11. Furstenberg, H. and Y. Kifer, “Random matrix products and measures on projective spaces,” Israel J. Math. 46, 12–32 (1983).MathSciNetMATHGoogle Scholar
  12. Goldsheid, I. Ya and G. A. Margulis, “Lyapunov indices of products of random matrices,” Russian Math. Surveys 44, No. 5, 11–71 (1989).MathSciNetCrossRefGoogle Scholar
  13. Grintsevichyus, A. K., “On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines,” Theory Probab. Appl. 19, 163–168 (1974).MATHCrossRefGoogle Scholar
  14. Guivarc’h, Y. and A. Raugi, “Frontiere de Furstenberg, proprietes de contractoin et theoremes de convergences,” Z Wahrscheinlichkeitstheorie verw. Gebiete 69, 187–242 (1985).MathSciNetMATHCrossRefGoogle Scholar
  15. Guivarc’h, Y. and A. Raugi, “Products of random matrices: convergence theorems,” in: Random Matrices and their Applications, Contemporary Mathematics, Vol. 50, pp. 31–54, AMS, Providence, R. I. (1986).Google Scholar
  16. Högnäs, G., “On products of random projections,” Acta Academiae Aboensis, Ser. B. 44, No. 5, 1–18 (1984).Google Scholar
  17. Högnäs, G., “A note on products of random matrices,” Stat. and Prob. Letters 5, 367–370 (1987).MATHCrossRefGoogle Scholar
  18. Högnäs, G., “Invariant measures and random walks on the semigroup of matrices,” in: Proc. of the Conf. on Markov processes and Stochastic Control, (H. Langer, editor), Gaussig, DDR, 11–15 (1988).Google Scholar
  19. Högnäs, G., Sequences of random transformations, Reports on Comp. Sc. and Math., Ser. A, No. 112, Abo Akademi University, Abo, Finland (1990).Google Scholar
  20. Hogans, G., A. Mukherjea, “Recurrent random walks and invariant measures on semigroups of n by n matrices,” Math. Z. 173, 69–94 (1980).MathSciNetCrossRefGoogle Scholar
  21. Högnäs, G., A. Mukherjea, “A mixed random walk on nonnegative matrices: A law of large numbers,” J. Theor. Prob. 8, No. 4, 973–990 (1995).MATHCrossRefGoogle Scholar
  22. Karlin, S., H. M. Taylor, A First Course in Stochastic Processes, Second Edition, Academic Press, New York—San Francisco—London (1975).Google Scholar
  23. Kesten, H., F. Spitzer, “Convergence in distribution of products of random matrices,” Z. Wahrscheinlichkeitstheorie verw. Gebiete 67, 363–386 (1984).MathSciNetMATHCrossRefGoogle Scholar
  24. Key, E., J. Theoret. Prob. 3, 477–488 (1990).MathSciNetMATHCrossRefGoogle Scholar
  25. Key, E., Probab. Th. Related Fields 75, 97–107 (1987).MathSciNetMATHCrossRefGoogle Scholar
  26. Kingman, J. F. C., “Subadditive ergodic theory,” Ann. Prob. 1, 883–909 (1973).MathSciNetMATHCrossRefGoogle Scholar
  27. Lo, C. C., A. Mukherjea, “Convergence in distribution of products of d by d random matrices,” J. Math. Anal. and Appl. 162, No. 1, 71–91 (1991).MathSciNetMATHCrossRefGoogle Scholar
  28. Mukherjea, A., “Limit theorems: Stochastic matrices, ergodic Markov chains and measures on semigroups,” in: Probabilistic Analysis and Related Topics, (A. T. Bharucha-Reid, editor), Vol. 2, pp. 143–203, Academic Press, New York (1979).Google Scholar
  29. Mukherjea, A., “Convergence in distribution of products of random matrices: A semigroup approach,” Trans. Amer. Math. Soc. 303, 395–411 (1987).MathSciNetMATHCrossRefGoogle Scholar
  30. Mukherjea, A., “The role of nonnegative idempotent matrices in certain problems in probability,” in: Proc. of Symp. in App. Math., Vol. 40, pp. 199–232, Amer. Math. Soc. Providence, RI (1990).Google Scholar
  31. Mukherjea, A., “Semigroups, attractors and products of random matrices,” in: Probability Measures on Groups X, (H. Heyer, editor), pp. 303–313, Plenum Press, New York (1991a).Google Scholar
  32. Mukherjea, A., “Tightness of products of i.i.d. random matrices,” Prob. Th. and Rel. Fields 87, 389–401 (1991b).MathSciNetMATHCrossRefGoogle Scholar
  33. Mukherjea, A., “Convergence in distribution of a Markov process generated by i.i.d. random matrices,” in: Diffusion Processes and Related Problems in Analysis (Vol. II), (M. A. Pinsky and V. Wihstutz, editors), pp. 171–200, Birkhäuser (1992a).Google Scholar
  34. Mukherjea, A., “Recurrent random walks in nonnegative matrices: attractors of certain iterated function systems,” Prob. Th. and Rel. Fields 91, 297–306 (1992b).MathSciNetMATHCrossRefGoogle Scholar
  35. Mukherjea, A., “Some remarks on products of random affine maps on (R+) d in: Contemporary Mathematics, Vol. 149, pp. 321–330, AMS, Providence, R. I. (1993a).Google Scholar
  36. Mukherjea, A., “Recurrent random walks in nonnegative matrices II,” Prob. Theory and Rel. Fields 96, 415–434 (1993b).MathSciNetMATHCrossRefGoogle Scholar
  37. Mukherjea, A., “Tightness of products of i.i.d. random matrices II,” Ann. Prob 22, 2223–2233 (1994).MathSciNetMATHCrossRefGoogle Scholar
  38. Mukherjea, A., A. Nakassis, “On the limit of the convolution iterates of a probability measure on n by n stochastic matrices,” J. Math. Anal. and Appl. 60, No. 2, 392–397 (1977).CrossRefGoogle Scholar
  39. Mukherjea, A., K. Pothoven, Real and Functional Analysis, Part B: Functional Analysis, Second Edition, Plenum Press, New York—London (1986).Google Scholar
  40. Nakassis, A., “Limit behavior of the convolution iterates of a probability measure on a semigroup of matrices,” J. Math. Anal. Appl. 70, 337–347 (1979).MathSciNetMATHCrossRefGoogle Scholar
  41. Oseledets, V. I., “A multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems,” Trans. Moscow Math. Soc. 19, 197–231 (1968).MATHGoogle Scholar
  42. Pincus, Trans. Amer. Math. Soc. 287, 65–89 (1985).MathSciNetMATHCrossRefGoogle Scholar
  43. Rosenblatt, M., “Products of i.i.d. stochastic matrices,” J. Math. Anal. Appl. 11, 1–10 (1965).MathSciNetCrossRefGoogle Scholar
  44. Rosenblatt, M., Markov Processes: Structure and Asymptotic Behavior, Springer, Berlin—HeidelbergNew York (1971).MATHCrossRefGoogle Scholar
  45. Sun, T. C., “Limits of convolutions of probability measures on the set of 2 by 2 stochastic matrices,” Bull. Inst. of Math. Academia Sinica 3, 235–248 (1975).MATHGoogle Scholar
  46. Sun, T. C., “Random walks on semigroups,” in: Random Matrices and their Applications, (J. E. Cohen, H. Kesten, and C. M. Newman, editors), Contemporary Mathematics, Vol. 50, p. 221239, AMS, Providence, R. I. (1986).Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Göran Högnäs
    • 1
  • Arunava Mukherjea
    • 2
  1. 1.Åbo Akademi UniversityÅboFinland
  2. 2.University of South FloridaTampaUSA

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