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Let (p k ) k∈N be a discrete probability distribution for which the counting function x → #{k ∈ N : p k ≥ 1/x} belongs to the de Haan class Π. Consider a deterministic weighted branching process generated by (p k ) k∈N . A nested Karlin's occupancy scheme is the sequence of Karlin balls-in-boxes schemes in which boxes of the jth level, j = 1, 2, . . . are identified with the jth 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 K n (j) the number of occupied (ever hit) boxes in the jth level. For each j ∈ N, we prove a functional limit theorem for the vector-valued process (K⌊e T +u ⌋ ) u∈R , properly normalized and centered, as T → ∞. The limit is a vector-valued process whose components are independent stationary Gaussian processes. An integral representation of the limit process is obtained.

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A nested Karlin's occupancy scheme is a symbiosis of classical Karlin's balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer j, identify the jth generation individuals with the jth generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree 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.Assume that there are n balls and that the discrete probability distribution responsible for the first generation is Weibull-like. Denote by K