Scaling limit of centered total-mass fluctuations

Determine whether the rescaled centered total-mass perturbation processes X_n(t)=n^{H_0}(Q_\kappa(t/n)-E[Q_\kappa(t/n)]) converge in distribution as n\to\infty to a stochastic process, and, if so, characterize the distribution of the limiting process.

Background

The variance asymptotic identifies t^{H_0} as the fluctuation scale of the total mass perturbation. The authors ask whether this scale yields a nontrivial process-level limit under temporal rescaling.

A limiting process and its law would help explain the fluctuation mechanism at and beyond the geometry–fluctuations threshold H_0=1/2.