Abstract
We prove two central limit theorems for real identically distribution random variables where the distribution is a complex-valued Borel measure μ. The results involve the weak convergence of the suitably scaledn-fold convolution of certain complex atomic or absolutely continuous measures μ of spectral radius 1 to ahypergaussian measure γβ whose Fourier-Stieltjes transform is exp(−θ2β for a positive integer β. The proof uses a generalization of the method of characteristic functions. Counter-examples are given to rather more general statements that had appeared previously in the literature. This research arose in connection with problems related to general tauberian theorems on the line for complexvalued summability methods which are discussed at the end of the paper. Some interesting examples are given generalizing well-known theorems related to Euler and Borel summability.
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