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Sur un théorème spectral et son application aux noyaux lipchitziens

Author: Hubert Hennion
Journal: Proc. Amer. Math. Soc. 118 (1993), 627-634
MSC: Primary 60J10; Secondary 28D99, 47A35, 47N30
MathSciNet review: 1129880
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Abstract: The aim of this note is to point out that the Ionescu Tulcea and Marinescu theorem can be reinforced, using a Nussbaum formula for the essential spectral radius of an operator. In this stronger version, this theorem is suitable for the spectral analysis of lipschitzian, positive, not necessarily markovian kernels, Ruelle theorem follows. As an application to Probability Theory, a large deviation theorem is proved.

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  • [1] Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674
  • [2] Felix E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/1961), 22–130. MR 0209909
  • [3] Wolfgang Doeblin and Robert Fortet, Sur des chaînes à liaisons complètes, Bull. Soc. Math. France 65 (1937), 132–148 (French). MR 1505076
  • [4] N. Dunford et J. T. Schwartz, Linear operators. Part I, Pure Appl. Math., Vol. VII, Interscience, New York, 1967.
  • [5] Y. Guivarc’h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), no. 1, 73–98 (French, with English summary). MR 937957
  • [6] Y. Guivarc’h and A. Raugi, Products of random matrices: convergence theorems, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 31–54. MR 841080, 10.1090/conm/050/841080
  • [7] Hubert Hennion, Dérivabilité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes à coefficients positifs, Ann. Inst. H. Poincaré Probab. Statist. 27 (1991), no. 1, 27–59 (French, with English summary). MR 1098563
  • [8] C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d’opérations non complètement continues, Ann. of Math. (2) 52 (1950), 140–147 (French). MR 0037469
  • [9] Gerhard Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 3, 461–478. MR 787608, 10.1007/BF00532744
  • [10] Émile Le Page, Théorèmes limites pour les produits de matrices aléatoires, Probability measures on groups (Oberwolfach, 1981) Lecture Notes in Math., vol. 928, Springer, Berlin-New York, 1982, pp. 258–303 (French). MR 669072
  • [11] S. V. Nagaev, Some limit theorems for stationary Markov chains, Teor. Veroyatnost. i Primenen. 2 (1957), 389–416 (Russian, with English summary). MR 0094846
  • [12] Roger D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473–478. MR 0264434
  • [13] David Ruelle, Thermodynamic formalism, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. The mathematical structures of equilibrium statistical mechanics. MR 2129258
  • [14] David Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 175–193 (1991). MR 1087395
  • [15] Helmut H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966. MR 0193469

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Keywords: Markov chains, limit theorems, Ruelle theorem
Article copyright: © Copyright 1993 American Mathematical Society