Archive for History of Exact Sciences

, Volume 32, Issue 3–4, pp 351–375 | Cite as

On the problem of random flights

  • Jacques Dutka


Random Flight 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bachelier, L. [1], “Théorie de la spéculation,” Annales Scientifiques de l'École Normale Supérieure, Ser. 3, Vol. 21 (1900), 21–86.Google Scholar
  2. Barber, M. N., & Ninham, B. W. [1], Random and Restricted Walks: Theory and Applications, New York, 1970.Google Scholar
  3. Beckmann, Peter [1], “The probability distribution of the vector sum of n-unit vectors with arbitrary phase distributions,” Acta Technica, C.S.A.V., Vol. 4 (1959), 323–335.Google Scholar
  4. Bennett, W. R. [1], “Distribution of the sum of randomly phased components,” Quarterly of Applied Mathematics, Vol. 5 (1947), 385–393.Google Scholar
  5. Bochner, S. [1], Vorlesungen über Fouriersche Integrale, 2nd ed., New York, 1948.Google Scholar
  6. Chandrasekhar, S. [1], “Stochastic problems in physics and astronomy,” Reviews of Modern Physics, Vol. 15 (1943), 1–89.Google Scholar
  7. Crofton, M. W. [1], “Question 1773,” Mathematical Questions with Their Solutions from the Educational Times, Vol. 4 (July–Dec. 1865), publ. 1866, 71–72.Google Scholar
  8. Dirichlet, P. G. L. [1], Werke, 2nd ed., New York, 1969.Google Scholar
  9. Durand, D., & Greenwood, J. A. [1], “Random unit vectors II: usefulness of Gram-Charlier and related series in approximating distributions,” Annals of Mathematical Statistics, Vol. 28 (1957), 978–986.Google Scholar
  10. Eyring, H. [1], “The resultant electric moment of complex molecules,” Physical Review, Ser. 2, Vol. 39 (1932), 746–748.Google Scholar
  11. Gauss, C. F. [1], “Schönes Theorem der Wahrscheinlichkeitsrechnung,” Werke, Bd. 7, 88–89, Leipzig, 1900.Google Scholar
  12. Glaisher, J. W. L. [1], “Remarks on a theorem in Laplace's Probabilités,” Messenger of Mathematics, N.S. Vol. 2 (1973), 62–64.Google Scholar
  13. Greenwood, J. A., & Durand, D. [1], “The distribution of length and components of the sum of n random unit vectors,” Annals of Mathematical Statistics, Vol. 26 (1955), 233–246.Google Scholar
  14. Grosjean, C. C. [1], “Solution of the non-isotropic random flight problem in the k-dimensional space,” Physica, Vol. 19 (1953), 29–45.Google Scholar
  15. Haldane, J. B. S. [1], “The addition of random vectors,” Sankhyā, Vol. 22 (1960), 213–220.Google Scholar
  16. Horner, F. [1], “A problem on the summation of simple harmonic functions of the same amplitude and frequency but of random phase,” The London, Edinburgh and Dublin Philosophical Magazine, Ser. 7, Vol. 37 (1946), 145–162.Google Scholar
  17. Hughes, B. W., & Prager, S. [1], “Random processes and random systems: An introduction,” Lecture Notes in Mathematics, No. 1035 (1983), 1–108.Google Scholar
  18. Johnson, N. L. [1], “Paths and chains of straight-line segments,” Technometrics, Vol. 8 (1966), 303–317.Google Scholar
  19. Kendall, M. G., & Stuart, M. G. [1], The Advanced Theory of Statistics, 4th ed., Vol. 1, New York, 1977.Google Scholar
  20. Kingman, J. F. C. [1], “Random walks with spherical symmetry,” Acta Mathematica, Vol. 109 (1963), 11–53.Google Scholar
  21. Kluyver, J. C. [1], “A local probability problem,” Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen te Amsterdam, Vol. 8 (1906), 341–350.Google Scholar
  22. Lagrange, J. L. [1], “Mémoire sur l'utilité de la méthode de prendre le milieu entre les résultats de plusieurs observations ...,” Oeuvres, F. 2, 173–234.Google Scholar
  23. Laplace, P. S. [1], Théorie Analytique des Probabilités, Troisième Ed., Paris, 1820.Google Scholar
  24. Liyange, L. H., Gulati, C. M., & Hill, J. M. [1], A Bibliography on Random Walks, 1980, Mathematics Department, University of Wollongong, Australia.Google Scholar
  25. Lord, R. D. [1], “A problem on random vectors,” The London, Edinburgh, and Dublin Philosophical Magazine, Ser. 7, Vol. 39 (1948), 66–71.Google Scholar
  26. Lord, R. D. [2], “The use of the Hankel transform in statistics. I. General theory and examples,” Biometrika, Vol. 41 (1954), 44–55.Google Scholar
  27. Lord, R. D. [3], “The use of the Hankel transform in statistics. II. Methods of computation, Biometrika, Vol. 41 (1954), 344–350.Google Scholar
  28. Mardia, K. V. [1], Statistics of Directional Data, London, 1972.Google Scholar
  29. Mardia, K. V. [2], “Statistics of directional data,” Journal of the Royal Statistical Society, Ser. B, Vol. 37 (1975), 349–393.Google Scholar
  30. Markoff, A. A. [1], Wahrscheinlichkeitsrechnung, Leipzig, 1912.Google Scholar
  31. Montroll, E. W., & West, B. J. [1], “On an enriched collection of stochastic processes,” Studies in Statistical Mechanics, Vol. 7, Ch. 2 (1979).Google Scholar
  32. Moran, P. A. P. [1], “The statistical distribution of the length of a rubber molecule,” Proceedings of the Cambridge Philosophical Society, Vol. 44 (1948), 342–344.Google Scholar
  33. Pearson, Karl [1], “The problem of the random walk,” Nature, July 27, 1905, 294.Google Scholar
  34. Pearson, Karl (with the assistance of John Blakeman) [2], “A mathematical theory of random migration,” Mathematical Contributions to the Theory of Evolution XV, Draper's Company Research Memoirs, Biometric Series, III (1906).Google Scholar
  35. Pólya, G. [1], “On some questions of the calculus of probability and on definite integrals” (in Hungarian), Mathematicai es Physikai Lapok, Vol. 22 (1913), 53–73, 163–215.Google Scholar
  36. Pólya, G. [2], “Wahrscheinlichkeitstheoretisches über die ‘Irrfahrt’,” Mitteilungen der physikalischen Gesellschaft zu Zürich, Bd. 19 (1919), 75–86.Google Scholar
  37. Quenouille, M. H. [1], “On the problem of random flights,” Proceedings of the Cambridge Philsolophical Society, Vol. 43 (1947), 581–582.Google Scholar
  38. Rayleigh (J. W. Strutt), Baron [1], “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase,” The London, Edinburgh, and Dublin Philosophical Magazine, Ser. 5, Vol. 10 (1880), 73–78.Google Scholar
  39. Rayleigh [2], Letter to the Editor, “The problem of the random walk,” Nature, Vol. 72 (1905), 318.Google Scholar
  40. Rayleigh [3], “On the problem of random vibrations and of random flights in one, two, or three dimensions,” The London, Edinburgh, and Dublin Philosophical Magazine, Ser. 6, Vol. 37 (1919), 321–347.Google Scholar
  41. Roberts, P. H., & Ursell, H. D. [1], “Random walk on a sphere and on a Riemannian manifold,” Transactions of the Royal Society of London, Ser. A, Vol. 252 (1960), 317–356.Google Scholar
  42. Ross, Ronald [1], “An address on the logical basis of the sanitary policy of mosquito reduction,” British Medical Journal, May 13, 1905, 1025–1029.Google Scholar
  43. Soper, H. E. [1], Frequency Arrays, Cambridge, 1922.Google Scholar
  44. Spitzer, F. [1], Principles of Random Walk, 2nd ed., New York, 1976.Google Scholar
  45. Stephens, M. A. [1], “Random walk on a circle,” Biometrika, Vol. 50 (1963), 385–390.Google Scholar
  46. Treloar, L. R. G. [1], “The statistical length of long-chain molecules,” Transactions of the Faraday Society, Vol. 42 (1946), 77–82.Google Scholar
  47. Van der Pol, Balth., & Bremmer, H. [1], Operational Calculus Based on the Two-Sided Laplace Integral, 2nd ed., Cambridge, 1955.Google Scholar
  48. Vincenz, S. A., & Brucksaw, J. McG. [1], “Note on the probability distribution of a small number of vectors,” Proceedings of the Cambridge Philosphical Society, Vol. 56 (1960), 21–26.Google Scholar
  49. Von Smoluchowski, M. [1], “Sur le chemin moyen parcouru par les molecules d'un gaz et sur son rapport avec la théorie de la diffusion,” Bulletin International d'Academie des Sciences de Cracovie, Classe des Sciences Mathématiques et Naturelles, Année 1906, 202–213.Google Scholar
  50. Von Smoluchowski, M. [2], “Essai d'une théorie cinétique du mouvement Brownien et des milieux troublés,” Bulletin International d'Académie des Sciences de Cracovie, Classe des Sciences Mathématiques et Naturelles, Année 1906, 577–602.Google Scholar
  51. Von Smoluchowski, M. [3], “Drei Vorträge über Diffusion, Brownische Bewegung und Koagulation von Kolloidteilchen,” Physikalische Zeitschrift, Bd. 17 (1916), 557–571.Google Scholar
  52. Watson, G. N. [1], A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge, 1952.Google Scholar
  53. Watson, G. S. [1], “Distributions on the circle and sphere,” Journal of Applied Probability, Vol. 19A (1982), 265–280.Google Scholar
  54. Watson, G. S. [2], Statistics on Spheres, New York, 1982.Google Scholar
  55. Weiss, H. G., & Rubin, R. J. [1], “Random walks: Theory and selected applications,” Advances in Chemical Physics, Vol. 52 (1983), 363–505.Google Scholar
  56. Weyl, H. [1], “Mean motion,” American Journal of Mathematics, Vol. 60 (1938), 889–896.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co. KG 1985

Authors and Affiliations

  • Jacques Dutka
    • 1
  1. 1.Audits & Surveys, Inc.New York City

Personalised recommendations