Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times

Electronic Journal of Probability

Volumn 4, Issue , 1999, Pages 1-33

  Pitman, Jim ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Alternating exponential random walk; Bessel polynomials; Critical binary random tree; Generalized Stirling numbers; Hermite fun; Mcdonald function; Products of gamma variables; Random partitions; Uniform order statistics; Vervaat s transformation

Indexed keywords

EID: 84978648707     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v4-48     Document Type: Article

References (80)

1. M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover, New York, 1965.
2. D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487-512, 1994.
3. D. Aldous and J. Pitman. A family of random trees with random edge lengths. Technical Report 526, Dept. Statistics, U.C. Berkeley, 1998. Available via http://www.stat.berkeley.edu/users/pitman.
4. D. J. Aldous and J. Pitman. Brownian trees and excursions. In preparation, 1999.
5. D.J. Aldous. The continuum random tree I. Ann. Probab., 19:1-28, 1991.
6. D.J. Aldous. The continuum random tree II: an overview. In M.T. Barlow and N.H. Bingham, editors, Stochastic Analysis, pages 23-70. Cambridge University Press, 1991.
7. D.J. Aldous. The continuum random tree III. Ann. Probab., 21:248-289, 1993.
8. D.J. Aldous and J. Pitman. The standard additive coalescent. Ann. Probab., 26:1703-1726, 1998.
9. D. J. Aldous and J. Pitman. Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. Henri Poincaré, 34:637-686, 1998.
10. L. Alili, C. Donati-Martin, and M. Yor. Une identite en loi remarquable pour l’excursion brownienne normalisee. In M. Yor, editor, Exponential functionals and principal values related to Brownian motion, pages 155-180. Biblioteca de la Revista Matematica Ibero-Americana, 1997.
11. J. Bertoin. Decomposition du mouvement brownien avec derive en un minimum local par juxtaposition de ses excursions positives et negatives. In Séminaire de Probabilités XXV, pages 330-344. SpringerVerlag, 1991. Lecture Notes in Math. 1485.
12. J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147-166, 1994.
13. Ph. Biane. Relations entre pont et excursion du mouvement Brownien reel. Ann. Inst. Henri Poincaré, 22:1-7, 1986.
14. Ph. Biane, J. F. Le Gall, and M. Yor. Un processus qui ressemble au pont brownien. In Séminaire de Probabilités XXI, pages 270-275. Springer, 1987. Lecture Notes in Math. 1247.
15. Ph. Biane and M. Yor. Quelques precisions sur le meandre brownien. Bull. Sci. Math., 112:101-109, 1988.
16. P. Billingsley. Probability and Measure. Wiley, New York, 1995. 3rd ed.
17. R. M. Blumenthal. Weak convergence to Brownian excursion. Ann. Probab., 11:798-800, 1983.
18. E. Bolthausen. On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab., 4:480-485, 1976.
19. L. Bondesson. Generalized gamma convolutions and related classes of distributions and densities. Springer-Verlag, 1992.
20. Ch. A. Charalambides and J. Singh. A review of the Stirling numbers, their generalizations and statistical applications. Commun. Statist.-Theory Meth., 17:2533-2595, 1988.
21. K. L. Chung. Excursions in Brownian motion. Arkiv fur Matematik, 14:155-177, 1976.
22. L. Comtet. Advanced Combinatorics. D. Reidel Pub. Co., Boston, 1974. (translated from French).
23. I. V. Denisov. A random walk and a Wiener process near a maximum. Theor. Prob. Appl., 28:821-824, 1984.
24. S. Dulucq and L. Favreau. A combinatorial model for Bessel polynomials. In Orthogonal polynomials and their applications (Erice, 1990), pages 243-249. Baltzer, Basel, 1991.
25. A. Erdelyi et al. Higher Transcendental Functions, volume II of Bateman Manuscript Project. McGrawHill, New York, 1953.
26. W. Feller. An Introduction to Probability Theory and its Applications, Vol 1, 3rd ed. Wiley, New York, 1968.
27. W. Feller. An Introduction to Probability Theory and its Applications, Vol 2, 2nd ed. Wiley, New York, 1971.
28. P. Fitzsimmons, J. Pitman, and M. Yor. Markovian bridges: construction, Palm interpretation, and splicing. In E. Qinlar, K.L. Chung, and M.J. Sharpe, editors, Seminar on Stochastic Processes, 1992, pages 101-134. Birkhauser, Boston, 1993.
29. R. K. Getoor. The Brownian escape process. Ann. Probab., 7:864-867, 1979.
30. L. Gordon. Bounds for the distribution of the generalized variance. The Annals of Statistics, 17:1684-1692, 1989.
31. L. Gordon. A stochastic approach to the gamma function. Amer. Math. Monthly, 101:858-865, 1994.
32. R. L. Graham, D.E. Knuth, and O. Patashnik. Concrete Mathematics: a foundation for computer science. 2nd ed. Addison-Wesley, Reading, Mass., 1989.
33. P. Greenwood and J. Pitman. Fluctuation identities for random walk by path decomposition at the maximum. Advances in Applied Probability, 12:291-293, 1978.
34. P. Greenwood and J. Pitman. Fluctuation identities for Levy processes and splitting at the maximum. Advances in Applied Probability, 12:893-902, 1980.
35. E. Grosswald. The Student t-distribution for odd degrees of freedom is infinitely divisible. Ann. Probability, 4(4):680-683, 1976.
36. E. Grosswald. The Student t-distribution of any degree of freedom is infinitely divisible. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36(2):103-109, 1976.
37. E. Grosswald. Bessel polynomials. Springer, Berlin, 1978.
38. L. C. Hsu and P. J.-S. Shiue. A unified approach to generalized Stirling numbers. Adv. in Appl. Math., 20(3):366-384, 1998.
39. I. A. Ibragimov and Yu. V. Linnik. Independent and Stationary Sequences of Random Variables. Groningen, Wolthers-Noordhof, 1971. [Translated from original in Russian (1965), Nauka, Moscow].
40. D.L. Iglehart. Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab., 2:608-619, 1974.
41. J. P. Imhof. Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab., 21:500-510, 1984.
42. M. E. H. Ismail and J. Pitman. Algebraic evaluations of some Euler integrals, duplication formulae for Appell’s hypergeometric function F\, and Brownian variations. Technical Report 554, Dept. Statistics, U.C. Berkeley, 1999. Available via http://www.stat.berkeley.edu/users/pitman.
43. K. Ito and H. P. McKean. Diffusion Processes and their Sample Paths. Springer, 1965.
44. N.L. Johnson, S. Kotz, and N. Balakrishnan. Discrete Multivariate Distributions. Wiley, New York, 1997.
45. W.D. Kaigh. An invariance principle for random walk conditioned by a late return to zero. Ann. Probab., 4:115-121, 1976.
46. G. Kersting. Symmetry properties of binary branching trees. Preprint, Fachbereich Mathematik, Univ. Frankfurt, 1997.
47. S. Kullback. An application of characteristic functions to the distribution problem of statistics. Ann. Math. Statist., 5:264-305, 1934.
48. J. F. Le Gall. Marches aratoires, mouvement Brownien et processus de branchement. In Séminaire de Probabilités XXIII, pages 258-274. Springer, 1989. Lecture Notes in Math. 1372.
49. J. F. Le Gall. The uniform random tree in a Brownian excursion. Probab. Th. Rel. Fields, 96:369-383, 1993.
50. N. N. Lebedev. Special Functions and their Applications. Prentice-Hall, Englewood Cliffs, N.J., 1965.
51. P. Levy. Sur certains processus stochastiques homogenes. Compositio Math., 7:283-339, 1939.
52. T. M. Liggett. An invariance principle for conditioned sums of random variables. J. Math. Mech., 18:559-570, 1968.
53. H. J. Malik. Exact distribution of the product of independent generalized gamma variables with the same shape parameter. The Annals of Mathematical Statistics, 39:1751-1752, 1968.
54. P. W. Millar. Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc., 226:365-391, 1977.
55. J. C. P. Miller. Tables of Weber parabolic cylinder functions. Her Majesty’s Stationery Office, London, 1955.
56. J. Neveu and J. Pitman. The branching process in a Brownian excursion. In Séminaire de Probabilités XXIII, pages 248-257. Springer, 1989. Lecture Notes in Math. 1372.
57. J. Neveu and J. Pitman. Renewal property of the extrema and tree property of a one-dimensional Brownian motion. In Séminaire de Probabilités XXIII, pages 239-247. Springer, 1989. Lecture Notes in Math. 1372.
58. J. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process. Advances in Applied Probability, 7:511-526, 1975.
59. J. Pitman. Exchangeable and partially exchangeable random partitions. Probab. Th. Rel. Fields, 102:145-158, 1995.
60. J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes-Monograph Series, pages 245-267. Institute of Mathematical Statistics, Hayward, California, 1996.
61. J. Pitman. Partition structures derived from Brownian motion and stable subordinators. Bernoulli, 3:79-96, 1997.
62. J. Pitman. The distribution of local times of Brownian bridge. Technical Report 539, Dept. Statistics, U.C. Berkeley, 1998. To appear in Séminaire de Probabilités XXXIII. Available via http://www.stat.berkeley.edu/users/pitman.
63. J. Pitman. A lattice path model for the Bessel polynomials. Technical Report 551, Dept. Statistics, U. C. Berkeley, 1999. Available via http://www.stat.berkeley.edu/users/pitman.
64. J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic Integrals, pages 285-370. Springer, 1981. Lecture Notes in Math. 851.
65. J. Pitman and M. Yor. Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. (3), 65:326-356, 1992.
66. J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab., 25:855-900, 1997.
67. a(-x). Bull. Amer. Math. Soc., 54:1115-6, 1948.
68. D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer, Berlin-Heidelberg, 1999. 3rd edition.
69. P. Salminen. Brownian excursions revisited. In Seminar on Stochastic Processes 1983, pages 161-187. Birkhauser Boston, 1984.
70. V. Seshadri. The inverse Gaussian distribution. The Clarendon Press, Oxford University Press, New York, 1993.
71. G. R. Shorack and J. A. Wellner. Empirical processes with applications to statistics. John Wiley & Sons, New York, 1986.
72. B. M. Steece. The distribution of the product and quotient for powers of two independent generalized gamma variates. Metron, 33:227-235, 1975.
73. L. Toscano. Numeri di Stirling generalizzati operatori differenziali e polinomi ipergeometrici. Commen-tationes Pontificia Academica Scientarum, 3:721-757, 1939.
74. L. Toscano. Some results for generalized Bernoulli, Euler, Stirling numbers. Fibonacci Quart., 16(2):103-112, 1978.
75. W. Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab., 7:143-149, 1979.
76. G.N. Watson. A treatise on the theory of Bessel functions. Cambridge: University Press, 1966.
77. S. S. Wilks. Certain generalizations in the analysis of variance. Biometrika, 24:471-494, 1932.
78. D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc., 76:871-873, 1970.
79. I D. Williams. Path decomposition and continuity of local time for one dimensional diffusions I. Proc. London Math. Soc. (3), 28:738-768, 1974.
80. I M. Yor. Some Aspects of Brownian Motion, Part I: Some Special Functionals. Lectures in Math., ETH Zürich. Birkhaüser, 1992.