Metric entropy of high dimensional distributions
Authors:
Ron Blei, Fuchang Gao and Wenbo V. Li
Journal:
Proc. Amer. Math. Soc. 135 (2007), 4009-4018
MSC (2000):
Primary 60G15, 46B50.
DOI:
https://doi.org/10.1090/S0002-9939-07-08935-6
Published electronically:
September 7, 2007
MathSciNet review:
2341952
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be the collection of all
-dimensional probability distribution functions on
,
. The metric entropy of
under the
norm is studied. The exact rate is obtained for
and bounds are given for
. Connections with small deviation probability for Brownian sheets under the sup-norm are established.
- 1. Artstein, S., Milman, V., and Szarek, S. (2004). Duality of metric entropy, Ann. of Math. 159, 1313-1328. MR 2113023 (2005h:47037)
- 2. Artstein, S., Milman, V., Szarek, S. J., and Tomczak-Jaegermann, N. (2004). On Convexified Packing and Duality, Geom. and Funct. Anal. 14, 1134-1141. MR 2105957 (2005h:47038)
- 3. Carl, B. and Stephani, I. (1990). Entropy, Compactness and Approximation of Operators, Cambridge University Press, Cambridge. MR 1098497 (92e:47002)
- 4. Dunker, T., Kuhn, T., Lifshits, M. A., and Linde, W. (1999). Metric entropy of integration operators and small ball probabilities for the Brownian sheet, J. Approx. Theory 101, 63-77. MR 1724026 (2001d:60032)
- 5. EDMUNDS, D. E. and TRIEBEL, H. (1996). Function Spaces, Entropy Numbers and Differential Operators, Cambridge Univ. Press., Cambridge. MR 1410258 (97h:46045)
- 6. Gao, F. (2004). Entropy of absolute convex hulls in Hilbert spaces, Bull. London Math. Soc. 36 (2004), 460 - 468. MR 2069008 (2005e:41071)
- 7. Gao, F. and Li, W.V. (2007). Small ball probabilities for Slepian Gaussian fields, Trans. Amer. Math. Soc. 359, 1339-1350. MR 2262853
- 8.
Kolmogorov, A.N. and Tihomirov, V.M. (1961).
-entropy and
-capacity of sets in function spaces, Uspehi Mat. Nauk. 14 (1959), 3-86. English transl. in Amer. Math. Soc. Transl. 17, 277-364. MR 0112032 (22:2890)
- 9. Kuelbs, J. and Li, W.V. (1993). Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal. 116, 133-157. MR 1237989 (94j:60078)
- 10. Li, W.V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab. 27, 1556-1578. MR 1733160 (2001c:60059)
- 11. Li, W.V. and Shao, Q.-M. (2001). Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods, Handbook of Statist., Vol. 19. 533-597. MR 1861734
- 12. Lorentz, G. (1966). Metric entropy and approximation, Bull. Amer. Math. Soc. 72, 903-937. MR 0203320 (34:3173)
- 13. Talagrand, M. (1994). The small ball problem for the Brownian sheet, Ann. Probab. 22, 1331-1354. MR 1303647 (95k:60049)
- 14. van der Vaart, A. and Wellner, J. Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996. MR 1385671 (97g:60035)
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Additional Information
Ron Blei
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
Email:
blei@math.uconn.edu
Fuchang Gao
Affiliation:
Department of Mathematics, University of Idaho, Moscow, Idaho 83844
Email:
fuchang@uidaho.edu
Wenbo V. Li
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email:
wli@math.udel.edu
DOI:
https://doi.org/10.1090/S0002-9939-07-08935-6
Received by editor(s):
May 23, 2006
Received by editor(s) in revised form:
August 25, 2006, and September 19, 2006
Published electronically:
September 7, 2007
Additional Notes:
The first author was supported in part by NSF Grant DMS-0405855.
The second author was supported in part by NSF Grant DMS-0505805.
Communicated by:
Richard C. Bradley
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.