Studentized Distributional Bands at Cancellation Points in Heavy-Tailed Cluster Samples

We study simultaneous inference for an observation-weighted distribution function when cluster sizes have a regularly varying tail with an index between one and two. Conditional cluster distributions vary along a common contrast, and cluster size may depend on the contrast coefficient. At zeros of the contrast, the leading heavy-tailed contribution cancels, and a Gaussian fluctuation remains. We derive local limits in shrinking neighborhoods of isolated zeros of arbitrary finite order and determine the limit of the supremum of the studentized empirical distribution function. The local range is the whole real line at an odd-order zero and a half-line at an even-order zero. This distinction yields different supremum laws. When all zeros have odd order, the squared limit is the sum of a squared stable self-normalized variable and the largest squared coordinate of an independent Gaussian vector. We prove that cluster subsampling consistently estimates the supremum law at each fixed model and yields asymptotically correct simultaneous coverage. The proofs control the denominator across the Gaussian and stable scales and account for estimation of the centering distribution. Explicit examples include informative cluster size. A finite-sample diagnostic examines coverage, band width, and sensitivity to the subsample size. The results do not assert uniform validity as the heterogeneity vanishes or the tail index approaches an endpoint.