A functional limit theorem for nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

Let (pk)k∈N be a discrete probability distribution for which the counting function x↦#{k∈N:pk≥1/x} belongs to de Haan class Π. Consider deterministic weighted branching process generated by (pk)k∈N. A nested Karlin's occupancy scheme is sequence of Karlin balls-in-boxes schemes in boxes jth level, j=1,2,… are identified with generation individuals and hitting probabilities corresponding weights. The collection balls same all generations, each ball starts at root moves along tree according following rule: transition from mother box daughter occurs given ratio Assuming there n balls, denote Kn(j) number occupied (ever hit) level. For j∈N, we prove functional limit theorem vector-valued (K⌊eT+u⌋(1),…,K⌊eT+u⌋(j))u∈R, properly normalized centered, as T→∞. whose components independent stationary Gaussian processes. An integral representation obtained.