Abstract
We study the moments of the distance traveled by a walk in the plane with unit steps in random directions. While this historically interesting random walk is well understood from a modern probabilistic point of view, our own interest is in determining explicit closed forms for the moment functions and their arithmetic values at integers when only a small number of steps is taken. As a consequence of a more general evaluation, a closed form is obtained for the average distance traveled in three steps. This evaluation, as well as its proof, rely on explicit combinatorial properties, such as recurrence equations of the even moments (which are lifted to functional equations). The corresponding general combinatorial and analytic features are collected and made explicit in the case of 3 and 4 steps. Explicit hypergeometric expressions are given for the moments of a 3-step and 4-step walk and a general conjecture for even length walks is made.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bailey, D.H., Borwein, J.M.: Hand-to-hand combat: Experimental mathematics with multi-thousand-digit integrals. J. Comput. Sci. (2010). doi:10.1016/j.jocs.2010.12.004. Available at: http://www.carma.newcastle.edu.au/~jb616/combat.pdf
Bailey, D.H., Borwein, J.M., Broadhurst, D.J., Glasser, M.L.: Elliptic integral evaluations of Bessel moments and applications. J. Phys. A, Math. Theor. 41, 5203–5231 (2008)
Barrucand, P.: Sur la somme des puissances des coefficients multinomiaux et les puissances successives d’une fonction de Bessel. C. R. Hebd. Séances Acad. Sci. 258, 5318–5320 (1964)
Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987)
Borwein, J.M., Straub, A., Wan, J.: Three-step and four-step random walk integrals. Exp. Math. (2011, in press). Available at: http://www.carma.newcastle.edu.au/~jb616/walks2.pdf
Borwein, J.M., Straub, A., Wan, J., Zudilin, W.: Densities of short uniform random walks. Canad. J. Math. (2011, in press). Available at: http://www.carma.newcastle.edu.au/~jb616/densities.pdf, http://arxiv.org/abs/1103.2995v1
Broadhurst, D.: Bessel moments, random walks and Calabi-Yau equations. Preprint, November 2009
Cools, R., Kuo, F.Y., Nuyens, D.: Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput. 28, 2162–2188 (2006)
Crandall, R.E.: Analytic representations for circle-jump moments. Preprint, October 2009
Donovan, P.: The flaw in the JN-25 series of ciphers, II. Cryptologia (2010, in press)
Fettis, F.E.: On a conjecture of Karl Pearson, pp. 39–54 in Rider Anniversary Volume (1963)
Hughes, B.D.: Random Walks and Random Environments, vol. 1. Oxford University Press, London (1995)
Kluyver, J.C.: A local probability problem. Ned. Acad. Wet. Proc. 8, 341–350 (1906)
Mattner, L.: Complex differentiation under the integral. Nieuw Arch. Wiskd. IV Ser. 5/2(1), 32–35 (2001)
Merzbacher, E., Feagan, J.M., Wu, T.-H.: Superposition of the radiation from N independent sources and the problem of random flights. Am. J. Phys. 45(10), 964–969 (1977)
Pearson, K.: The random walk. Nature 72, 294 (1905)
Pearson, K.: The problem of the random walk. Nature 72, 342 (1905)
Pearson, K.: A Mathematical Theory of Random Migration. Mathematical Contributions to the Theory of Evolution XV. Drapers (1906)
Petkovsek, M., Wilf, H., Zeilberger, D.: A=B, 3rd edn. AK Peters, Wellesley (2006)
Rayleigh, L.: The problem of the random walk. Nature 72, 318 (1905)
Richmond, L.B., Shallit, J.: Counting abelian squares. Electron. J. Comb. 16, R72 (2009)
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com/sequences, 2009
Titchmarsh, E.: The Theory of Functions, 2nd edn. Oxford University Press, London (1939)
Verrill, H.A.: Some congruences related to modular forms. Preprint MPI-1999-26, Max-Planck-Institut (1999)
Verrill, H.A.: Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations (2004). arXiv:math/0407327v1 [math.CO]
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1941)
Wilf, H.: Generatingfunctionology, 2nd edn. Academic Press, San Diego (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Part of D. Nuyens’s work done while a research associate at the Department of Mathematics and Statistics, University of New South Wales, Australia.
The work of the third author, as a graduate student, was partially funded by NSF-DMS 0070567.
Rights and permissions
About this article
Cite this article
Borwein, J.M., Nuyens, D., Straub, A. et al. Some arithmetic properties of short random walk integrals. Ramanujan J 26, 109–132 (2011). https://doi.org/10.1007/s11139-011-9325-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-011-9325-y