An extension of random walks on the dual of SU(2) (Q1820513)

In this paper, we announced a generalization of a result obtained by \textit{R. Eymard} and \textit{B. Roynette} [Lect. Notes Math. 497, 108-152 (1975; Zbl 0333.60065)]. We show how a class of random walks on the set \({\mathbb{N}}\) of all nonnegative integers can be defined via ultraspherical polynomials \(\{C_ n^{\lambda}\}\) with parameter \(\lambda\geq 1\). With \(\lambda =1\), these random walks are the same as those on the dual of SU(2) as defined by Eymard and Roynette. The main theorem states that, if \(\lambda\geq 1\) and \(\mu\) is an aperiodic measure on \({\mathbb{N}}\), then the random walk of law \(\mu\) via the ultraspherical polynomials \(\{C_ n^{\lambda}\}\) is transient. A proof of this result appeared in the author's paper in Bull. Malays. Math. Soc., II. Ser. 8, 63-71 (1985; Zbl 0592.60053).