Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Total positivity, absorption probabilities and applications


Author: Samuel Karlin
Journal: Trans. Amer. Math. Soc. 111 (1964), 33-107
MSC: Primary 60.40; Secondary 60.62
DOI: https://doi.org/10.1090/S0002-9947-1964-0168010-2
MathSciNet review: 0168010
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [1] G. Baxter and M. Donsker, On the supremum functional for processes with stationary independent increments, Trans. Amer. Math. Soc. 85 (1957), 73-88. MR 0084900 (18:944c)
  • [2] S. Bochner, Harmonic analyses and the theory of probability, Univ. of California Press, Berkeley, Calif., 1955.
  • [3] K. L. Chung, Markoff chains, Springer, Berlin, 1960. MR 0116388 (22:7176)
  • [4] D. A. Darling and A. J. F. Siegart, The first passage problem for a continuous Markoff process, Ann. Math. Statist. 24 (1953), 624-639. MR 0058908 (15:449f)
  • [5] E. B. Dynkin, One dimensional continuous strong Markov processes, Theory of Probability and its Applications 4 (1959), 1-53. (Russian) MR 0121850 (22:12580)
  • [6] A. Edrei, On the generating functions of totally positive sequences, J. Analyse Math. 2 (1952), 104-109. MR 0053175 (14:732e)
  • [7] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77 (1954), 1-31. MR 0063607 (16:150d)
  • [8] -, On second order differential operators, Ann. of Math. (2) 61 (1951), 90-105. MR 0068082 (16:824g)
  • [9] F. Gantmacher and M. Krein, Oscillation matrices and vibrations of mechanical systems, Moscow, 1950.
  • [10] W. Hahn, Über Orthogonalpolynome, die p-differenzgleichungen genugen, Math. Nachr. 2 (1949), 4-34. MR 0030647 (11:29b)
  • [11] I. I. Hirschman, Hankel transforms and variation diminishing kernels, Bull. Amer. Math. Soc. 66 (1960), 40-42. MR 0110926 (22:1794)
  • [12] I. I. Hirschman and D. V. Widder, The convolution transform, Princeton Univ. Press, Princeton, N. J., 1955. MR 0073746 (17:479c)
  • [13] S. Karlin, Decision theory for Pólya type distributions, case of two actions. I, Proc. Third Berkeley Symposium on Probability and Statistics, Vol. 1, pp. 115-129. Univ. of California Press, Berkeley, Calif., 1956. MR 0084917 (18:947c)
  • [14] -, Pólya type distributions. II, Ann. Math. Statist. 28 (1957), 281-308. MR 0088148 (19:475a)
  • [15] -, Total positivity and applications to probability and statistics, Stanford Univ. Press, Stanford, Calif., (to appear).
  • [16] S. Karlin and J. L. McGregor, The differential equations of birth and death processes and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957), 489-546. MR 0091566 (19:989d)
  • [17] -, The classification of birth and death processes, Trans. Amer. Math. Soc. 86 (1957), 366-401. MR 0094854 (20:1363)
  • [18] -, Classical diffusion processes and total positivity, J. Math. Anal. Appl. 1 (1960) 163-183. MR 0121844 (22:12574)
  • [19] -, Random walks, Illinois J. Math. 3 (1959), 66-81. MR 0100927 (20:7352)
  • [20] -, Coincidence probabilities, Pacific J. Math. 9 (1959), 1141-1164. MR 0114248 (22:5072)
  • [21] -, Coincidence properties of birth and death processes, Pacific J. Math. 9 (1959), 1109-1140. MR 0114247 (22:5071)
  • [22] -, A characterization of the birth and death processes, Proc. Nat. Acad. Sci. U. S. A. 45 (1959), 375-379. MR 0102129 (21:923)
  • [23] -, Total positivity of fundamental solutions of parabolic equations, Proc. Amer. Math. Soc. 13 (1962), 136-139. MR 0132925 (24:A2761)
  • [24] S. Karlin and F. Proschan, Pólya type distributions of convolutions, Ann. Math. Statist. 31 (1960), 721-736. MR 0150797 (27:784)
  • [25] S. Karlin and H. Rubin, The theory of decision procedures for distributions with monotone ikelihood ratio, Ann. Math. Statist. 27 (1956), 272-299. MR 0081593 (18:425f)
  • [26] S. Karlin and G. Szegö, Determinant properties of certain orthogonal polynomials, J. Analyse Math. (1961).
  • [27] P. Levy, Processus stochastiques et mouvement Brownian, Gauthier-Villars, Paris, 1948. MR 0190953 (32:8363)
  • [28] H. P. McKean, Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc. 82 (1956), 519-548. MR 0087012 (19:285b)
  • [29] S. Orey, Sums arising in the theory of Markoff chains, Univ. of Minnesota, Tech. Rep. 3, 1961.
  • [30] G. Pólya and G. Szegö, Aufgaben und Lehrasatze aus der Analysis, Vols. I, II, Berlin, 1925.
  • [31] G. Black and F. Proschan, On optimal redundancy, Operations Res. 7 (1959), 581-588. MR 0108389 (21:7105)
  • [32] I. J. Schoenberg, On smoothing operations and their generating functions, Bull. Amer. Math. Soc. 59 (1953), 199-230. MR 0056042 (15:16d)
  • [33] -, On Pólya frequency functions, J. Analyse Math. 1 (1951), 331-374. MR 0047732 (13:923c)
  • [34] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. Vol. 23, Amer. Math. Soc., Providence, R. I., 1939.
  • [35] K. L. Chung and P. Erdös, Probability limit theorems assuming only the first moment. I, Mem. Amer. Math. Soc. No. 6 (1951), 19 pp. MR 0040612 (12:722g)
  • [36] G. Pólya, Festschrift, Stanford Univ. Press, Stanford, Calif., 1962.
  • [37] S. Orey, Strong ratio limit property, Bull. Amer. Math. Soc. 67 (1961), 571. MR 0132600 (24:A2440)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60.40, 60.62

Retrieve articles in all journals with MSC: 60.40, 60.62


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1964-0168010-2
Article copyright: © Copyright 1964 American Mathematical Society

American Mathematical Society