Advertisement

Random Walks on Semigroups

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

Abstract

The term random walk suggests stochastic motion in space, a succession of random steps combined in some way. In Chapter 3 we interpret the term very narrowly: We require the steps to be independent and to have the same probability distribution. The walk is then a succession of products of those steps. Later on we apply our results to slightly more general situations, e.g., cases where steps depend on each other in a Markovian way. Thus our study of random walks is synonymous with the study of products of independent identically distributed random elements of a semigroup. We study the most basic notions for these processes which are of course discrete-time Markov chains with the semigroups as state spaces. We deal with, for example, communication relations, irreducibility questions, recurrence versus transience, periodicity, and ergodicity. Generally speaking these probabilistic notions have an algebraic counterpart, in the sense that probabilistic properties of a random walk cannot be satisfied unless the semigroup supporting the random walk has a certain algebraic structure. The situation is very similar to that in Chapter 2 where we saw, for example, that only completely simple semigroups with a compact group factor support limit points of a tight convolution sequence of measures (see Theorem 2.28).

Keywords

Markov Chain Probability Measure Random Walk Haar Measure Simple Semigroup 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baldi, P., “Caractérisation des groupes de Lie connexes récurrents,” Ann. Inst. Henri Poincaré 17, 281–308 (1981).MathSciNetzbMATHGoogle Scholar
  2. Berbee, H., “Recurrence and transience for random walks with stationary increments,” Z Wahrscheinlichkeitstheorie verw. Gebiete 56, 531–536 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  3. Berstel, J., D. Perrin, Theory of Codes, Academic Press, Orlando (1985).Google Scholar
  4. Bougerol, P., J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, Boston-Basel-Stuttgart (1985).zbMATHCrossRefGoogle Scholar
  5. Bougerol, P., “Tightness of products of random matrices and stability of linear stochastic systems,” Ann. Probab. 15, 40–74 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  6. Breiman, L., Probability, Addison-Wesley, Reading-Menlo Park-London-Don Mills (1968).zbMATHGoogle Scholar
  7. Brissaud, M., “Sur les marches aléatoires dans les demi-groupes topologiques,” C. R. Acad. Sci. Paris, Sér. A 268, 1286–1289 (1969).MathSciNetzbMATHGoogle Scholar
  8. Brunel, A., D. Revuz, “Un critère probabiliste de compacité des groupes,” Ann. Prob. 2, 745–746 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  9. Chung, K. L., W. H. Fuchs, “On the distribution of values of sums of random variables,” Mem. Amer. Math. Soc. 6, (1951).Google Scholar
  10. Chung, K. L., Markov Chains with Stationary Transition Probabilities, Second Edition, Springer-Verlag, Berlin-Heidelberg-New York (1967).Google Scholar
  11. Cigler, J. “Über die Grenzverteilung von Summen Markowscher Ketten auf endlichen Gruppen I,” Z. Wahrscheinlichkeitstheorie verw. Gebiete 1 415–420 (1963).Google Scholar
  12. Cohen, J. E., H. Kesten, C. M. Newman (editors), Random Matrices and their Applications, Contemporary Mathematics 50, American Mathematical Society, Providence, Rhode Island (1986).zbMATHGoogle Scholar
  13. Dekking, F. M., “On transience and recurrence of generalized random walks,” Z. Wahrscheinlichkeitstheorie verw. Gebiete 61, 459–465 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  14. Dudley, R. M., “Pathological topologies and random walks on abelian groups,” Proc. Amer. Math. Soc. 15, 231–238 (1964).MathSciNetzbMATHCrossRefGoogle Scholar
  15. Furstenberg, H., H. Kesten, “Products of random matrices,” Ann. Math. Stat. 31, 457–469 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  16. Grenander, U., Probabilities on Algebraic Structures, Almqvist and Wiksell, Stockholm (1963).zbMATHGoogle Scholar
  17. Guivarc’h Y., M. Keane, B. Roynette, Marches aléatoires sur les groupes de Lie, Lecture Notes in Mathematics 624, Springer, Berlin-Heidelberg-New York (1977).Google Scholar
  18. Högnäs, G., “An ergodic random walk on a compact semigroup,” Acta Acad. Aboensis, Ser. B 33 (9), (1973a).Google Scholar
  19. Högnäs, G., “An note on random walks on a compact semigroup,” Acta Acad. Aboensis, Ser. B 33 (10), (1973b).Google Scholar
  20. Högnäs, G., “Marches aléatoires sur un demi-groupe compact,” Ann. Inst. Henri Poincaré 10, 115–154 (1974a).zbMATHGoogle Scholar
  21. Högnäs, G., “Remarques sur les marches aléatoires dans un demi-groupe avec un idéal compact ayant une probabilité positive,” Ann. Inst. Henri Poincaré 10, 345–354 (1974b).Google Scholar
  22. Högnäs, G., On random walks with continuous components, Aarhus Universitet, Matematisk Institut, Preprint Series 1976/77, No. 26 (1977b).Google Scholar
  23. Högnäs, G., “A note on the product of random elements of a semigroup,” Mb. Math. 85, 317–321 (1978).zbMATHGoogle Scholar
  24. Högnäs, G., “On random walks with continuous components,” Semigroup Forum 17, 75–93 (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  25. Högnäs, G., “Recurrence and transience of Markov random walks,” Acta Acad. Aboensis, Ser. B 44 (4), (1984a).Google Scholar
  26. Högnäs, G., “On products of random projections,” Acta Acad. Aboensis, Ser. B 44 (5), (1984b).Google Scholar
  27. Högnäs, G., “Markov random walks on groups,” Math. Scand. 58, 35–45 (1986).MathSciNetzbMATHGoogle Scholar
  28. Högnäs, G., “A note on products of random matrices,” Stat. and Prob. Letters 5, 367–370 (1987).zbMATHCrossRefGoogle Scholar
  29. Högnäs, G., “A note on the semigroup of analytic mappings with a common fixed point,” in: Probability Measures on Groups IX, (H. Heyer, editor) (Lecture Notes in Mathematics 1379, p. 135, Springer, Berlin-Heidelberg-New York (1989).Google Scholar
  30. Högnäs, G., A. Mukherjea, “Recurrent random walks and invariant measures on semigroups of n x n matrices,” Math. Z. 173, 69–94 (1980).MathSciNetzbMATHCrossRefGoogle Scholar
  31. Hewitt, E., K. A. Ross, Abstract Harmonic Analysis, Vol. I, Springer-Verlag, Berlin-GottingenHeidelberg (1963).zbMATHCrossRefGoogle Scholar
  32. Heyer, H., Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin-HeidelbergNew York (1977).zbMATHCrossRefGoogle Scholar
  33. Husain, T., Introduction to Topological Groups, W. B. Saunders Company, Philadelphia (1966).Google Scholar
  34. Karlin, S., H. M. Taylor, A First Course in Stochastic Processes, Second Edition, Academic Press, New York-San Francisco-London (1975).Google Scholar
  35. Kesten, H., “The Martin boundary of recurrent random walks on countable groups,” in: Proc. 5th Berkeley Symp. on Mathematical Statistics and Probability II, p. 51 (1967).Google Scholar
  36. Koutsky, Z., “Einige Eigenschaften der modulo k addierten Markowschen Ketten,” in: Transactions of the 2nd Prague Conference on Information Theory, Random Functions and Statistical Decision Theory, (Czechoslovak Academy of Sciences, Prague), p. 263–278 (1959).Google Scholar
  37. Larisse, J., “Marches au hasard sur les demi-groupes discrets; I, II, III,” Ann. Inst. Henri Poincaré 8, 107–125, 127–173, 229–240 (1972).MathSciNetGoogle Scholar
  38. Lo, C. C., A. Mukherjea, “Convergence in distribution of products of d x d random matrices,” J. Math. Anal. Appl. 162, 71–91 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  39. Loynes, R. M., “Products of independent random elements in a topological group,” Z Wahrscheinlichkeitstheorie verve Gebiete 1, 446–455 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  40. Martin-Löf, P., “Probability theory on discrete semigroups,” Z Wahrscheinlichkeitstheorie verve. Gebiete 4, 78–102 (1965).CrossRefGoogle Scholar
  41. Mukherjea A., T. C. Sun, N. A. Tserpes, “Random walks on compact semigroups,” Proc. Amer. Math. Soc. 39, 599–605 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  42. Mukherjea, A., N. A. Tserpes, `Bilateral random walks on compact semigroups,“ Proc. Amer. Math. Soc. 47, 457–466 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  43. Mukherjea, A., N. A. Tserpes, Measures on Topological Semigroups: Convolution Products and Random Walks, Lecture Notes in Mathematics 547, Springer, Berlin-Heidelberg-New York (1976).Google Scholar
  44. Mukherjea, A., “Limit theorems for probability measures on noncompact groups and semigroups,” Z Wahrscheinlichkeitstheorie verw. Gebiete 33, 273–284 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  45. Mukherjea, A., “Convergence in distribution of products of random matrices: a semigroup approach,” Trans. Amer. Math. Soc. 303, 395–411 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  46. Muthsam, H., “Über die Summe Markoffscher Ketten auf Halbgruppen,” Mh. Math. 76, 43–54 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  47. Niemi, S. and E. Nummelin, “Central limit theorems for Markov random walks,” Soc. Sci. Fenn. Comment. Phys.-Math. 54 (1982).Google Scholar
  48. Nummelin, E., General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge (1984).CrossRefGoogle Scholar
  49. Pólya, G., “Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz,” Math. Ann. 84, 149–160 (1921).MathSciNetzbMATHCrossRefGoogle Scholar
  50. Revuz, D., Markov Chains, North Holland, Amsterdam-Oxford (1975).zbMATHGoogle Scholar
  51. Rosenblatt, M., Markov Processes: Structure and Asymptotic Behavior, Springer-Verlag, BerlinHeidelberg-New York (1971).zbMATHCrossRefGoogle Scholar
  52. Rosenblatt, M., “Invariant and subinvariant measures of transition probability functions acting on continuous functions,” Z Wahrscheinlichkeitstheorie verve Gebiete 25, 209–221 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  53. Roynette, B., “Marches aléatoires sur les groupes de Lie,” in: Ecole d’Eté de Probabilités de Saint-Flour VII - 1977, Lecture Notes in Mathematics 678, Springer, Berlin-Heidelberg-New York (1978).Google Scholar
  54. Rudin, W., Fourier Analysis on Groups, Interscience Publishers, New York-London (1962). Schmetterer, L., “Über die Summe Markov’scher Ketten auf Halbgruppen,” Mh. Math. 71, 223–230 (1967).CrossRefGoogle Scholar
  55. Sun, T. C., A. Mukherjea and N. A. Tserpes, “On recurrent random walks on semigroups,” Trans. Amer. Math. Soc. 185, 213–227 (1973).MathSciNetGoogle Scholar
  56. Sun, T. C., “Random walks on semigroups,” in: Random Matrices and Their Applications, (J. E. Cohen, H. Kesten, C. M. Newman, editors), Contemporary Mathematics, Vol. 50, pp. 221–239, American Mathematical Society, Providence, Rhode Island (1986).Google Scholar
  57. Tortrat, A., “Lois de probabilité dans les semi-groupes topologiques complètement réguliers,” C. R. Acad. Sci. Paris, Sér. A 261, 3941–3944 (1965).MathSciNetzbMATHGoogle Scholar
  58. Tortrat, A., “Lois tendues tt sur un demi-groupe topologique complètement simple X,” Z Wahrscheinlichkeitstheorie verw. Gebiete 6, 145–160 (1966).Google Scholar
  59. Tuominen, P., R. L. Tweedie, “Markov chains with continuous components,” Proc. London Math. Soc. (3) 38, 89–114 (1979).MathSciNetzbMATHGoogle Scholar
  60. Wolff, M., “Über Produkte abhängiger zufälliger Veränderlicher mit Werten in einer kompakten Halbgruppe,” Z Wahrscheinlichkeitstheorie verve. Gebiete 35, 253–264 (1976).MathSciNetzbMATHCrossRefGoogle 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

Personalised recommendations