×

Compactness in the theory of large deviations. (English) Zbl 0824.60019

Large deviation principles are expressed as the vague or narrow convergence of sequences of the set functions called capacities. As an application, a short proof of Gärtner-Ellis criterion for the large deviation principle is given. The capacity methods introduced in the paper are also used in forthcoming papers [G. L. O’Brien, Sequences of capacities, with connections to large-deviation theory, J. Theor. Probab. (to appear) and W. Bryc and A. Dembo, Large deviations and strong mixing, Ann. Inst. Henri Poincaré, Probab. Stat. (to appear)].

MSC:

60F10 Large deviations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baxter, J. R.; Jain, N. C., Convexity and compactness in large deviation theory (1993), preprint
[2] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[3] Bryc, W., Large deviations by the asymptotic value method, (Pinsky, M. A., Diffusion Processes and Related Problems in Analysis, Vol. 1 (1990), Birkhäuser: Birkhäuser Boston), 447-472
[4] de Acosta, A., Upper bounds for large deviations of dependent random vectors, Z. Wahrsch. verw. Geb., 69, 551-565 (1985) · Zbl 0547.60033
[5] Dembo, A.; Zeitouni, O., Large Deviations Techniques and Applications (1993), A.K. Peters: A.K. Peters Wellesley, MA, formerly published by Jones and Bartlett, Boston · Zbl 0793.60030
[6] Deuschel, J.-D.; Stroock, D. W., Large Deviations (1989), Academic Press: Academic Press Boston · Zbl 0682.60018
[7] Dinwoodie, I. H., Identifying a large deviation rate function, Ann. Probab., 21, 216-231 (1993) · Zbl 0777.60024
[8] Ekeland, I.; Temam, R., Convex Analysis and Variational Problems (1976), North Holland: North Holland Amsterdam
[9] Ellis, R. S., Large deviations for a general class of random vectors, Ann. Probab, 12, 1-12 (1984) · Zbl 0534.60026
[10] Ellis, R. S., Entropy, Large Deviations, and Statistical Mechanics (1985), Springer: Springer New York · Zbl 0566.60097
[11] Gärtner, J., On large deviations from the invariant measure, Theory Probab. Appl., 22, 24-39 (1977) · Zbl 0375.60033
[12] Gerrit, B., Varadhan’s theorem for capacities, (Rept. 9347 (1993), Department of Mathematics, Catholic University of Nijmegen: Department of Mathematics, Catholic University of Nijmegen Nijmegen, Netherlands)
[13] Lynch, J.; Sethuraman, J., Large deviations for processes with independent increments, Ann. Probab, 15, 610-627 (1987) · Zbl 0624.60045
[14] O’Brien, G. L., Sequences of capacities, with connections to large-deviation theory, J. Theoretical Probab. (1995), to appear in
[15] O’Brien, G. L.; Vervaat, W., Capacities, large deviations and loglog laws, (Cambanis, S.; Samorodnitsky, G.; Taqqu, M. S., Stable Processes and Related Topics (1991), Birkhäuser: Birkhäuser Boston), 43-83
[16] Pukhalskii, A., On functional principle of large deviations, (Sazonov, V.; Shervashidze, T., New Trends in Probability and Statistics (1991), VSP: VSP Moscow), 198-218 · Zbl 0767.60024
[17] Pukhalskii, A., The method of stochastic exponentials for large deviations, Stochastic Process. Appl. (1994), to appear
[18] Rockafellar, R. T., Convex Analysis (1970), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0229.90020
[19] Varadhan, S. R.S., Asymptotic probabilities and differential equations, Comm. Pure Appl. Math, 19, 261-286 (1966) · Zbl 0147.15503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.