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Sharp asymptotics of large deviations in ℝd

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Abstract

Given a sequence {X1}i=1,2,3,... of i.i.d. random variables taking values in ℝd,d≥2, letS n =Σ n i=1 X t=1. For Λ a Borel set in ℝd having smooth boundary, witha=infx∈ΛI(x) the minimal value of the large deviation rate functionI(x) over Λ, we find, under suitable hypotheses, asymptotic results asn→∞, of the form

$$P(S_n \in n\Gamma ) = n^\gamma e^{ - na} (d_0 + o(1))$$

where the constant γ depends sensitively on the geometry of Λ and the dimensiond, and takes values −∞<γ≤(d−2/2). For fixeda=infx∈ΛI(x), we construct examples having any specific γ in this range.

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References

  1. Bahadur, R. R., and Rao, R. Ranga (1960). On deviations of the sample mean.Ann. Math. Statist. 31, 43–54.

    Google Scholar 

  2. Bhattacharya, R. N., and Rao, R. Ranga (1976).Normal Approximation and Asymptotic Expansions. John Wiley & Sons.

  3. Bolthausen, E. (1986). Laplace approximations for sums of independent random vectors Part I.Prob. Th. Rel. Fields 72, 305–318.

    Google Scholar 

  4. Bolthausen, E. (1987). Laplace approximations for sums of independent random vectors Part II.Prob. Th. Rel. Fields 76, 167–206.

    Google Scholar 

  5. Borovkov, A. A., and Rogozin, B. A. (1965). On the multi-dimensional central limit theorem.Theory of Prob. Appl. 10, 55–62.

    Google Scholar 

  6. Bruna, J., Nagel, A., and Wainger, S. (1988). Convex hypersurfaces and Fourier transforms.Annals of Math. 127, 333–365.

    Google Scholar 

  7. Chaganty, N. R., and Sethuraman, J. (1986). Multidimensional large deviation local limit theorems.J. Multivariate Anal. 20, 190–204.

    Google Scholar 

  8. Ellis, Richard S. (1985).Entropy, Large Deviations, and Statistical Mechanics. Springer-Verlag.

  9. Ellis, Richard, and Rosen, Jay (1982). Asymptotic analysis of Gaussian integrals I. Isolated minimum points.Trans. of AMS 273, 447–481.

    Google Scholar 

  10. Ellis, Richard, and Rosen, Jay (1981). Asymptotic analysis of Gaussian integrals II. Manifold of minimum points.Commun. Math. Phys. 82, 153–181.

    Google Scholar 

  11. Herz, C. S. (1962). On the number of lattice points in a convex set.Amer. J. Math. 84, 126–133. See also: Fourier transforms related to convex sets.Annals of Math. 75, 81–92.

    Google Scholar 

  12. Iltis, Michael G. (1991). Asymptotics of large deviations for I.I.D. and Markov-additive random variables inR d. PhD Thesis, University of Wisconsin, Madison.

    Google Scholar 

  13. Iltis, Michael G. (1993). Asymptotics of large deviations for Markov-additive chains, preprint.

  14. Kuelbs, James, and Li, Wenbo V. (1994). Some large deviation results for Gaussian measures.Proc. of the Probability in Banach Space. Conf. IX Denmark August 1993. Birkhäuser:Progress in Probability, Vol. 35.

  15. Kusuoka, S., and Tamura, Y. (1991). Precise estimates for large deviations of Donsker-Varadhan type.J. Fac. Sc. Univ. Tokyo 38, 533–565.

    Google Scholar 

  16. Littman, Walter (1963). Fourier transforms of surface-carried measures and differentiability of surface averages.Bull. Amer. Math. Soc. 69, 766–770.

    Google Scholar 

  17. Ney, P. (1983). Dominating points and the asymptotics of large deviations for random walk onR d.Ann. Prob. 11, 158–167.

    Google Scholar 

  18. Ney, P. (1984). Convexity and large deviations.Ann. Prob. 12, 903–906 (this is an addendum to [18]).

    Google Scholar 

  19. Randol, Burton (1966). A lattice point problem.Trans. Amer. Math. Soc. 121, 257–268.

    Google Scholar 

  20. Randol, Burton (1969). On the asymptotic behavior of the Fourier transform of the indicator function of a convex set.Trans. Amer. Soc. 139, 271–278, 279–285.

    Google Scholar 

  21. Reeds, James (1980). Correction terms for multinomial large deviations.Asymptotic Theory of Statistical Tests and Estimation. Academic Press, New York, pp. 287–305.

    Google Scholar 

  22. Richter, V. (1958). Multi-dimensional local limit theorems for large deviations.Theory of Prob. Appl. 3, 100–106.

    Google Scholar 

  23. Rockafellar, R. Tyrell (1970).Convex Analysis. Princeton University Press.

  24. Saulis, L., and Statulevicius, V. A. (1991).Limit Theorems for Large Deviations. Kluwer Academic Publishers.

  25. Svensson, Ingvar (1971). Estimates for the Fourier transform of the characteristic function of a convex set.Arkiv for Matematik 9, 11–22.

    Google Scholar 

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  1. Department of Mathematics, Van Vleck Hall, University of Wisconsin, 53706, Madison, Wisconsin

    Michael Iltis

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  1. Michael Iltis
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Iltis, M. Sharp asymptotics of large deviations in ℝd . J Theor Probab 8, 501–522 (1995). https://doi.org/10.1007/BF02218041

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  • DOI: https://doi.org/10.1007/BF02218041

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