Patterns in Random Walks and Brownian Motion

  • Jim PitmanEmail author
  • Wenpin Tang
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)


We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.


Brownian Motion Simple Random Walk Brownian Bridge Wiener Space Matching Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to express our gratitude to Patrick Fitzsimmons for posing the question whether one can find the distribution of Vervaat bridges by a random spacetime shift of Brownian motion. We thank Steven Evans for helpful discussion on potential theory, and Davar Koshnevisan for remarks on additive Lévy processes. We also thank an anonymous referee for his careful reading and helpful suggestions.


  1. 1.
    D. Aldous, The continuum random tree, III. Ann. Probab. 21(1), 248–289 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    R.F. Bass, The measurability of hitting times. Electron. Commun. Probab. 15, 99–105 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    R.F. Bass, Correction to The measurability of hitting times. Electron. Commun. Probab. 16, 189–191 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    I. Benjamini, R. Pemantle, Y. Peres, Martin capacity for Markov chains. Ann. Probab. 23(3), 1332–1346 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge 1996)zbMATHGoogle Scholar
  6. 6.
    J. Bertoin, Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Stat. 38(3), 319–340 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Bertoin, Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics, vol. 102 (Cambridge University Press, Cambridge, 2006)Google Scholar
  8. 8.
    J. Bertoin, L. Chaumont, J. Pitman, Path transformations of first passage bridges. Electron. Commun. Probab. 8, 155–166 (electronic) (2003)Google Scholar
  9. 9.
    Ph. Biane, M. Yor, Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. (2) 111(1), 23–101 (1987)Google Scholar
  10. 10.
    Ph. Biane, M. Yor, Quelques précisions sur le méandre brownien. Bull. Sci. Math. (2) 112(1), 101–109 (1988)Google Scholar
  11. 11.
    P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968)zbMATHGoogle Scholar
  12. 12.
    S. Breen, M.S. Waterman, N. Zhang, Renewal theory for several patterns. J. Appl. Probab. 22(1), 228–234 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    R.V. Chacon, D.S. Ornstein, A general ergodic theorem. Ill. J. Math. 4, 153–160 (1960)zbMATHMathSciNetGoogle Scholar
  14. 14.
    M.-H. Chang, T. Pang, M. Pemy, Optimal control of stochastic functional differential equations with a bounded memory. Stochastics 80(1), 69–96 (2008)zbMATHMathSciNetGoogle Scholar
  15. 15.
    K.L. Chung, Excursions in Brownian motion. Ark. Mat. 14(2), 155–177 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    E. Çınlar, Probability and Stochastics. Graduate Texts in Mathematics, vol. 261 (Springer, New York, 2011)Google Scholar
  17. 17.
    R. Cont, D-A. Fournié, Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259(4), 1043–1072 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Cont, D-A. Fournié, A functional extension of the Ito formula. C. R. Math. Acad. Sci. Paris 348(1–2), 57–61 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Cont, D-A. Fournié, Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41(1), 109–133 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    C. Dellacherie, Capacités et Processus Stochastiques. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 67 (Springer, Berlin, 1972)Google Scholar
  21. 21.
    C. Dellacherie, P.-A. Meyer, Probabilités et potentiel, in Chapitres I à IV, Édition entièrement refondue, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV. Actualités Scientifiques et Industrielles, No. 1372. (Hermann, Paris, 1975), pp. x+291Google Scholar
  22. 22.
    C. Dellacherie, P-A. Meyer, Probabilités et Potentiel. Chapitres IX à XI: Théorie discrete du potentiel. Édition entièrement refondue, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XVIII. Actualités Scientifiques et Industrielles, 141 (Hermann, Paris, 1983)Google Scholar
  23. 23.
    M.D. Donsker, An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc. 1951(6), 12 (1951)Google Scholar
  24. 24.
    L.E. Dubins, On a theorem of Skorohod. Ann. Math. Stat. 39, 2094–2097 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    B. Dupire, Functional Itô calculus. Bloomberg Portfolio Research Paper (2009-04) (2009)Google Scholar
  26. 26.
    A. Dvoretzky, P. Erdös, S. Kakutani, Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged. 12(Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B), 75–81 (1950)Google Scholar
  27. 27.
    A. Dvoretzky, P. Erdős, S. Kakutani, S.J. Taylor, Triple points of Brownian paths in 3-space. Proc. Camb. Philos. Soc. 53, 856–862 (1957)zbMATHCrossRefGoogle Scholar
  28. 28.
    S.N. Evans, On the Hausdorff dimension of Brownian cone points. Math. Proc. Camb. Philos. Soc. 98(2), 343–353 (1985)zbMATHCrossRefGoogle Scholar
  29. 29.
    W. Feller, An Introduction to Probability Theory and Its Applications, vol. I, 3rd edn. (Wiley, New York, 1968)Google Scholar
  30. 30.
    P.J. Fitzsimmons, R.K. Getoor, On the potential theory of symmetric Markov processes. Math. Ann. 281(3), 495–512 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    D.-A. Fournié, Functional Ito calculus and applications. Thesis (Ph.D.)–Columbia University (ProQuest LLC, Ann Arbor, 2010)Google Scholar
  32. 32.
    J.C. Fu, W.Y.W. Lou, Distribution Theory of Runs and Patterns and Its Applications. A Finite Markov Chain Embedding Approach (World Scientific, River Edge, 2003)Google Scholar
  33. 33.
    M. Fukushima, Basic properties of Brownian motion and a capacity on the Wiener space. J. Math. Soc. Jpn. 36(1), 161–176 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    H.U. Gerber, S.Y.R. Li, The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stoch. Process. Appl. 11(1), 101–108 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    P. Greenwood, J. Pitman, Construction of local time and Poisson point processes from nested arrays. J. Lond. Math. Soc. (2) 22(1), 182–192 (1980)Google Scholar
  36. 36.
    P. Greenwood, J. Pitman, Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab. 12(4), 893–902 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    L.J. Guibas, A.M. Odlyzko, String overlaps, pattern matching, and nontransitive games. J. Combin. Theory Ser. A 30(2), 183–208 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    J.-P. Imhof, Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21(3), 500–510 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    S. Kakutani, On Brownian motions in n-space. Proc. Imp. Acad. Tokyo 20, 648–652 (1944)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    O. Kallenberg, Foundations of Modern Probability, 2nd edn. Probability and Its Applications (New York) (Springer, New York, 2002)zbMATHCrossRefGoogle Scholar
  41. 41.
    H. Kesten, Hitting Probabilities of Single Points for Processes with Stationary Independent Increments. Memoirs of the American Mathematical Society, vol. 93 (American Mathematical Society, Providence, 1969)Google Scholar
  42. 42.
    D. Khoshnevisan, Brownian sheet and quasi-sure analysis, in Asymptotic Methods in Stochastics. Fields Institute Communications, vol. 44 (American Mathematical Society, Providence, 2004), pp. 25–47Google Scholar
  43. 43.
    D. Khoshnevisan, Y. Xiao, Level sets of additive Lévy processes. Ann. Probab. 30(1), 62–100 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    D. Khoshnevisan, Y. Xiao, Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes. Proc. Am. Math. Soc. 131(8), 2611–2616 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    D. Khoshnevisan, Y. Xiao, Additive Lévy processes: capacity and Hausdorff dimension, in Fractal Geometry and Stochastics III. Progress in Probability, vol. 57 (Birkhäuser, Basel, 2004), pp. 151–170Google Scholar
  46. 46.
    D. Khoshnevisan, Y. Xiao, Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33(3), 841–878 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    D. Khoshnevisan, Y. Xiao, Harmonic analysis of additive Lévy processes. Probab. Theory Relat. Fields 145(3–4), 459–515 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    D. Khoshnevisan, Y. Xiao, Y. Zhong, Local times of additive Lévy processes. Stoch. Process. Appl. 104(2), 193–216 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    D. Khoshnevisan, Y. Xiao, Y. Zhong, Measuring the range of an additive Lévy process. Ann. Probab. 31(2), 1097–1141 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    F.B. Knight, A predictive view of continuous time processes. Ann. Probab. 3(4), 573–596 (1975)zbMATHCrossRefGoogle Scholar
  51. 51.
    F.B. Knight, Foundations of the Prediction Process. Oxford Studies in Probability, vol. 1. Oxford Science Publications (The Clarendon Press/Oxford University Press, New York, 1992)Google Scholar
  52. 52.
    J.-M. Labarbe, J.-F. Marckert, Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks. Electron. J. Probab. 12(9), 229–261 (2007)zbMATHMathSciNetGoogle Scholar
  53. 53.
    J.-F. Le Gall, The uniform random tree in a Brownian excursion. Probab. Theory Relat. Fields 96(3), 369–383 (1993)zbMATHCrossRefGoogle Scholar
  54. 54.
    J.-F. Le Gall, Hitting probabilities and potential theory for the Brownian path-valued process. Ann. Inst. Fourier (Grenoble) 44(1), 277–306 (1994)Google Scholar
  55. 55.
    S.Y.R. Li, A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Probab. 8(6), 1171–1176 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    T. Lupu, J. Pitman, W. Tang, The Vervaat transform of Brownian bridges and Brownian motion (2013) [arXiv:1310.3889]Google Scholar
  57. 57.
    Z.M. Ma, M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Universitext (Springer, Berlin, 1992)CrossRefGoogle Scholar
  58. 58.
    B. Maisonneuve, Exit systems. Ann. Probab. 3(3), 399–411 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    P.-A. Meyer, La théorie de la prédiction de F. Knight, in Séminaire de Probabilités, X (Première partie, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975). Lecture Notes in Mathematics, vol. 511 (Springer, Berlin, 1976), pp. 86–103Google Scholar
  60. 60.
    P.-A. Meyer, Appendice: Un résultat de D. Williams, in Séminaire de Probabilités de Strasbourg, vol. 16 (1982), pp. 133–133Google Scholar
  61. 61.
    P.W. Millar, Zero-one laws and the minimum of a Markov process. Trans. Am. Math. Soc. 226, 365–391 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    P.W. Millar, A path decomposition for Markov processes. Ann. Probab. 6(2), 345–348 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    S.E.A. Mohammed, Stochastic Functional Differential Equations. Research Notes in Mathematics, vol. 99 (Pitman [Advanced Publishing Program], Boston, 1984)Google Scholar
  64. 64.
    I. Monroe, On embedding right continuous martingales in Brownian motion. Ann. Math. Stat. 43, 1293–1311 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  65. 65.
    I. Monroe, Processes that can be embedded in Brownian motion. Ann. Probab. 6(1), 42–56 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    A.M. Mood, The distribution theory of runs. Ann. Math. Stat. 11, 367–392 (1940)MathSciNetCrossRefGoogle Scholar
  67. 67.
    P. Mörters, Y. Peres, Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge, 2010)Google Scholar
  68. 68.
    J. Neveu, J. Pitman, Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion, in Séminaire de Probabilités, XXIII. Lecture Notes in Mathematics, vol. 1372 (Springer, Berlin, 1989), pp. 239–247Google Scholar
  69. 69.
    J. Obłój, The Skorokhod embedding problem and its offspring. Probab. Surv. 1, 321–390 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    J.W. Pitman, Lévy systems and path decompositions, in Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981). Progr. Prob. Statist., vol. 1 (Birkhäuser, Boston, 1981), pp. 79–110Google Scholar
  71. 71.
    J. Pitman, Combinatorial Stochastic Processes. Lecture Notes in Mathematics, vol. 1875 (Springer, Berlin, 2006) [Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard]Google Scholar
  72. 72.
    J. Pitman, M. Winkel, Growth of the Brownian forest. Ann. Probab. 33(6), 2188–2211 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    J. Pitman, W. Tang, The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift (2014) [arXiv: 1411.0040]Google Scholar
  74. 74.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 293 (Springer, Berlin, 1999)Google Scholar
  75. 75.
    H. Rost, Markoff-Ketten bei sich füllenden Löchern im Zustandsraum. Ann. Inst. Fourier (Grenoble) 21(1), 253–270 (1971)Google Scholar
  76. 76.
    H. Rost, The stopping distributions of a Markov Process. Invent. Math. 14, 1–16 (1971)zbMATHMathSciNetCrossRefGoogle Scholar
  77. 77.
    R.Y. Rubinstein, D.P. Kroese, Simulation and the Monte Carlo Method, 2nd edn. Wiley Series in Probability and Statistics (Wiley-Interscience [John Wiley & Sons], Hoboken, 2008)Google Scholar
  78. 78.
    L.A. Shepp, Radon-Nikodým derivatives of Gaussian measures. Ann. Math. Stat. 37, 321–354 (1966)zbMATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    L.A. Shepp, First passage time for a particular Gaussian process. Ann. Math. Stat. 42, 946–951 (1971)zbMATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    A.V. Skorokhod, Studies in the Theory of Random Processes (Addison-Wesley, Reading, 1965) [Translated from the Russian by Scripta Technica, Inc.]zbMATHGoogle Scholar
  81. 81.
    D. Slepian, First passage time for a particular gaussian process. Ann. Math. Stat. 32(2), 610–612 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    R.P. Stanley, Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999) [With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin]Google Scholar
  83. 83.
    B. Tsirelson, Brownian local minima, random dense countable sets and random equivalence classes. Electron. J. Probab. 11(7), 162–198 (electronic) (2006)Google Scholar
  84. 84.
    M.A. Van Leeuwen, Some simple bijections involving lattice walks and ballot sequences (2010) [arXiv:1010.4847]Google Scholar
  85. 85.
    W. Vervaat, A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7(1), 143–149 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    J. Von Neumann, Various techniques used in connection with random digits. Appl. Math. Ser. 12(36–38), 1 (1951)Google Scholar
  87. 87.
    A. Wald, J. Wolfowitz, On a test whether two samples are from the same population. Ann. Math. Stat. 11, 147–162 (1940)MathSciNetCrossRefGoogle Scholar
  88. 88.
    D. Williams, Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. Lond. Math. Soc. (3) 28, 738–768 (1974)Google Scholar
  89. 89.
    M. Yang, On a theorem in multi-parameter potential theory. Electron. Commun. Probab. 12, 267–275 (electronic) (2007)Google Scholar
  90. 90.
    M. Yang, Lebesgue measure of the range of additive Lévy processes. Probab. Theory Relat. Fields 143(3–4), 597–613 (2009)zbMATHCrossRefGoogle Scholar
  91. 91.
    J-Y. Yen, M. Yor, Local Times and Excursion Theory for Brownian Motion. A Tale of Wiener and Itô measures. Lecture Notes in Mathematics, vol. 2088 (Springer, Cham, 2013)Google Scholar

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Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

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