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On random walks on wreath products. (English) Zbl 1021.60004

Let \(K\) and \(B\) be groups with finite generators. Considering \(H\) as an \(H\) set, we make a twisted and restricted wreath product \(G= K\text{ wr }H\). Let \(\nu\) (resp. \(\mu\)) be a symmetric probability measure whose support generates \(K\) (resp. \(H\)) and assume that the support of \(\mu\) contains the unit element of \(H\). Considering these measures as measures on \(G\), define the measure \(q =\nu\ast\mu\ast\nu\) on \(G\). In this article, the asymptotic behavior of \(q^{2n}(e)\), where \(q^{2n}\) is the \(2n\)-convolution of \(q\) and \(e\) is the unit element of \(G\), is considered for various algebraic structures of groups \(K\) and \(H\). In formula (3.6) (p. 964) \(V(m)\) is put to \(m\), however it must be \(2^{m}\), so Lemma 3.8 (p. 964), which is crucial in the following arguments, must be reexamined. In my try, it does not hold. Also many inconsistencies of symbols are to be found.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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