The local limit problem and Hölder-continuity
The local limit problem is investigated for sequences \((p_ n)\) of probability densities with stable limit densities \(q_{\alpha}\) having characteristic exponent \(\alpha\in (0,2]\). It is shown that certain continuity-properties (Hölder-continuity) are necessary and - under appropriate additionl conditions - sufficient for \[ \sup_{x}| p_ n(x)-q_{\alpha}(x)| \to 0\quad as\quad n\to \infty. \] In this sense the speed of convergence is also studied.
scientific article; zbMATH DE number 4078369
scientific article