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Patterns in Random Walks and Brownian Motion

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In Memoriam Marc Yor - Séminaire de Probabilités XLVII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2137))

Abstract

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.

Keywords

AMS 2010 Mathematics Subject Classification: 60C05, 60G17, 60J65.

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References

  1. D. Aldous, The continuum random tree, III. Ann. Probab. 21(1), 248–289 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. R.F. Bass, The measurability of hitting times. Electron. Commun. Probab. 15, 99–105 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. R.F. Bass, Correction to The measurability of hitting times. Electron. Commun. Probab. 16, 189–191 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Benjamini, R. Pemantle, Y. Peres, Martin capacity for Markov chains. Ann. Probab. 23(3), 1332–1346 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge 1996)

    MATH  Google Scholar 

  6. J. Bertoin, Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Stat. 38(3), 319–340 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Bertoin, Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics, vol. 102 (Cambridge University Press, Cambridge, 2006)

    Google Scholar 

  8. J. Bertoin, L. Chaumont, J. Pitman, Path transformations of first passage bridges. Electron. Commun. Probab. 8, 155–166 (electronic) (2003)

    Google Scholar 

  9. Ph. Biane, M. Yor, Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. (2) 111(1), 23–101 (1987)

    Google Scholar 

  10. Ph. Biane, M. Yor, Quelques précisions sur le méandre brownien. Bull. Sci. Math. (2) 112(1), 101–109 (1988)

    Google Scholar 

  11. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968)

    MATH  Google Scholar 

  12. S. Breen, M.S. Waterman, N. Zhang, Renewal theory for several patterns. J. Appl. Probab. 22(1), 228–234 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  13. R.V. Chacon, D.S. Ornstein, A general ergodic theorem. Ill. J. Math. 4, 153–160 (1960)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  15. K.L. Chung, Excursions in Brownian motion. Ark. Mat. 14(2), 155–177 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  16. E. Çınlar, Probability and Stochastics. Graduate Texts in Mathematics, vol. 261 (Springer, New York, 2011)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Cont, D-A. Fournié, Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41(1), 109–133 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. C. Dellacherie, Capacités et Processus Stochastiques. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 67 (Springer, Berlin, 1972)

    Google Scholar 

  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+291

    Google Scholar 

  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. M.D. Donsker, An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc. 1951(6), 12 (1951)

    Google Scholar 

  24. L.E. Dubins, On a theorem of Skorohod. Ann. Math. Stat. 39, 2094–2097 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  25. B. Dupire, Functional Itô calculus. Bloomberg Portfolio Research Paper (2009-04) (2009)

    Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  28. S.N. Evans, On the Hausdorff dimension of Brownian cone points. Math. Proc. Camb. Philos. Soc. 98(2), 343–353 (1985)

    Article  MATH  Google Scholar 

  29. W. Feller, An Introduction to Probability Theory and Its Applications, vol. I, 3rd edn. (Wiley, New York, 1968)

    Google Scholar 

  30. P.J. Fitzsimmons, R.K. Getoor, On the potential theory of symmetric Markov processes. Math. Ann. 281(3), 495–512 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  31. D.-A. Fournié, Functional Ito calculus and applications. Thesis (Ph.D.)–Columbia University (ProQuest LLC, Ann Arbor, 2010)

    Google Scholar 

  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. M. Fukushima, Basic properties of Brownian motion and a capacity on the Wiener space. J. Math. Soc. Jpn. 36(1), 161–176 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. P. Greenwood, J. Pitman, Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab. 12(4), 893–902 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  37. L.J. Guibas, A.M. Odlyzko, String overlaps, pattern matching, and nontransitive games. J. Combin. Theory Ser. A 30(2), 183–208 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  39. S. Kakutani, On Brownian motions in n-space. Proc. Imp. Acad. Tokyo 20, 648–652 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  40. O. Kallenberg, Foundations of Modern Probability, 2nd edn. Probability and Its Applications (New York) (Springer, New York, 2002)

    Book  MATH  Google Scholar 

  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. D. Khoshnevisan, Brownian sheet and quasi-sure analysis, in Asymptotic Methods in Stochastics. Fields Institute Communications, vol. 44 (American Mathematical Society, Providence, 2004), pp. 25–47

    Google Scholar 

  43. D. Khoshnevisan, Y. Xiao, Level sets of additive Lévy processes. Ann. Probab. 30(1), 62–100 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 151–170

    Google Scholar 

  46. D. Khoshnevisan, Y. Xiao, Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33(3), 841–878 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  47. D. Khoshnevisan, Y. Xiao, Harmonic analysis of additive Lévy processes. Probab. Theory Relat. Fields 145(3–4), 459–515 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  48. D. Khoshnevisan, Y. Xiao, Y. Zhong, Local times of additive Lévy processes. Stoch. Process. Appl. 104(2), 193–216 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  49. D. Khoshnevisan, Y. Xiao, Y. Zhong, Measuring the range of an additive Lévy process. Ann. Probab. 31(2), 1097–1141 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  50. F.B. Knight, A predictive view of continuous time processes. Ann. Probab. 3(4), 573–596 (1975)

    Article  MATH  Google Scholar 

  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. 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)

    MATH  MathSciNet  Google Scholar 

  53. J.-F. Le Gall, The uniform random tree in a Brownian excursion. Probab. Theory Relat. Fields 96(3), 369–383 (1993)

    Article  MATH  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  56. T. Lupu, J. Pitman, W. Tang, The Vervaat transform of Brownian bridges and Brownian motion (2013) [arXiv:1310.3889]

    Google Scholar 

  57. Z.M. Ma, M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Universitext (Springer, Berlin, 1992)

    Book  Google Scholar 

  58. B. Maisonneuve, Exit systems. Ann. Probab. 3(3), 399–411 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  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. 86–103

    Google Scholar 

  60. P.-A. Meyer, Appendice: Un résultat de D. Williams, in Séminaire de Probabilités de Strasbourg, vol. 16 (1982), pp. 133–133

    Google Scholar 

  61. P.W. Millar, Zero-one laws and the minimum of a Markov process. Trans. Am. Math. Soc. 226, 365–391 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  62. P.W. Millar, A path decomposition for Markov processes. Ann. Probab. 6(2), 345–348 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  63. S.E.A. Mohammed, Stochastic Functional Differential Equations. Research Notes in Mathematics, vol. 99 (Pitman [Advanced Publishing Program], Boston, 1984)

    Google Scholar 

  64. I. Monroe, On embedding right continuous martingales in Brownian motion. Ann. Math. Stat. 43, 1293–1311 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  65. I. Monroe, Processes that can be embedded in Brownian motion. Ann. Probab. 6(1), 42–56 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  66. A.M. Mood, The distribution theory of runs. Ann. Math. Stat. 11, 367–392 (1940)

    Article  MathSciNet  Google Scholar 

  67. P. Mörters, Y. Peres, Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge, 2010)

    Google Scholar 

  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. 239–247

    Google Scholar 

  69. J. Obłój, The Skorokhod embedding problem and its offspring. Probab. Surv. 1, 321–390 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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. 79–110

    Google Scholar 

  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. J. Pitman, M. Winkel, Growth of the Brownian forest. Ann. Probab. 33(6), 2188–2211 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 293 (Springer, Berlin, 1999)

    Google Scholar 

  75. H. Rost, Markoff-Ketten bei sich füllenden Löchern im Zustandsraum. Ann. Inst. Fourier (Grenoble) 21(1), 253–270 (1971)

    Google Scholar 

  76. H. Rost, The stopping distributions of a Markov Process. Invent. Math. 14, 1–16 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  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. L.A. Shepp, Radon-Nikodým derivatives of Gaussian measures. Ann. Math. Stat. 37, 321–354 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  79. L.A. Shepp, First passage time for a particular Gaussian process. Ann. Math. Stat. 42, 946–951 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  80. A.V. Skorokhod, Studies in the Theory of Random Processes (Addison-Wesley, Reading, 1965) [Translated from the Russian by Scripta Technica, Inc.]

    MATH  Google Scholar 

  81. D. Slepian, First passage time for a particular gaussian process. Ann. Math. Stat. 32(2), 610–612 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. M.A. Van Leeuwen, Some simple bijections involving lattice walks and ballot sequences (2010) [arXiv:1010.4847]

    Google Scholar 

  85. W. Vervaat, A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7(1), 143–149 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  86. J. Von Neumann, Various techniques used in connection with random digits. Appl. Math. Ser. 12(36–38), 1 (1951)

    Google Scholar 

  87. A. Wald, J. Wolfowitz, On a test whether two samples are from the same population. Ann. Math. Stat. 11, 147–162 (1940)

    Article  MathSciNet  Google Scholar 

  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. M. Yang, On a theorem in multi-parameter potential theory. Electron. Commun. Probab. 12, 267–275 (electronic) (2007)

    Google Scholar 

  90. M. Yang, Lebesgue measure of the range of additive Lévy processes. Probab. Theory Relat. Fields 143(3–4), 597–613 (2009)

    Article  MATH  Google Scholar 

  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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Acknowledgements

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.

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  1. Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA, 94720-3860, USA

    Jim Pitman & Wenpin Tang

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  1. Jim Pitman
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Correspondence to Jim Pitman .

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Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandœuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Alain Rouault

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Pitman, J., Tang, W. (2015). Patterns in Random Walks and Brownian Motion. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_4

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