Abstract
Let (S n∶n>-1) be a random walk on a hypergroup (ℝ + , *), i.e., a Markov chain with transition kernelN(x, A) = εx * μ(A), where μ is a fixed probability measure on ℝ + such that the second moment exists. Then depending on the growth of the hypergroup two situations can occur: when (ℝ + , *) is of exponential growth then it is shown thatS n is asymptotically normal. In the case of polynomial growth {more precisely, if the densityA of the Haar measure of (ℝ + , *) satisfies limχ→∞[χA′(χ)/A(χ)]=β}, the normalized variablesS n/[n Var(μ)/(β+1)]1/2 converge to a Rayleigh distributionρ β with parameter β.
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Zeuner, H. The central limit theorem for Chébli-Trimèche hypergroups. J Theor Probab 2, 51–63 (1989). https://doi.org/10.1007/BF01048268
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DOI: https://doi.org/10.1007/BF01048268