Letter to the Editor the Higher Moments of the Number of Returns of a Simple Random Walk

We consider a simple random walk starting at 0 and leading to 0 after 2n steps. By a generating functions approach we achieve closed formulae for the moments of the random variables `number of visits to the origin' . GENERATING FUNCTIONS ; CATALAN NUMBERS AMS 1991 SUBJECT CLASSIFICATION : PRIMARY 60J15 Let Xk , k = 1, 2, • • • be independent and identically distributed random variables with P{Xk = 1 } = P{Xk = 1} = Z . Consider the simple random walk n, S,,, = > Xk with So = 0 and Sz„ = 0, k=1 i .e . a simple random walk starting at 0 and leading to 0 after 2n steps. Let the variable T be the number of visits to the origin . In [2] the higher moments of this random variable were expressed as sums where the number of terms increases with n . The authors also gave asymptotic formulae by means of a Mellin transform approximation of the sums . In a following paper [4] the higher moments are described by certain recurrence relations with `full history', i .e. using all moments of smaller order . The aim of this note is, motivated by a comment in [2], to give closed-form expressions (i .e . the number of terms is independent of n) for the moments in question . Our generating functions approach would also allow one to get the asymptotics in an elementary way . We mention two other problems where this kind of approach can be used . In order to get a suitable expression for the generating function we decompose the family V of random walks in question according to their returns. Noting that between any two consecutive returns a walk is either positive (W + ) or negative (W_) we have Adv. Appl. Prob. 26, 561-563 (1994) Printed in N. Ireland © Applied Probability Trust 1994 (1) <W=(W+ + V)*, where the asterisk denotes the combinatorial construction of forming finite sequences of elements of the concerned set of objects . It is well known that the generating function of `W+ (or W) involves the Catalan Received 21 April 1993; revision received 7 July 1993 . * Postal address : Department of Algebra and Discrete Mathematics, Wiedner Hauptstr . 8-10/118, Technical University of Vienna, A-1040 Wien, Austria . 561