Moderate deviations and associated Laplace approximations for sums of independent random vectors
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- by A. de Acosta PDF
- Trans. Amer. Math. Soc. 329 (1992), 357-375 Request permission
Abstract:
Let $\{ {X_j}\}$ be an i.i.d. sequence of Banach space valued r.v.’s and let ${S_n} = \sum \nolimits _{j = 1}^n {{X_j}}$. For certain positive sequences ${b_n} \to \infty$, we determine the exact asymptotic behavior of $E{\operatorname {exp}}\{ (b_n^2/n)\Phi ({S_n}/{b_n})\}$, where $\Phi$ is a smooth function. We also prove a large deviation principle for $\{ \mathcal {L}({S_n}/{b_n})\}$.References
- A. de Acosta, Upper bounds for large deviations of dependent random vectors, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 4, 551–565. MR 791911, DOI 10.1007/BF00532666
- A. de Acosta, On large deviations of sums of independent random vectors, Probability in Banach spaces, V (Medford, Mass., 1984) Lecture Notes in Math., vol. 1153, Springer, Berlin, 1985, pp. 1–14. MR 821973, DOI 10.1007/BFb0074942
- Alejandro de Acosta, Aloisio Araujo, and Evarist Giné, On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces, Probability on Banach spaces, Adv. Probab. Related Topics, vol. 4, Dekker, New York, 1978, pp. 1–68. MR 515429
- Alejandro de Acosta and Evarist Giné, Convergence of moments and related functionals in the general central limit theorem in Banach spaces, Z. Wahrsch. Verw. Gebiete 48 (1979), no. 2, 213–231. MR 534846, DOI 10.1007/BF01886874
- A. de Acosta and J. Kuelbs, Some results on the cluster set $C(\{S_{n}/a_{n}\})$ and the LIL, Ann. Probab. 11 (1983), no. 1, 102–122. MR 682803
- Aloisio Araujo and Evarist Giné, The central limit theorem for real and Banach valued random variables, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 576407
- R. Azencott, Grandes déviations et applications, Eighth Saint Flour Probability Summer School—1978 (Saint Flour, 1978), Lecture Notes in Math., vol. 774, Springer, Berlin, 1980, pp. 1–176 (French). MR 590626
- R. R. Bahadur and S. L. Zabell, Large deviations of the sample mean in general vector spaces, Ann. Probab. 7 (1979), no. 4, 587–621. MR 537209
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- E. Bolthausen, Laplace approximations for sums of independent random vectors, Probab. Theory Relat. Fields 72 (1986), no. 2, 305–318. MR 836280, DOI 10.1007/BF00699109
- E. Bolthausen, Laplace approximations for sums of independent random vectors. II. Degenerate maxima and manifolds of maxima, Probab. Theory Related Fields 76 (1987), no. 2, 167–206. MR 906774, DOI 10.1007/BF00319983
- Christer Borell, Gaussian Radon measures on locally convex spaces, Math. Scand. 38 (1976), no. 2, 265–284. MR 436303, DOI 10.7146/math.scand.a-11634
- A. A. Borovkov and A. A. Mogul′skiĭ, Probabilities of large deviations in topological spaces. I, Sibirsk. Mat. Zh. 19 (1978), no. 5, 988–1004, 1213 (Russian). MR 508496
- M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. III, Comm. Pure Appl. Math. 29 (1976), no. 4, 389–461. MR 428471, DOI 10.1002/cpa.3160290405
- Richard S. Ellis, Entropy, large deviations, and statistical mechanics, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 271, Springer-Verlag, New York, 1985. MR 793553, DOI 10.1007/978-1-4613-8533-2
- William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1984. Translated from the Russian by Joseph Szücs. MR 722136, DOI 10.1007/978-1-4684-0176-9
- Anders Martin-Löf, A Laplace approximation for sums of independent random variables, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 101–115. MR 643791, DOI 10.1007/BF00575528
- V. V. Petrov, Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown. MR 0388499
- 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
- V. V. Yurinskiĭ, Exponential inequalities for sums of random vectors, J. Multivariate Anal. 6 (1976), no. 4, 473–499. MR 428401, DOI 10.1016/0047-259X(76)90001-4
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 329 (1992), 357-375
- MSC: Primary 60F10; Secondary 60B12
- DOI: https://doi.org/10.1090/S0002-9947-1992-1046015-4
- MathSciNet review: 1046015