Abstract
We show how the essential spectral radius r e (Q) of a bounded positive kernel Q, acting on bounded functions, is linked to the lower approximation of Q by certain absolutely continuous kernels. The standart Doeblin’s condition can be interpreted in this context, and, when suitably reformulated, it leads to a formula for r e (Q). This results may be used to characterize the Markov kernels having a quasi-compact action on a space of measurable functions bounded with respect to some test function, when no irreducibilty and aperiodicity are assumed.
References
Brunel A., Revuz D. (1974). Quelques applications probabilistes de la quasi-compacité. Ann. Inst. H. Poincaré 10(3): 301–337
Conze J.-P., Raugi A. (1990). Fonctions harmoniques pour un opérateur de transition et applications. Bull. Soc. math. France 118: 273–310
Dunford et, N. Schwartz, J.T.: Linear operators. Part. I. Pure and applied mathematics. vol. 7. Interscience
Fortet R. (1978). Condition de Doeblin et quasi-compacité. Ann. Inst. Henri Poincaré 14(4): 379–390
Hennion H. (1993). Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Am. Math. Soc. Com 118(2): 627–634
Hennion, H.: Quasi-compacité. Cas des noyaux lipschitziens et des noyaux markoviens, Séminaire de Probabilités de Rennes (1995)
Hennion, H.: Quasi-compactness and absolutely continuous kernels. Applications to Markov chains. ArXiv, math.PR/0606680
Hennion, H., Hervé, L.: Limit theorems for Markov chains and stochastic properties of dynamical systems by Quasi-compactness, L.N. 1766, (2001)
Hervé, L.: Limit theorems for geometrically ergodic Markov chains, preprint, IRMAR
IonescuTulcea et C.T., Marinescu G. (1950). Théorie ergodique pour des classes d’opérateurs non complètement continus. Ann. Math. 52(2): 140–147
Meyn S.P., Tweedie R.L. (1993). Markov chains and stochastic stability. Springer, Berlin Heidelberg New York
Meyer-Nieberg P. (1991). Banach lattices. Springer, Berlin Heidelberg New York
Nagaev S.V. (1957). Some limit theorems for stationary Markov chains. Theory of Probability and its applications. 11(4): 378–406
Neveu, J.: Bases Mathématiques du calcul des Probabilités, Masson (1964)
Nummelin E., Tweedie R.L. (1978). Geometric ergodicity and R-positivity for general Markov chains. Ann. Prob. 6: 404–420
Nussbaum R.D. (1970). The radius of essential spectrum. Duke Math. J. 37: 473–478
Revuz, D.: Markov chains, North-Holland (1975)
Wu L. (2000). Uniformly integrable operators and large deviations for Markov chains. J. Funct. Anal. 172: 301–376
Wu L. (2004). Essential spectral radius for Markov semi-groups (I): discrete time case. Prob. Theory Relat. Fields 128: 255–321
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Hennion, H. Quasi-compactness and absolutely continuous kernels. Probab. Theory Relat. Fields 139, 451–471 (2007). https://doi.org/10.1007/s00440-006-0048-8
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DOI: https://doi.org/10.1007/s00440-006-0048-8
Keywords
- Markov chain
- Quasi-compactness
- Positive operator
Mathematics Subject Classification (2000)
- 60J05
- 47B07
- 47B65