Summary
We consider expressions of the form Z n =E(exp nF(S n /n)) where S n is the sum of n i.i.d. random vectors with values in a Banach space and F is a smooth real valued function. By results of Donsker-Varadhan and Bahadur-Zabell one knows that lim (1/n) log Z n =sup x F(x)-h(x) where h is the so-called entropy function. In an earlier paper a more precise evaluation of Z n is given in the case where there was a unique point maximizing F-h and the curvature at the maximum was nonvanishing. The present paper treats the more delicate problem where these conditions fail to hold.
References
Azencott, R.: Grandes déviations et applications. Ecole d'été Saint-Flour VIII 1978. Lect. Notes Math. 774 (1980)
Azencott, R.: Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman. Sém de Probabilités XVI, Supplément; Géométrie différentielle stochastique. Lect. Notes Math. 921, 237–285 (1982)
Bahadur, R.R., Zabell, S.L.: Large deviations of the sample mean in general vector spaces. Ann. Probab. 7, 537–621 (1979)
Bhattacharya, R.N., Rao, R.N.: Normal approximations and asymptotic expansions. New York: Wiley 1976
Bolthausen, E.: Laplace approximations for sums of independent random vectors. Probab. Th. Rel. Fields 72, 305–318 (1986)
Bolthausen, E.: Laplace approximations for Markov process expectations. To appear in Ann. Probab. (1986)
Borell, C.: Gaussian radon measures on locally convex spaces. Math. Scand. 38, 265–284 (1976)
Csiszar, I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3, 146–158 (1975)
Dawson, D.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31, 29–85 (1983)
Dawson, D., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Preprint (1987)
de Acosta, A., Giné F.: Convergence of moments and related functionals in the general central limit theorem in Banach spaces. Z. Wahrscheinlichkeitstheor. Verw. Geb. 48, 213–231 (1979)
de Acosta, A.: Upper bounds for large deviations of dependent random vectors. Z. Wahrscheinlich-keitstheor. Verw. Geb. 69, 551–565 (1985)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time III. Comm. Pure Appl. Math. 29, 389–461 (1976)
Ellis, R.S., Newman, C.M.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, 117–139 (1978)
Ellis, R.S., Rosen, J.S.: Asymptotic analysis of Gauss integrals I, II: Trans. Am. Math. Soc. 273, 447–481 (1982) and Comm. Math. Phys. 82, 153–181 (1982)
Götze, F.: On Edgeworth expansions in Banach spaces. Ann. Probab. 9, 852–859 (1981)
Jain, N.C.: Central limit theorem in Banach spaces. Proc. First Conf. on Prob. in Banach Spaces 1975. Lect. Notes in Math. 598 (1976)
Kuo, H.-H.: Gaussian measures in Banach spaces. Lect. Notes Math. 463 (1975)
Kusuoka and Tamura: The convergence of Gibbs measures associated with mean field potentials. J. Fac. Sci. Univ. Tokyo, Sect. 1A, 31, 223–245 (1984)
Martin-Löf, A.: Laplace approximation for sums of independent random variables. Z. Wahrschein-lichkeitstheor. Verw. Geb. 59, 101–115 (1982)
Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane-Emden equation. J. Stat. Phys. 29, 561–578 (1982)
Ney, P.: Dominating points and the asymptotics of large deviations for random walk on ℝd. Ann. Probab. 11, 158–167 (1983)
Varadhan, S.R.S.: Large deviations and applications. SIAM, Philadelphia 1984
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Part of this work has been done when the author was at the Forschungsinstitut für Mathematik in Zürich. I would like to thank the members of the institute and especially Hans Föllmer for the kind hospitality
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Bolthausen, E. Laplace approximations for sums of independent random vectors. Probab. Th. Rel. Fields 76, 167–206 (1987). https://doi.org/10.1007/BF00319983
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DOI: https://doi.org/10.1007/BF00319983
Keywords
- Entropy
- Banach Space
- Stochastic Process
- Probability Theory
- Statistical Theory