Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some limit theorems for nonhomogeneous Markoff processes


Author: A. Fuchs
Journal: Trans. Amer. Math. Soc. 86 (1957), 511-531
MSC: Primary 60.00
MathSciNet review: 0094848
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We intend to study some problems related to the asymptotic behaviour of a physical system the evolution of which is markovian. The typical example of such an evolution is furnished by an homogeneous discrete chain with a finite number of possible states considered first by A. A. Markoff. In §1 we recall briefly the main results of this theory and in §2 we treat its obvious generalization to the continuous parameter case. In §3 we pass to the proper object of this paper and we establish a limit theorem for time-homogeneous Markoff processes. This limit theorem is then extended to the nonhomogeneous case under some supplementary conditions (§4). Finally we give an application of this theory to random functions connected with a Markoff process (§5).


References [Enhancements On Off] (What's this?)

  • [1] A. A. Markoff, Extension of the law of large numbers to dependent events, Isvestia Soc. Phys. Math. Kazan vol. 15 no. 4 (1906) pp. 135-156.
  • [2] M. Fréchet, Recherches théoriques modernes sur le calcul des probabilités II: Méthode des fonctions arbitraires: Théorie des événements en chaîne dans le cas d'un nombre fini d'états possibles, Paris, Gauthier-Villars, 1938.
  • [3] J. Kaucky, Quelques remarques sur les chaînes de Markoff, Spisy Vydávané Přírodovědeckou Fakultou Masarykovy University no. 131 (1930).
  • [4] M. Konečny, Sur la théorie des chaînes de Markoff, Spisy Vydávané Přírodovědeckou Fakultou Masarykovy University no. 147 (1931).
  • [5] A. A. Kolmogoroff, Anfangsgründe der Markoff'schen Ketten mit unendlich vielen moglichen Zuständen, Rec. Math. Moscou (Mat. Sbornik) vol. 1 (43) (1936) pp. 607-610.
  • [6] J. L. Doob, Markoff chains—denumerable case, Trans. Amer. Math. Soc. 58 (1945), 455–473. MR 0013857 (7,210b), http://dx.doi.org/10.1090/S0002-9947-1945-0013857-4
  • [7] Paul Lévy, Systèmes markoviens et stationnaires. Cas dénombrable, Ann. Sci. École Norm. Sup. (3) 68 (1951), 327–381 (French). MR 0047961 (13,959c)
  • [8] J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896 (15,445b)
  • [9] J. Neveu, Thèse, Paris, 1955 (unpublished).
  • [10] J. P. Vigier, Thèse, Paris, 1954 (unpublished).
  • [11] A. Blanc-Lapierre and Robert Fortet, Théorie des fonctions aléatoires. Applications à divers phénomènes de fluctuation, Masson et Cie, Paris, 1953 (French). Avec un chapitre sur la mécanique des fluides par J. Kampé de Fériet. MR 0061780 (15,883d)
  • [12] Kôsaku Yosida and Shizuo Kakutani, Operator-theoretical treatment of Markoff’s process and mean ergodic theorem, Ann. of Math. (2) 42 (1941), 188–228. MR 0003512 (2,230e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60.00

Retrieve articles in all journals with MSC: 60.00


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1957-0094848-2
PII: S 0002-9947(1957)0094848-2
Article copyright: © Copyright 1957 American Mathematical Society