Permutation bootstrap consistency conjecture for the max self-similarity estimator

Let $X(1),\ldots,X(N)$ be an i.i.d. sample, and let $\widehat H_i$, $i=1,\ldots,M$, be a permutation bootstrap sample of the estimator $\widehat H(j_1,j_2)$. Let the scales $j_1$, $j_2$ and the permutation sample size $M$ depend on the sample size $N$, with each tending to infinity as $N\to\infty$. Permutation bootstrap consistency conjecture. Under certain conditions on the rates of growth of $j_1$, $j_2$ and $M$, the empirical distribution of the permutation bootstrap sample $\widehat H_i$, $i=1,\ldots,M$, yields asymptotically consistent confidence intervals for $\widehat H$. The preceding discussion explains that the bootstrap statistics are exchangeable and have the same distribution as the original estimator, motivating their use as a proxy for its sampling distribution; the conjecture concerns the asymptotic validity of the resulting confidence intervals, but does not specify the required growth conditions.