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How many Laplace transforms of probability measures are there?


Authors: Fuchang Gao, Wenbo V. Li and Jon A. Wellner
Journal: Proc. Amer. Math. Soc. 138 (2010), 4331-4344
MSC (2010): Primary 46B50, 60G15, 60G52; Secondary 62G05
DOI: https://doi.org/10.1090/S0002-9939-2010-10448-3
Published electronically: May 24, 2010
MathSciNet review: 2680059
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Abstract | References | Similar Articles | Additional Information

Abstract: A bracketing metric entropy bound for the class of Laplace transforms of probability measures on $ [0,\infty)$ is obtained through its connection with the small deviation probability of a smooth Gaussian process. Our results for the particular smooth Gaussian process seem to be of independent interest.


References [Enhancements On Off] (What's this?)

  • 1. ALZER, H. and BERG, C. (2002).
    Some classes of completely monotonic functions.
    Ann. Acad. Sci. Fenn. Math. 27 445-460. MR 1922200 (2003e:26013)
  • 2. ARTSTEIN, S., MILMAN, V., SZAREK, S. and TOMCZAK-JAEGERMANN, N. (2004).
    On convexified packing and entropy duality.
    Geom. Funct. Anal. 14 1134-1141. MR 2105957 (2005h:47038)
  • 3. AURZADA, F., IBRAGIMOV, I., LIFSHITS, M. and VAN ZANTEN J.H. (2009).
    Small deviations of smooth stationary Gaussian processes.
    Theory of Probability and Its Applications 53 697-707.
  • 4. BALL, K. and PAJOR, A. (1990).
    The entropy of convex bodies with ``few'' extreme points.
    In Geometry of Banach spaces (Strobl, 1989), vol. 158 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 25-32. MR 1110183 (93b:46024)
  • 5. BLEI, R., GAO, F. and LI, W. V. (2007).
    Metric entropy of high dimensional distributions.
    Proc. Amer. Math. Soc. 135 4009-4018. MR 2341952 (2008g:60010)
  • 6. BOURGAIN, J., PAJOR, A., SZAREK, S. J. and TOMCZAK-JAEGERMANN, N. (1989).
    On the duality problem for entropy numbers of operators.
    In Geometric aspects of functional analysis (1987-88), vol. 1376 of Lecture Notes in Math., Springer, Berlin, 50-63. MR 1008716 (90k:47043)
  • 7. CARL, B. (1997).
    Metric entropy of convex hulls in Hilbert spaces.
    Bull. London Math. Soc. 29 452-458. MR 1446564 (98g:46023)
  • 8. CARL, B., KYREZI, I. and PAJOR, A. (1999).
    Metric entropy of convex hulls in Banach spaces.
    J. London Math. Soc. (2) 60 871-896. MR 1753820 (2001c:46019)
  • 9. DUDLEY, R. M. (1987).
    Universal Donsker classes and metric entropy.
    Ann. Probab. 15 1306-1326. MR 905333 (88g:60081)
  • 10. FELLER, W. (1971).
    An introduction to probability theory and its applications. Vol. II.
    Second edition, John Wiley & Sons Inc., New York. MR 0270403 (42:5292)
  • 11. GAO, F. (2004).
    Entropy of absolute convex hulls in Hilbert spaces.
    Bull. London Math. Soc. 36 460-468. MR 2069008 (2005e:41071)
  • 12. GAO, F. (2008).
    Entropy estimate for $ k$-monotone functions via small ball probability of integrated Brownian motion.
    Electron. Commun. Probab. 13 121-130. MR 2386068 (2008m:60063)
  • 13. GAO, F. and WELLNER, J. A. (2009).
    On the rate of convergence of the maximum likelihood estimator of a $ k$-monotone density.
    Science in China, Series A: Mathematics 52 1525-1538. MR 2520591
  • 14. JEWELL, N. P. (1982).
    Mixtures of exponential distributions.
    Ann. Statist. 10 479-484. MR 653523 (83f:62057)
  • 15. KRATTENTHALER, C. (1999).
    Advanced determinant calculus.
    Sém. Lothar. Combin. 42 Art. B42q, 67 pp. (electronic).
    The Andrews Festschrift (Maratea, 1998). MR 1701596 (2002i:05013)
  • 16. 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)
  • 17. 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)
  • 18. LI, W. V. and LINDE, W. (2000).
    Metric entropy of convex hulls in Hilbert spaces.
    Studia Math. 139 29-45. MR 1763043 (2001h:60063)
  • 19. TOMCZAK-JAEGERMANN, N. (1987).
    Dualité des nombres d'entropie pour des opérateurs à valeurs dans un espace de Hilbert.
    C. R. Acad. Sci. Paris Sér. I Math. 305 299-301. MR 910364 (89c:47027)
  • 20. VAN DER VAART, A. W. and WELLNER, J. A. (1996).
    Weak convergence and empirical processes.
    With applications to statistics.
    Springer Series in Statistics, Springer-Verlag, New York. MR 1385671 (97g:60035)
  • 21. WIDDER, D. V. (1941).
    The Laplace Transform.
    Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J. MR 0005923 (3:232d)
  • 22. WILLIAMSON, R. E. (1956).
    Multiply monotone functions and their Laplace transforms.
    Duke Math. J. 23 189-207. MR 0077581 (17:1061d)

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Additional Information

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

Jon A. Wellner
Affiliation: Department of Statistics, University of Washington, Seattle, Washington 98195
Email: jaw@stat.washington.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10448-3
Keywords: Laplace transform, bracketing metric entropy, completely monotone functions, smooth Gaussian process, small deviation probability
Received by editor(s): September 15, 2009
Received by editor(s) in revised form: February 2, 2010
Published electronically: May 24, 2010
Additional Notes: The second author was supported in part by NSF grant DMS-0805929
The third author was supported in part by NSF Grant DMS-0804587 and NIH/NIAID Grant 5 R37 A1029168
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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