Abstract
The techniques of convex analysis have come to play an increasingly important role in the theory of large deviations (see, e.g., Bahadur and Zabell, 1979; Ellis, 1985; de Acosta, 1988). The purpose of this brief note is to point out an interesting connection between a basic form of convergence commonly employed in convex analysis (“Mosco convergence”), and two theorems of fundamental importance in the theory of large deviations.
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References
de Acosta, A. (1985). Upper bounds for large deviations of dependent random vectors. Z. Wahrscheinlichkeitstheorie verw. Gebiete. 69, 551–564.
de Acosta, A. (1988). Large deviations for vector-valued functionals of a Markov chain: lower bounds. Ann. Probab. 16, 925–960.
Bahadur, R. R., and Zabell, S. L. (1979). Large deviations of the sample mean in general vector spaces. Ann. Probab. 7, 587–621.
Baum, L.E, Katz, M., and Read, R. R. (1962). Exponential convergence rates for the law of large numbers. Trans. Amer. Math. Soc. 102, 187–199.
Dinwoodie, I. H. and Zabell, S. L. (1992). Large deviations for exchangeable sequences. Ann. Probab. 20. In press.
Dinwoodie, I. H. and Zabell, S. L. (1993). Large deviations for sequences of mixtures. Festschrift in honor of R. R. Bahadur. To appear.
Ellis, R. S. (1984). Large deviations for a general class of random vectors. Ann. Probab. 12, 1–16.
Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer-Verlag, New York.
Gärtner, J. (1977). On large deviations from the invariant measure. Theory Probab. Appl. 22, 24–39.
Lynch, J. (1978). A curious converse of Siever’s theorem. Ann. Probab. 6, 169–173.
Mosco, U. (1971). On the continuity of the Young-Fenchel transform. J. Math. Analysis Appl. 35, 518–535.
Plachky, D. (1971). On a theorem of G. L. Sievers. Ann. Math. Statist. 42, 1442–1443.
Plachky, D. and Steinebach, J. (1975). A theorem about probabilities of large deviations with applications to queuing theory. Period. Math. Hungar. 5, 343–345.
Sievers, G. L. (1969). On the probabilities of large deviations and exact slopes. Ann. Math. Statist. 40, 1908–1921.
Steinebach, J. (1978). Convergence rates of large deviation probabilities in the multidimensional case. Ann. Probab. 6, 751–759.
Zabell, S. L. (1992). Mosco convergence in locally convex spaces. J. Funct. Analysis. To appear.
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Zabell, S.L. (1992). Mosco Convergence and Large Deviations. In: Dudley, R.M., Hahn, M.G., Kuelbs, J. (eds) Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Progress in Probability, vol 30. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0367-4_16
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DOI: https://doi.org/10.1007/978-1-4612-0367-4_16
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