Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times

Jim Pitman (University of California, Berkeley)

Abstract


For a random process $X$ consider the random vector defined by the values of $X$ at times $0 < U_{n,1} < ... < U_{n,n} < 1$ and the minimal values of $X$ on each of the intervals between consecutive pairs of these times, where the $U_{n,i}$ are the order statistics of $n$ independent uniform $(0,1)$ variables, independent of $X$. The joint law of this random vector is explicitly described when $X$ is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at $n$ independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae are Brownian representions of various special functions, including Bessel polynomials, some hypergeometric polynomials, and the Hermite function. Various combinatorial identities involving random partitions and generalized Stirling numbers are also obtained.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-33

Publication Date: April 26, 1999

DOI: 10.1214/EJP.v4-48

References

  1. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions. Dover, New York, 1965. Math. Review 94b:00012.
  2. D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487-512, 1994. Math. Review 95k:60055.
  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. Math. Review 91i:60024.
  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. Math. Review 93f:60010.
  7. D.J. Aldous. The continuum random tree III. Ann. Probab., 21:248-289, 1993. Math. Review 94c:60015.
  8. D.J. Aldous and J. Pitman. The standard additive coalescent. Ann. Probab., 26:1703-1726, 1998. Math. Review CMP 1 675 063.
  9. D.J. Aldous and J. Pitman. Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. Henri Poincaré, 34:637-686, 1998. Math. Review CMP 1 641 670.
  10. L. Alili, C. Donati-Martin, and M. Yor. Une identité en loi remarquable pour l'excursion brownienne normalisée. In M. Yor, editor, Exponential functionals and principal values related to Brownian motion, pages 155-180. Biblioteca de la Revista Matemática Ibero-Americana, 1997. Math. Review CMP 1 648 659.
  11. J. Bertoin. Décomposition du mouvement brownien avec dérive en un minimum local par juxtaposition de ses excursions positives et negatives. In Séminaire de Probabilités XXV, pages 330-344. Springer-Verlag, 1991. Lecture Notes in Math. 1485. Math. Review 93k:60204.
  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 réel. Ann. Inst. Henri Poincaré, 22:1-7, 1986. Math. Review 87k:60186.
  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. Math. Review 89d:60145.
  15. Ph. Biane and M. Yor. Quelques précisions sur le méandre brownien. Bull. Sci. Math., 112:101-109, 1988. Math. Review 89i:60156.
  16. P. Billingsley. Probability and Measure. Wiley, New York, 1995. 3rd ed. Math. Review 95k:60001.
  17. R. M. Blumenthal. Weak convergence to Brownian excursion. Ann. Probab., 11:798-800, 1983. Math. Review 85e:60083.
  18. E. Bolthausen. On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab., 4:480-485, 1976. Math. Review 54 #3782.
  19. L. Bondesson. Generalized gamma convolutions and related classes of distributions and densities. Springer-Verlag, 1992. Math. Review 94g:60031.
  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. Math. Review 89d:62017.
  21. K. L. Chung. Excursions in Brownian motion. Arkiv fur Matematik, 14:155-177, 1976. Math. Review 57 #7791.
  22. L. Comtet. Advanced Combinatorics. D. Reidel Pub. Co., Boston, 1974. (translated from French). Math. Review 57 #124.
  23. I. V. Denisov. A random walk and a Wiener process near a maximum. Theor. Prob. Appl., 28:821-824, 1984. Math. Review 85f:60117.
  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. Math. Review 93j:05173.
  25. A. Erdélyi et al. Higher Transcendental Functions, volume II of Bateman Manuscript Project. McGraw-Hill, New York, 1953. Math. Review 84h:33001a ( Math. Review of Errata)
  26. W. Feller. An Introduction to Probability Theory and its Applications, Vol 1, 3rd ed. Wiley, New York, 1968. Math. Review 37 #3604.
  27. W. Feller. An Introduction to Probability Theory and its Applications, Vol 2, 2nd ed. Wiley, New York, 1971. Math. Review 42 #5292.
  28. P. Fitzsimmons, J. Pitman, and M. Yor. Markovian bridges: construction, Palm interpretation, and splicing. In E. Cinlar, K.L. Chung, and M.J. Sharpe, editors, Seminar on Stochastic Processes, 1992, pages 101-134. Birkhäuser, Boston, 1993. Math. Review 95i:60070.
  29. R. K. Getoor. The Brownian escape process. Ann. Probab., 7:864-867, 1979. Math. Review 80h:60102.
  30. L. Gordon. Bounds for the distribution of the generalized variance. The Annals of Statistics, 17:1684 - 1692, 1989. Math. Review 91a:62127.
  31. L. Gordon. A stochastic approach to the gamma function. Amer. Math. Monthly, 101:858-865, 1994. Math. Review 95k:33003.
  32. R. L. Graham, D.E. Knuth, and O. Patashnik. Concrete Mathematics: a foundation for computer science. 2nd ed. Addison-Wesley, Reading, Mass., 1989. Math. Review 91f:00001.
  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. Math. Review number not available.
  34. P. Greenwood and J. Pitman. Fluctuation identities for Lévy processes and splitting at the maximum. Advances in Applied Probability, 12:893-902, 1980. Math. Review 81j:60084.
  35. E. Grosswald. The Student t-distribution for odd degrees of freedom is infinitely divisible. Ann. Probability, 4(4):680-683, 1976. Math. Review 53 #14591.
  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. Math. Review 54 #14037.
  37. E. Grosswald. Bessel polynomials. Springer, Berlin, 1978. Math. Review 80i:33013.
  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. Math. Review 99c:11020.
  39. I.A. Ibragimov and Yu. V. Linnik. Independent and Stationary Sequences of Random Variables. Gröningen, Wolthers-Noordhof, 1971. [Translated from original in Russian (1965), Nauka, Moscow]. Math. Review 48 #1287.
  40. D.L. Iglehart. Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab., 2:608-619, 1974. Math. Review 50 #14939.
  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. Math. Review 85j:60152.
  42. M. E. H. Ismail and J. Pitman. Algebraic evaluations of some Euler integrals, duplication formulae for Appell's hypergeometric function F1, 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. Math. Review 49 #9963 .
  44. N.L. Johnson, S. Kotz, and N. Balakrishnan. Discrete Multivariate Distributions. Wiley, New York, 1997. Math. Review CMP 1 429 617.
  45. W.D. Kaigh. An invariance principle for random walk conditioned by a late return to zero. Ann. Probab., 4:115-121, 1976. Math. Review 54 #3786.
  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. Math. Review number not available.
  48. J. F. Le Gall. Marches aléatoires, mouvement Brownien et processus de branchement. In Séminaire de Probabilités XXIII, pages 258-274. Springer, 1989. Lecture Notes in Math. 1372. Math. Review 91e:60241.
  49. J. F. Le Gall. The uniform random tree in a Brownian excursion. Probab. Th. Rel. Fields, 96:369-383, 1993. Math. Review 94e:60073.
  50. N. N. Lebedev. Special Functions and their Applications. Prentice-Hall, Englewood Cliffs, N.J., 1965. Math. Review 30 #4988.
  51. P. L&eacut;vy. Sur certains processus stochastiques homogénes. Compositio Math., 7:283-339, 1939. Math. Review 1,150a.
  52. T. M. Liggett. An invariance principle for conditioned sums of random variables. J. Math. Mech., 18:559-570, 1968. Math. Review 38 #6649.
  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. Math. Review 37 #7024.
  54. P. W. Millar. Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc., 226:365-391, 1977. Math. Review 55 #6579.
  55. J. C. P. Miller. Tables of Weber parabolic cylinder functions. Her Majesty's Stationery Office, London, 1955. Math. Review 17,1012c.
  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. Math. Review 91e:60240.
  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. Math. Review 91e:60239 .
  58. J. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process. Advances in Applied Probability, 7:511-526, 1975. Math. Review 51 #11677.
  59. J. Pitman. Exchangeable and partially exchangeable random partitions. Probab. Th. Rel. Fields, 102:145-158, 1995. Math. Review 96e:60059.
  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. Math. Review 99c:60017.
  61. J. Pitman. Partition structures derived from Brownian motion and stable subordinators. Bernoulli, 3:79-96, 1997. Math. Review 99c:60078.
  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. Math. Review 82j:60149 .
  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. Math. Review 93e:60152.
  66. J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab., 25:855-900, 1997. Math. Review 98f:60147.
  67. H. Pollard. The compleletly monotonic character of the Mittag-Leffler function $E_\alpha(-x)$. Bull. Amer. Math. Soc., 54:1115-6, 1948. Math. Review 10,295e.
  68. D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer, Berlin-Heidelberg, 1999. 3rd edition. Math. Review number not available.
  69. P. Salminen. Brownian excursions revisited. In Seminar on Stochastic Processes 1983, pages 161-187. Birkháuser Boston, 1984. Math. Review 88m:60209.
  70. V. Seshadri. The inverse Gaussian distribution. The Clarendon Press, Oxford University Press, New York, 1993. Math. Review CMP 1 622 488.
  71. G. R. Shorack and J. A. Wellner. Empirical processes with applications to statistics. John Wiley & Sons, New York, 1986. Math. Review 88e:60002.
  72. B. M. Steece. The distribution of the product and quotient for powers of two independent generalized gamma variates. Metron, 33:227 - 235, 1975. Math. Review 57 #14224.
  73. L. Toscano. Numeri di Stirling generalizzati operatori differenziali e polinomi ipergeometrici. Commentationes Pontificia Academica Scientarum, 3:721-757, 1939. Math. Review number not available.
  74. L. Toscano. Some results for generalized Bernoulli, Euler, Stirling numbers. Fibonacci Quart., 16(2):103-112, 1978. Math. Review 58 #5257.
  75. W. Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab., 7:143-149, 1979. Math. Review 80b:60107.
  76. G.N. Watson. A treatise on the theory of Bessel functions. Cambridge: University Press, 1966. Math. Review 96i:33010.
  77. S. S. Wilks. Certain generalizations in the analysis of variance. Biometrika, 24:471-494, 1932. Math. Review number not available.
  78. D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc., 76:871-873, 1970. Math. Review 41 #2777.
  79. D. Williams. Path decomposition and continuity of local time for one dimensional diffusions I. Proc. London Math. Soc. (3), 28:738-768, 1974. Math. Review 50 #3373.
  80. M. Yor. Some Aspects of Brownian Motion, Part I: Some Special Functionals. Lectures in Math., ETH Zürich. Birkhaüser, 1992. Math. Review 93i:60155.
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.