Some arithmetic properties of short random walk integrals. (English) Zbl 1233.60024

Borwein, Jonathan M.; Nuyens, Dirk; Straub, Armin; Wan, James

Ramanujan J. 26, No. 1, 109-132 (2011).

Summary: 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.

MSC:

60G50	Sums of independent random variables; random walks
33C20	Generalized hypergeometric series, \({}_pF_q\)
05A10	Factorials, binomial coefficients, combinatorial functions

Keywords:

random walks; hypergeometric functions; high-dimensional integration; analytic continuation

Software:

OEIS

References:

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