How many Laplace transforms of probability measures are there?

Gao, Fuchang; Li, Wenbo; Wellner, Jon A.

doi:10.1090/S0002-9939-2010-10448-3

How many Laplace transforms of probability measures are there?
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by Fuchang Gao, Wenbo V. Li and Jon A. Wellner PDF

Proc. Amer. Math. Soc. 138 (2010), 4331-4344 Request permission

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

Horst Alzer and Christian Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445–460. MR 1922200
S. Artstein, V. Milman, S. Szarek, and N. Tomczak-Jaegermann, On convexified packing and entropy duality, Geom. Funct. Anal. 14 (2004), no. 5, 1134–1141. MR 2105957, DOI 10.1007/s00039-004-0486-3
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.
Keith Ball and Alain Pajor, The entropy of convex bodies with “few” extreme points, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 25–32. MR 1110183
Ron Blei, Fuchang Gao, and Wenbo V. Li, Metric entropy of high dimensional distributions, Proc. Amer. Math. Soc. 135 (2007), no. 12, 4009–4018. MR 2341952, DOI 10.1090/S0002-9939-07-08935-6
J. Bourgain, A. Pajor, S. J. Szarek, and N. Tomczak-Jaegermann, On the duality problem for entropy numbers of operators, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 50–63. MR 1008716, DOI 10.1007/BFb0090048
Bernd Carl, Metric entropy of convex hulls in Hilbert spaces, Bull. London Math. Soc. 29 (1997), no. 4, 452–458. MR 1446564, DOI 10.1112/S0024609397003044
Bernd Carl, Ioanna Kyrezi, and Alain Pajor, Metric entropy of convex hulls in Banach spaces, J. London Math. Soc. (2) 60 (1999), no. 3, 871–896. MR 1753820, DOI 10.1112/S0024610799008005
R. M. Dudley, Universal Donsker classes and metric entropy, Ann. Probab. 15 (1987), no. 4, 1306–1326. MR 905333, DOI 10.1214/aop/1176991978
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
Fuchang Gao, Entropy of absolute convex hulls in Hilbert spaces, Bull. London Math. Soc. 36 (2004), no. 4, 460–468. MR 2069008, DOI 10.1112/S0024609304003121
Fuchang Gao, Entropy estimate for $k$-monotone functions via small ball probability of integrated Brownian motion, Electron. Commun. Probab. 13 (2008), 121–130. MR 2386068, DOI 10.1214/ECP.v13-1357
FuChang Gao and Jon A. Wellner, On the rate of convergence of the maximum likelihood estimator of a $k$-monotone density, Sci. China Ser. A 52 (2009), no. 7, 1525–1538. MR 2520591, DOI 10.1007/s11425-009-0102-y
Nicholas P. Jewell, Mixtures of exponential distributions, Ann. Statist. 10 (1982), no. 2, 479–484. MR 653523
C. Krattenthaler, Advanced determinant calculus, Sém. Lothar. Combin. 42 (1999), Art. B42q, 67. The Andrews Festschrift (Maratea, 1998). MR 1701596
James Kuelbs and Wenbo V. Li, Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal. 116 (1993), no. 1, 133–157. MR 1237989, DOI 10.1006/jfan.1993.1107
Wenbo V. Li and Werner Linde, Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab. 27 (1999), no. 3, 1556–1578. MR 1733160, DOI 10.1214/aop/1022677459
Wenbo V. Li and Werner Linde, Metric entropy of convex hulls in Hilbert spaces, Studia Math. 139 (2000), no. 1, 29–45. MR 1763043
Nicole Tomczak-Jaegermann, Dualité des nombres d’entropie pour des opérateurs à valeurs dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 7, 299–301 (French, with English summary). MR 910364
Aad W. van der Vaart and Jon A. Wellner, Weak convergence and empirical processes, Springer Series in Statistics, Springer-Verlag, New York, 1996. With applications to statistics. MR 1385671, DOI 10.1007/978-1-4757-2545-2
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
R. E. Williamson, Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189–207. MR 77581, DOI 10.1215/S0012-7094-56-02317-1