The Annals of Probability

Large deviation probabilities and dominating points for open convex sets: nonlogarithmic behavior

J. Kuelbs

Full-text: Open access

Abstract

The existence of a dominating point for an open convex set and a corresponding representation formula for large deviation probabilities are established in the infinite-dimensional setting under conditions which are both necessary and sufficient and follow from those used previously in $\mathbb{R}^d$ . A precise nonlogarithmic estimate of large deviation probabilities applicable to Gaussian measures is also included.

Article information

Source
Ann. Probab. Volume 28, Number 3 (2000), 1259-1279.

Dates
First available: 18 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1019160334

Mathematical Reviews number (MathSciNet)
MR1797312

Digital Object Identifier
doi:10.1214/aop/1019160334

Zentralblatt MATH identifier
1023.60006

Subjects
Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F10: Large deviations

Keywords
Large deviation probabilities dominating points for open convex sets nonlogarithmic behavior

Citation

Kuelbs, J. Large deviation probabilities and dominating points for open convex sets: nonlogarithmic behavior. The Annals of Probability 28 (2000), no. 3, 1259--1279. doi:10.1214/aop/1019160334. http://projecteuclid.org/euclid.aop/1019160334.


Export citation

References

  • [1] Bahadur, R. R. and Zabell, S. (1979). Large deviations of the sample mean in general vector spaces. Ann. Probab. 7 587-621.
  • [2] de Acosta, A. and Kuelbs, J. (1983). Limit theorems for moving averages of independent random variables.Wahrsch. Verw. Gebiete 64 67-123.
  • [3] de Acosta, A. and Ney, P. (1998). Large deviation lower bounds for arbitrary additive functionals of a Markovchain. Ann. Probab. 26 1660-1682.
  • [4] Dembo, A. and Kuelbs, J. (1998). Refined Gibbs conditioning principle for certain infinite dimensional statistics. Studia Sci. Math. Hungar. 34 107-126.
  • [5] Dembo, A. and Zeitouni, O. (1993). Large Deviation Techniques and Applications. Jones and Bartlett, Boston.
  • [6] Deuschel, J. and Stroock, D. (1989). Large Deviations. Academic Press, Boston.
  • [7] Donsker, M. D. and Varadhan, S. R. S. (1976). Asymptotic evaluation of certain Markov process expectations for large time III. Comm. Pure Appl. Math. 29 389-461.
  • [8] Dunford, N. and Schwartz, J. T. (1964). Linear Operators I. Interscience, New York.
  • [9] Einmahl, U. and Kuelbs, J. (1996). Dominating points and large deviations for random vectors. Probab. Theory Related Fields 105 529-543.
  • [10] Ney, P. (1983). Dominating points and the asymptotics of large deviations for random walks on d. Ann. Probab. 11 158-167.
  • [11] Ney, P. (1984). Convexity and large deviations. Ann. Probab. 12 903-906.
  • [12] Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
  • [13] Schaefer, H. H. (1970). Topological Vector Spaces. Springer, Berlin.