Characterization of the class of upward first passage time distributions of birth and death processes and related results

Makoto YAMAZATO

doi:10.2969/jmsj/04030477

Makoto YAMAZATO

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JOURNAL FREE ACCESS

1988 Volume 40 Issue 3 Pages 477-499

DOI https://doi.org/10.2969/jmsj/04030477

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Published: 1988 Received: January 19, 1987 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) L. Bondesson, Classes of infinitely divisible distributions and densities, Z. Wahrsch. Verw. Gebiete, 57 (1981), 39-71. Correction and addendum, 59 (1982), 277.
2) B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, 1954.
3) I. A. Ibragimov, On the composition of unimodal distributions, Theory Probab. Appl., 1 (1956), 255-260.
4) S. Karlin, Total Positivity, Vol. 1, Stanford Univ. Press, 1968.
5) J. Keilson, On the unimodality of passage time densities in birth-death processes, Statist. Neerlandica, 35 (1981), 49-55.
6) J. T. Kent, Some probabilistic properties of Bessel functions, Ann. Probab., 6 (1978), 760-770.
7) J. T. Kent, The spectral decomposition of a diffusion hitting time, Ann. Probab., 10 (1982), 207-219.
8) F. Knight, Characterization of the Lévy measures of inverse local times of gap diffusion, Progress in Probability and Statistics, Vol. 1, Birkhauser, Boston, 1981, pp. 58-78.
9) S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, Lecture Notes in Mathematics, 923, Springer, 1982, pp. 235-259.
10) L. Lorch, Elementary comparison techniques for certain classes of Sturm-Liouville equations, Differential Equations (Proc. Internat. Conf., Uppsala, 1977), 125-133, Sympos. Univ. Upsaliensiss, Ann. Quingenteisimum Celebrantis 7, Almqvist and Wiksell, Stockholm, 1977.
11) U. Rösler, Unimodality of passage times for one-dimensional strong Markov processes, Ann. Probab., 8 (1980), 853-859.
12) F. W. Steutel, Preservation of Infinite Divisibility under Mixing and Related Topics, Math. Centre Tracts, 33, Amsterdam, 1970.
13) T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Univ. Toulouse, 8 (1894), 122; 9(1895), 47.
14) C. Stone, Limit theorems for random walks, birth and death processes, and diffusion processes, Illinois J. Math., 7 (1963), 638-660.
15) G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge Univ. Press, London, 1962.
16) D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, 1972.
17) M. Yamazato, On strongly unimodal infinitely divisible distributions, Ann. Probab., 10 (1982), 589-601.

Right : [1] L. Bondesson, Classes of infinitely divisible distributions and densities, Z. Wahrsch. Verw. Gebiete, 57 (1981), 39-71. Correction and addendum, 59 (1982), 277.
[2] B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, 1954.
[3] I. A. Ibragimov, On the composition of unimodal distributions, Theory Probab. Appl., 1 (1956), 255-260.
[4] S. Karlin, Total Positivity, Vol. 1, Stanford Univ. Press, 1968.
[5] J. Keilson, On the unimodality of passage time densities in birth-death processes, Statist. Neerlandica, 35 (1981), 49-55.
[6] J. T. Kent, Some probabilistic properties of Bessel functions, Ann. Probab., 6 (1978), 760-770.
[7] J. T. Kent, The spectral decomposition of a diffusion hitting time, Ann. Probab., 10 (1982), 207-219.
[8] F. Knight, Characterization of the Lévy measures of inverse local times of gap diffusion, Progress in Probability and Statistics, Vol. 1, Birkhauser, Boston, 1981, pp. 58-78.
[9] S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, Lecture Notes in Mathematics, 923, Springer, 1982, pp. 235-259.
[10] L. Lorch, Elementary comparison techniques for certain classes of Sturm-Liouville equations, Differential Equations (Proc. Internat. Conf., Uppsala, 1977), 125-133, Sympos. Univ. Upsaliensiss, Ann. Quingenteisimum Celebrantis 7, Almqvist and Wiksell, Stockholm, 1977.
[11] U. Rösler, Unimodality of passage times for one-dimensional strong Markov processes, Ann. Probab., 8 (1980), 853-859.
[12] F. W. Steutel, Preservation of Infinite Divisibility under Mixing and Related Topics, Math. Centre Tracts, 33, Amsterdam, 1970.
[13] T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Univ. Toulouse, 8 (1894), 122; 9 (1895), 47.
[14] C. Stone, Limit theorems for random walks, birth and death processes, and diffusion processes, Illinois J. Math., 7 (1963), 638-660.
[15] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge Univ. Press, London, 1962.
[16] D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, 1972.
[17] M. Yamazato, On strongly unimodal infinitely divisible distributions, Ann. Probab., 10 (1982), 589-601.

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