Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-08T16:23:44.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Correction to a paper by A. G. Pakes

Part of: Distribution theory - Probability Integral transforms, operational calculus

Published online by Cambridge University Press:  09 April 2009

Christian Berg
Christian Berg
Affiliation:
Universitesparken 5DK-2100 København Ø, Denmark e-mail: berg@math.ku.dk
Rights & Permissions [Opens in a new window]

Absrtact

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Staring form a probability σ on the half-line moments of any order A. G. Pakes has defined probabilities σr, by length biasing order r and gr, by the stationary-excess operation of order r, r = 1, 2,…Examples are given to show that σ can bt determined in the Stieltjes sence while σ1 and g1 are indeterminate in the Stieltjes sence. This shows that a statement in a recent paper by Pakes does not hold.

Keywords

Indeterminate moment problemslength biasingstationary excess-operation

MSC classification

Secondary: 44A60: Moment problems 60E05: Distributions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 1 , February 2004 , pp. 67 - 74
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Akhiezer, N. I., The classical moment problem and some related questions in analysis (Oliver and Boyd, Edinburgh, 1965).Google Scholar
[2]Berg, C., ‘Recent results about moment problems’, in: Probability Measures on Groups and Related Structures XI, Priceedings Oberwolfach 1994 (ed. Heyer, H.) (World Scientific, Singapore, 1995).Google Scholar
[3]Berg, C. and Christensen, J. P. R., ‘Density questions in the classical theory of moments’, Ann. Inst. Fourier 31 (1981), 99–114.CrossRefGoogle Scholar
[4]Berg, C. and Duran, A. J., ‘The index of determinacy for measures and the l2-non of orthonormal polymials’, Trans. Amer. Math. Soc. 347 (1995), 2795–2811.Google Scholar
[5]Berg, C., ‘When does a discrete differential perturbation of a sequence oif orthonormal polynomials belong ℓ2?’, J. Funct. Anal. 136 (1996), 127–153.CrossRefGoogle Scholar
[6]Berg, C. and Thill, M., ‘A density index for the Stieltjes momemt problem’, in: Orthogonal polynomials and their applications (eds. Brezinski, C., Gori, L. and Ronveaux, A.), IMACS ann. Comput. Appl. Math. 9 (Baltzer, Basel, 1991), pp. 185–188.Google Scholar
[7]Berg, C. and Thill, M., ‘Rotation invariant moment problems’, Acta Math. 167 (1991), 207–227.CrossRefGoogle Scholar
[8]Berg, C. and Valent, G., ‘The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes’, Methods and Applications of Analysis 1 (1994), 169–209.CrossRefGoogle Scholar
[9]Berg, C. and Valent, G., ‘Nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes’, J. Comp. Appl. Math. 57 (1995), 29–43.CrossRefGoogle Scholar
[10]Ismail, M. E. H., Letessier, J., Masson, D. and Valent, G., ‘Birth and death processes and orthoginal polynomials’, in: Orthogonal polymials: theory and practice, NATO ASI series C 294 (Kluwer Academic Publishers, The Nerherlands, 1990), pp. 229–255.CrossRefGoogle Scholar
[11]Pakes, A. G., ‘Remarks on the converse Carleman and Krien criteria for the classical moment problem’, J. Austral. Math. Sco. 71 (2001), 81–104.CrossRefGoogle Scholar
[12]Valent, G., ‘Orthogonal polynomials for a quartic birth and death process’, in: Proceedings of the Granada conference 1991, J. Comput. Appl. Math. 49 (North Holland, Amsterdam, 1993), pp. 281–288.Google Scholar
[13]Valent, G., ‘Asymptotic analysis of some associated orthogonal polynomials connected with elliptic functions’, SIAM J. Math. Anal. 25 (1994), 749–775.CrossRefGoogle Scholar
[14]Valent, G., ‘Exact solutions of some quadratic and quartic birth and death processes and related orthogonal polynomials’, J. Comput. Appl. Math. 67 (1996), 103–127.CrossRefGoogle Scholar