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Densities of Short Uniform Random Walks

  Published:2011-11-03
 Printed: Oct 2012
  • Jonathan M. Borwein,
    CARMA, University of Newcastle, Australia
  • Armin Straub,
    Tulane University, New Orleans, LA, USA
  • James Wan,
    CARMA, University of Newcastle, Australia
  • Wadim Zudilin,
    CARMA, University of Newcastle, Australia
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Abstract

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.
Keywords: random walks, hypergeometric functions, Mahler measure random walks, hypergeometric functions, Mahler measure
MSC Classifications: 60G50, 33C20, 34M25, 44A10 show english descriptions Sums of independent random variables; random walks
Generalized hypergeometric series, ${}_pF_q$
Formal solutions, transform techniques
Laplace transform
60G50 - Sums of independent random variables; random walks
33C20 - Generalized hypergeometric series, ${}_pF_q$
34M25 - Formal solutions, transform techniques
44A10 - Laplace transform
 

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