On generating functions of Hausdorff moment sequences
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- by Jian-Guo Liu and Robert L. Pego PDF
- Trans. Amer. Math. Soc. 368 (2016), 8499-8518 Request permission
Abstract:
The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-\infty ,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$. Also we provide a simple analytic proof that for any real $p$ and $r$ with $p>0$, the Fuss-Catalan or Raney numbers $\frac {r}{pn+r}\binom {pn+r}{n}$, $n=0,1,\ldots$, are the moments of a probability distribution on some interval $[0,\tau ]$ if and only if $p\ge 1$ and $p\ge r\ge 0$. The same statement holds for the binomial coefficients $\binom {pn+r-1}n$, $n=0,1,\ldots$.References
- N. Alexeev, F. Götze, and A. Tikhomirov, Asymptotic distribution of singular values of powers of random matrices, Lith. Math. J. 50 (2010), no. 2, 121–132. MR 2653641, DOI 10.1007/s10986-010-9074-4
- Julius Bendat and Seymour Sherman, Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), 58–71. MR 82655, DOI 10.1090/S0002-9947-1955-0082655-4
- Persi Diaconis and David Freedman, The Markov moment problem and de Finetti’s theorem. I, Math. Z. 247 (2004), no. 1, 183–199. MR 2054525, DOI 10.1007/s00209-003-0636-6
- William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556, DOI 10.1007/978-3-642-65755-9
- William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
- 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
- Alexander Gnedin and Jim Pitman, Moments of convex distribution functions and completely alternating sequences, Probability and statistics: essays in honor of David A. Freedman, Inst. Math. Stat. (IMS) Collect., vol. 2, Inst. Math. Statist., Beachwood, OH, 2008, pp. 30–41. MR 2459948, DOI 10.1214/193940307000000374
- Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
- Uffe Haagerup and Sören Möller, The law of large numbers for the free multiplicative convolution, Operator algebra and dynamics, Springer Proc. Math. Stat., vol. 58, Springer, Heidelberg, 2013, pp. 157–186. MR 3142036, DOI 10.1007/978-3-642-39459-1_{8}
- B. G. Hansen and F. W. Steutel, On moment sequences and infinitely divisible sequences, J. Math. Anal. Appl. 136 (1988), no. 1, 304–313. MR 972601, DOI 10.1016/0022-247X(88)90133-3
- Wojciech Młotkowski, Fuss-Catalan numbers in noncommutative probability, Doc. Math. 15 (2010), 939–955. MR 2745687, DOI 10.1016/j.cnsns.2009.05.004
- Wojciech Młotkowski and Karol A. Penson, Probability distributions with binomial moments, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), no. 2, 1450014, 32. MR 3212684, DOI 10.1142/S0219025714500143
- Wojciech Młotkowski, Karol A. Penson, and Karol Życzkowski, Densities of the Raney distributions, Doc. Math. 18 (2013), 1573–1596. MR 3158243
- K. A. Penson and K. \ifmmode Ż\elseŻyczkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions, Phys. Rev. E 83 (2011), 061118.
- Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, $A=B$, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk. MR 1379802
- John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
- Oliver Roth, Stephan Ruscheweyh, and Luis Salinas, A note on generating functions for Hausdorff moment sequences, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3171–3176. MR 2407081, DOI 10.1090/S0002-9939-08-09460-4
- Stephan Ruscheweyh, Luis Salinas, and Toshiyuki Sugawa, Completely monotone sequences and universally prestarlike functions, Israel J. Math. 171 (2009), 285–304. MR 2520111, DOI 10.1007/s11856-009-0050-9
- René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2010. Theory and applications. MR 2598208
- Thomas Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab. 19 (2014), no. 16, 25. MR 3164769, DOI 10.1214/EJP.v19-3058
- F. W. Steutel, Note on completely monotone densities, Ann. Math. Statist. 40 (1969), 1130–1131. MR 254949, DOI 10.1214/aoms/1177697626
- Fred W. Steutel and Klaas van Harn, Infinite divisibility of probability distributions on the real line, Monographs and Textbooks in Pure and Applied Mathematics, vol. 259, Marcel Dekker, Inc., New York, 2004. MR 2011862
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
Additional Information
- Jian-Guo Liu
- Affiliation: Departments of Physics and Mathematics, Duke University, Durham, North Carolina 27708
- MR Author ID: 233036
- ORCID: 0000-0002-9911-4045
- Email: jliu@phy.duke.edu
- Robert L. Pego
- Affiliation: Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
- MR Author ID: 137455
- ORCID: 0000-0001-8502-2820
- Email: rpego@cmu.edu
- Received by editor(s): January 29, 2014
- Received by editor(s) in revised form: February 26, 2014, and October 13, 2014
- Published electronically: February 2, 2016
- Additional Notes: This material is based upon work supported by the National Science Foundation under grants DMS 1211161 and RNMS11-07444 (KI-Net) and partially supported by the Center for Nonlinear Analysis (CNA) under National Science Foundation grant 0635983.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8499-8518
- MSC (2010): Primary 44A60; Secondary 60E99, 62E10, 05A15
- DOI: https://doi.org/10.1090/tran/6618
- MathSciNet review: 3551579