  We consider the hard-edge scaling of the Mittag-Leffler ensemble confined to
a fixed disk inside the droplet. Our primary emphasis is on fluctuations of
rotationally-invariant additive statistics that depend on the radius and thus
give rise to radius-dependent stochastic processes. For the statistics
originating from bounded measurable functions, we establish a central limit
theorem in the appropriate functional space. By assuming further regularity, we
are able to extend the result to a vector functional central limit theorem that
additionally includes the first hitting "time" of the radius-dependent
statistic. The proof of the first theorem involves an approximation by
exponential random variables alongside a coupling technique. The proof of the
second result rests heavily on Skorohod's almost sure representation theorem
and builds upon a result of Galen Shorack (1973).
