SUPPORTS OF CONVOLUTIONS OF IDENTICAL DISTRIBUTIONS.
Herman Rubin · Defense Technical Information Center (DTIC) · 1967
Simple examples are given of convolutions of identical singular univariate distributions whose convolution is absolutely continuous. Furthermore, it is shown that there exist infinitely divisible univariate distributions F and G such that the Hausdorff dimensions of the supports (lowest-dimensional sets of probability 1) of F(t), G(t), and F(t) * G(t) have any preassigned behavior consistent with non-decrease under convolution. Absolute continuity can be included as a dimension greater than 1. Furthermore, the Levy-Khintchine representations of F and G may be required to be purely discrete or purely signular. (Author)