  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.
