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

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