## Densities of idempotent measures and large deviations

HTML articles powered by AMS MathViewer

- by Marianne Akian PDF
- Trans. Amer. Math. Soc.
**351**(1999), 4515-4543 Request permission

## Abstract:

Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle.## References

- M. Akian,
*Theory of cost measures: Convergence of decision variables*, Rapport de Recherche 2611, INRIA, 1995. - —,
*On the continuity of the Cramer transform*, Rapport de Recherche 2841, INRIA, 1996. - M. Akian, J. P. Quadrat, and M. Viot,
*Bellman processes*, 11th International Conference on Analysis and Optimization of Systems: Discrete Event Systems, Lecture Notes in Control and Information Sciences, vol. 199, Springer Verlag, 1994. - —,
*Duality between probability and optimization*, Idempotency (J. Gunawardena, ed.), Cambridge University Press, 1998. - H. Attouch,
*Variational convergence for functions and operators*, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**773850** - François Louis Baccelli, Guy Cohen, Geert Jan Olsder, and Jean-Pierre Quadrat,
*Synchronization and linearity*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1992. An algebra for discrete event systems. MR**1204266** - F. Bellalouna,
*Un point de vue linéaire sur la programmation dynamique. Détection de ruptures dans le cadre des problèmes de fiabilité*, Thèse, Université Paris-IX Dauphine, Paris, 1992. - Patrick Billingsley,
*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396** - Włodzimierz Bryc,
*Large deviations by the asymptotic value method*, Diffusion processes and related problems in analysis, Vol. I (Evanston, IL, 1989) Progr. Probab., vol. 22, Birkhäuser Boston, Boston, MA, 1990, pp. 447–472. MR**1110177** - Cahit Arf,
*Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper*, J. Reine Angew. Math.**181**(1939), 1–44 (German). MR**18**, DOI 10.1515/crll.1940.181.1 - P. Del Moral,
*Résolution particulaire des problèmes d’estimation et d’optimisation non-linéaires*, Thèse, Université Paul Sabatier, Toulouse, 1994. - P. Del Moral, T. Thuillet, G. Rigal, and G. Salut,
*Optimal versus random processes: The non-linear case*, Rapport de recherche, LAAS, 1990. - Amir Dembo and Ofer Zeitouni,
*Large deviations techniques and applications*, Jones and Bartlett Publishers, Boston, MA, 1993. MR**1202429** - Ryszard Engelking,
*General topology*, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR**1039321** - Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott,
*A compendium of continuous lattices*, Springer-Verlag, Berlin-New York, 1980. MR**614752** - T. Jiang,
*The metric of large deviation convergence*, Preprint, September 1995. - V. N. Kolokoltsov,
*Maslov’s arithmetic in general topology*, Geometry, Topology and Applications. Moskov. Instrument. Inst. (1990), 64–68 (Russian). - V. N. Kolokoltsov and V. P. Maslov,
*The general form of the endomorphisms in the space of continuous functions with values in a numerical commutative semiring (with the operation $\oplus =\max$)*, Soviet Math. Dokl.**36**(1988), no. 1, 55–59. - B. M. Miller,
*Optimization of dynamical systems with generalized control*, Avtomat. i Telemekh.**6**(1989), 23–34 (Russian, with English summary); English transl., Automat. Remote Control**50**(1989), no. 6, 733–742. MR**1016198** - V. Maslov,
*Méthodes opératorielles*, Éditions Mir, Moscow, 1987 (French). Translated from the Russian by Djilali Embarek. MR**1085245** - V. P. Maslov and V. N. Kolokoltsov,
*Idempotent analysis and its applications to optimal control theory*, Nauka, Moscow, 1994 (Russian). - V. P. Maslov and S. N. Samborskiĭ (eds.),
*Idempotent analysis*, Advances in Soviet Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1992. MR**1203781**, DOI 10.1090/advsov/013 - V. P. Maslov and S. N. Samborskiĭ,
*Stationary Hamilton-Jacobi and Bellman equations (existence and uniqueness of solutions)*, Idempotent analysis, Adv. Soviet Math., vol. 13, Amer. Math. Soc., Providence, RI, 1992, pp. 119–133. MR**1203788**, DOI 10.1007/BF01084111 - T. Norberg and W. Vervaat,
*Capacities on non-Hausdorff spaces*, Probability and lattices, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1997. - George L. O’Brien,
*Sequences of capacities, with connections to large-deviation theory*, J. Theoret. Probab.**9**(1996), no. 1, 19–35. MR**1371069**, DOI 10.1007/BF02213733 - George L. O’Brien and Wim Vervaat,
*Capacities, large deviations and loglog laws*, Stable processes and related topics (Ithaca, NY, 1990) Progr. Probab., vol. 25, Birkhäuser Boston, Boston, MA, 1991, pp. 43–83. MR**1119351** - Endre Pap,
*On nonadditive set functions*, Atti Sem. Mat. Fis. Univ. Modena**39**(1991), no. 1, 345–360. MR**1111778** - —,
*Solution of nonlinear differential and difference equations*, EUFIT’93 (Aachen), Sept 7–10, 1993, pp. 498–503. - —,
*The Lebesgue decomposition of the null-additive fuzzy measures*, Zb.-Rad.-Prirod.-Mat.-Fak.-Ser.-Mat.**24**(1994), no. 1, 129–137. - A. Puhalskii,
*Large deviations of semimartingales via convergence of the predictable characteristics*, Stochastic and Stochastics Reports**49**(1994), 27–85. - A. Puhalskii,
*Large deviation analysis of the single server queue*, Queueing Systems Theory Appl.**21**(1995), no. 1-2, 5–66. MR**1372048**, DOI 10.1007/BF01158574 - Jean-Pierre Quadrat,
*Théorèmes asymptotiques en programmation dynamique*, C. R. Acad. Sci. Paris Sér. I Math.**311**(1990), no. 11, 745–748 (French, with English summary). MR**1081638** - D. W. Stroock,
*An introduction to the theory of large deviations*, Universitext, Springer-Verlag, New York, 1984. MR**755154**, DOI 10.1007/978-1-4613-8514-1

## Additional Information

**Marianne Akian**- Affiliation: INRIA, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
- Email: marianne.akian@inria.fr
- Received by editor(s): June 16, 1995
- Received by editor(s) in revised form: April 17, 1997
- Published electronically: July 19, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 4515-4543 - MSC (1991): Primary 28B15, 49J52; Secondary 06B35, 60F10
- DOI: https://doi.org/10.1090/S0002-9947-99-02153-4
- MathSciNet review: 1466943