Conditional weak laws in Banach spaces
Author:
Ana Meda
Journal:
Proc. Amer. Math. Soc. 131 (2003), 25972609
MSC (2000):
Primary 60F10, 60B10, 60F05, 60G50
Published electronically:
November 27, 2002
MathSciNet review:
1974661
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a separable Banach space. Let be centered i.i.d. random vectors taking values on with law , , and let Under suitable conditions it is shown for every open and convex set that converges to zero (exponentially), where is the dominating point of As applications we give a different conditional weak law of large numbers, and prove a limiting aposteriori structure to a specific Gibbs twisted measure (in the direction determined solely by the same dominating point).
 1.
De Acosta, A. (1985). On large deviations of sums of independent random vectors. In Probability in Banach Spaces V. Lecture Notes in Math. Springer. (1153) 114. MR 87f:60035
 2.
Van Campenhout, J. M., and Cover, T. M. (1981). Maximum entropy and conditional probability. IEEE Trans. Inform. Theory. (IT27) 483489. MR 83h:94008
 3.
Csiszár, I. (1984). Sanov property, generalized projection and a conditional limit theorem. Ann. Probab. (12) 768793. MR 86h:60065
 4.
Dembo, A., and Kuelbs, J. (1998). Refined Gibbs conditioning principle for certain infinite dimensional statistics. Studia Sci. Math. Hungar. (34) 107126. MR 2000b:60068
 5.
Dembo, A., and Zeitouni, O. (1998). Large Deviation Techniques and Applications. 2nd ed. Springer, New York. MR 99d:60030
 6.
Dembo, A., and Zeitouni, O. (1996). Refinements of the Gibbs conditioning principle. Probab. Theory Related Fields. (104) 114. MR 97k:60078
 7.
Dinwoodie, I. H. (1992). Mesures dominantes et Théorème de Sanov. Ann. Inst. H. Poincaré Probab. Statist. (28) 365373. MR 93h:60038
 8.
Donsker, M. D., and Varadhan, S. R. S. (1976). Asymptotic evaluation of certain Markov processes expectations for large time III. Comm. Pure Appl. Math. (29) 389461. MR 55:1492
 9.
Einmahl, U., and Kuelbs, J. (1996). Dominating points and large deviations for random vectors. Probab. Theory Related Fields. (105) 529543. MR 97k:60008
 10.
Iscoe, I., Ney, P., and Nummelin, E. (1985). Large deviations for uniformly recurrent Markov additive processes. Adv. Appl. Math. (6) 373412. MR 88b:60077
 11.
Kuelbs, J. (2000). Large deviation probabilities and dominating points for open, convex sets: nonlogarithmic behavior. Ann. Probab. (28) 12591279. MR 2001k:60003
 12.
Lehtonen, T., and Nummelin, E. (1988). On the convergence of empirical distributions under partial observations. Ann. Acad. Sci. Fenn. Math. Ser. A.I. (13) 219223. MR 90e:60035
 13.
Lehtonen, T., and Nummelin, E. (1990). Level I theory of large deviations in the ideal gas. Int'l J. Theor. Phys. (29) 621635. MR 91f:82033
 14.
Meda, A. (1998). Conditional Laws and Dominating Points. University of Wisconsin Ph.D. Thesis.
 15.
Meda, A., and Ney, P. (1998). A conditioned law of large numbers for Markov additive chains. Studia Sci. Math. Hungar. (34) 305316.
 16.
Meda, A., and Ney, P. (1999). The Gibbs conditioning principle for Markov chains, in Perplexing Problems in Probability. M. Bramson and R. Durret, Ed., Ser. Progress in Probability. Birkhäuser, Boston. (44) 385398. MR 2001i:60037
 17.
Ney, P. (1983). Dominating points and the asymptotics of large deviations on . Ann. Probab. (11) 158167. MR 85b:60026
 18.
Ney, P. (1984). Convexity and large deviations. Ann. Probab. (12) 903906. MR 85j:60046
 19.
Nummelin, E. (1987). A conditional weak law of large numbers. In Proc. of Seminar on Stability Problems for Stoch. Models Suhumi, USSR. Lecture Notes in Math. (1142) 259262. MR 91f:60057
 1.
 De Acosta, A. (1985). On large deviations of sums of independent random vectors. In Probability in Banach Spaces V. Lecture Notes in Math. Springer. (1153) 114. MR 87f:60035
 2.
 Van Campenhout, J. M., and Cover, T. M. (1981). Maximum entropy and conditional probability. IEEE Trans. Inform. Theory. (IT27) 483489. MR 83h:94008
 3.
 Csiszár, I. (1984). Sanov property, generalized projection and a conditional limit theorem. Ann. Probab. (12) 768793. MR 86h:60065
 4.
 Dembo, A., and Kuelbs, J. (1998). Refined Gibbs conditioning principle for certain infinite dimensional statistics. Studia Sci. Math. Hungar. (34) 107126. MR 2000b:60068
 5.
 Dembo, A., and Zeitouni, O. (1998). Large Deviation Techniques and Applications. 2nd ed. Springer, New York. MR 99d:60030
 6.
 Dembo, A., and Zeitouni, O. (1996). Refinements of the Gibbs conditioning principle. Probab. Theory Related Fields. (104) 114. MR 97k:60078
 7.
 Dinwoodie, I. H. (1992). Mesures dominantes et Théorème de Sanov. Ann. Inst. H. Poincaré Probab. Statist. (28) 365373. MR 93h:60038
 8.
 Donsker, M. D., and Varadhan, S. R. S. (1976). Asymptotic evaluation of certain Markov processes expectations for large time III. Comm. Pure Appl. Math. (29) 389461. MR 55:1492
 9.
 Einmahl, U., and Kuelbs, J. (1996). Dominating points and large deviations for random vectors. Probab. Theory Related Fields. (105) 529543. MR 97k:60008
 10.
 Iscoe, I., Ney, P., and Nummelin, E. (1985). Large deviations for uniformly recurrent Markov additive processes. Adv. Appl. Math. (6) 373412. MR 88b:60077
 11.
 Kuelbs, J. (2000). Large deviation probabilities and dominating points for open, convex sets: nonlogarithmic behavior. Ann. Probab. (28) 12591279. MR 2001k:60003
 12.
 Lehtonen, T., and Nummelin, E. (1988). On the convergence of empirical distributions under partial observations. Ann. Acad. Sci. Fenn. Math. Ser. A.I. (13) 219223. MR 90e:60035
 13.
 Lehtonen, T., and Nummelin, E. (1990). Level I theory of large deviations in the ideal gas. Int'l J. Theor. Phys. (29) 621635. MR 91f:82033
 14.
 Meda, A. (1998). Conditional Laws and Dominating Points. University of Wisconsin Ph.D. Thesis.
 15.
 Meda, A., and Ney, P. (1998). A conditioned law of large numbers for Markov additive chains. Studia Sci. Math. Hungar. (34) 305316.
 16.
 Meda, A., and Ney, P. (1999). The Gibbs conditioning principle for Markov chains, in Perplexing Problems in Probability. M. Bramson and R. Durret, Ed., Ser. Progress in Probability. Birkhäuser, Boston. (44) 385398. MR 2001i:60037
 17.
 Ney, P. (1983). Dominating points and the asymptotics of large deviations on . Ann. Probab. (11) 158167. MR 85b:60026
 18.
 Ney, P. (1984). Convexity and large deviations. Ann. Probab. (12) 903906. MR 85j:60046
 19.
 Nummelin, E. (1987). A conditional weak law of large numbers. In Proc. of Seminar on Stability Problems for Stoch. Models Suhumi, USSR. Lecture Notes in Math. (1142) 259262. MR 91f:60057
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Additional Information
Ana Meda
Affiliation:
Departmento de Matemáticas, Cub. 132, Facultad de Ciencias, UNAM, Circuito Exterior s/n, Ciudad Universitaria, Coyoacán 04510, México D. F., México
Email:
amg@hp.fciencias.unam.mx
DOI:
http://dx.doi.org/10.1090/S0002993902067850
PII:
S 00029939(02)067850
Keywords:
Conditional laws,
dominating point,
large deviations,
Banach spaces
Received by editor(s):
July 16, 2000
Received by editor(s) in revised form:
March 21, 2002
Published electronically:
November 27, 2002
Additional Notes:
The author was supported in part by Grant PAPIITDGAPA IN115799 of UNAM, and the final version was written while holding a Postdoctoral position at IMP, México
Communicated by:
Claudia M. Neuhauser
Article copyright:
© Copyright 2002
American Mathematical Society
