Exact Hausdorff measure on the boundary of a Galton--Watson tree

A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton--Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions $\phi$ according to whether $\phi$-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes's conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved.Comment: Published at http://dx.doi.org/10.1214/009117906000000629 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On convolution closure properties of subexponentiality approaching from densities

Non-closedness of subexponentiality by the convolution operation is well-known. We go a step further and show that subexponentiality and non-subexponentiality are generally changeable by the convolution. We also give several conditions, by which (non-) subexponentiality is kept. Most results are given with densities, which are easily converted to those for distributions. As a by-product, we give counterexamples to several past results, which were used to derive the non-closedness of the convolution, and modify the original proof.Comment: 23 page