Arkiv för Matematik

  • Ark. Mat.
  • Volume 42, Number 2 (2004), 239-257.

A transformation from Hausdorff to Stieltjes moment sequences

Christian Berg and Antonio J. Durán

Full-text: Open access

Abstract

We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T[(an) ${}_{n=1}^{∞}$ ]n = 1/a1 ... an. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to q-series.

Note

This work was done while the first author was visiting University of Sevilla supported by the Secretaría de Estado de Educación y Universidades, Ministerio de Ciencia, Cultura y Deporte de España, SAB2000-0142. The work of the second author has been supported by DGES ref. BFM-2000-206-C04-02 and FQM 262 (Junta de Andalucía).

Article information

Source
Ark. Mat., Volume 42, Number 2 (2004), 239-257.

Dates
Received: 8 April 2003
Revised: 30 May 2003
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898852

Digital Object Identifier
doi:10.1007/BF02385478

Mathematical Reviews number (MathSciNet)
MR2101386

Zentralblatt MATH identifier
1057.44002

Rights
2004 © Institut Mittag-Leffler

Citation

Berg, Christian; Durán, Antonio J. A transformation from Hausdorff to Stieltjes moment sequences. Ark. Mat. 42 (2004), no. 2, 239--257. doi:10.1007/BF02385478. https://projecteuclid.org/euclid.afm/1485898852


Export citation

References

  • Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Hafner Publ., New York, 1965.
  • Berg, C., Correction to a paper by A. G. Pakes, J. Austral. Math. Soc 76 (2004), 67–73.
  • Berg, C., On a generalized Gamma convolution related to the q-calculus, in Theory and Applications of Special Functions. (Ismail, M. E. H. and Koelink, E., eds.), Kluwer, Dordrecht, 2004.
  • Berg, C., On powers of Stieltjes moment sequences I, Submitted.
  • Berg, C., Christensen, J. P. R. and Ressel, P., Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions, Graduate Texts in Math. 100, Springer-Verlag, New York, 1984.
  • Berg, C. and Forst, G., Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Math. und ihrer Grenzgebiete 87, Springer-Verlag, New York-Heidelberg, 1975.
  • Berg, C. and Thill, M., Rotation invariant moment problems, Acta Math. 167 (1991), 207–227.
  • Berg, C. and Valent, G., The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods Appl. Anal. 1 (1994), 169–209.
  • Bertoin, J., Lévy Processes. Cambridge Tracts in Math. 121, Cambridge Univ. Press, Cambridge, 1996.
  • Bertoin, J., Biane, P. and Yor, M., Poissonian exponential functionals, q-series, q-integrals, and the moment problem for the log-normal distribution. To appear in Progress in Probability, Birkhäuser, Basel-Boston, 2004.
  • Bertoin, J. and Yor, M., On subordinators, self-similar Markov processes and some factorizations of the exponential variable, Electron. Comm. Probab. 6 (2001), 95–106.
  • Bertoin, J. and Yor, M., On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes, Ann. Fac. Sci. Toulouse Math. 11 (2002), 33–45.
  • Carmona, P., Petit, F. and Yor, M., Sur les fonctionelles exponentielles de certains processus de Lévy, Stochastics Stochastics Rep. 47 (1994), 71–101.
  • Carmona, P., Petit, F. and Yor, M., On the distribution and asymptotic results for exponential functionals of Lévy processes, in Exponential Functionals and Principal Values Related to Brownian Motion, Rev. Mat. Iberoamericana, Madrid, pp. 73–130, 1997.
  • Chihara, T. S., Indeterminate symmetric moment problems, J. Math. Anal. Appl. 85 (1982), 331–346.
  • Christiansen, J. S., The moment problem associated with the Stieltjes-Wigert polynomials, J. Math. Anal. Appl. 277 (2003), 218–245.
  • Euler, L., Introductio in analysin infinitorum, Book I, Marcum-Michaelem Bousquet & Socios, Lausanne, 1748. English transl.: Introduction to Analysis of the Infinite, Book I, Springer-Verlag, New York, 1988.
  • Gasper, G. and Rahman, M., Basic Hypergeometric Series. Encyclopedia of Math. and its Appl. 35, Cambridge Univ. Press, Cambridge, 1990.
  • Hausdorff, F., Momentprobleme für ein endliches Intervall, Math. Z. 16 (1923), 220–248.
  • Jacobsen, M. and Yor, M., Multi-self-similar Markov processes on R ${}_{+}^{n}$ and their Lamperti representations, Probab. Theory Related Fields 126 (2003), 1–28.
  • Lamperti, J., Semi-stable Markov processes, Z. Wahrsch. Verw. Gebiete 22 (1972), 205–225.
  • Shohat, J. A. and Tamarkin, J. D., The Problem of Moments, Amer. Math. Soc. Math. Surveys, 2, Amer. Math. Soc., New York, 1943.
  • Stieltjes, T.-J., Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8 (1894), J1-J122; 9 (1895), A5–A47.
  • Urbanik, K., Functionals on transient stochastic processes with independent increments, Studia Math. 103 (1992), 299–315.
  • Widder, D. V., The Laplace Transform, Princeton Math. Ser. 6, Princeton Univ. Press, Princeton, NJ, 1941.