Asymptotic normality of nonparametric kernel type deconvolution density estimators: crossing the Cauchy boundary

We derive asymptotic normality of kernel type deconvolution density estimators. In particular, we consider deconvolution problems where the known component of the convolution has a symmetric λ-stable distribution with 0 < λ ≤ 2. It turns out that the limit behavior changes if the exponent parameter λ passes the value 1, the case of Cauchy deconvolution.