Abstract
LetG 0 be a (not necessarily Abelian) discrete group, and letp 0 be a probability onG 0. Form the direct sumG of a countable number of copies ofG 0, and letp be a probability onG which is a convex combination of copies ofp 0 on the factorsG 0. We consider the associated random walk onG, and study the asymptotic behavior of the probabilityp *n (e) of returning to the identity elemente aftern steps. This behavior depends heavily on the choice of the convex combination, and is considerably more complicated than the behavior for finite direct sums described in, for example, Refs. 4 and 5. A by-product of the main results is a description of the asymptotic behavior of the moments of the Cantor distribution and of the moment-generating function of symmetric Bernoulli convolutions.
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Cartwright, D.I. Random walks on direct sums of discrete groups. J Theor Probab 1, 341–356 (1988). https://doi.org/10.1007/BF01048724
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DOI: https://doi.org/10.1007/BF01048724