On the Hausdorff dimension of some random Cantor sets (Q1881800)

The authors study random Cantor set \(C_{X,p}\) obtained by deleting intervals of random lengths \(X_n^{-p}\), \(n\geq1\), from the unit segment. Note that \(X_1\) is the length of the segment deleted at the first step, \(X_2\) and \(X_3\) are the lengths of two segments deleted from the left and right remaining segments, etc. It is shown that, if the sequence of random variables \(X_n\) satisfies \[ \sum_{k=1}^\infty P\{| X_k-X| >h(k)\}<\infty \] with a monotone function \(h\) such that \(0<h(x)<x\), \(h(x)/x\to0\) as \(x\to\infty\) and \(\sum h(2^j)2^{-j}<\infty\), then the Hausdorff dimension of \(C_{X,p}\) equals \(p\) almost surely and \(C_{X,p}\) has a nontrivial Hausdorff measure of order \(p^{-1}\).