Let r, r₁ be integer nonnegative numbers such that r < r₁. Let η₁, ... η_N be a general allocation scheme of (r₁ − r)n + rN particles by N cells, defined by independent random variables ξ₁, ... ξ_N which have power series distribution defined by the series Σ_k=0^∞b_kβ^k / k!. Denote: the random process X_n,N^{r₁}(t) = Σ^[tN]_i=1 I_{ηi=r1}, 0 ≤ t ≤ 1, F_n is an empirical process with the parameter n. It proved that if n, r and r₁ are fixed numbers and the condition Ar(r₁): b_k = 0, k < r, b_r > 0, b_k = 0, r < k < r₁, b_r1 > 0, is valid, then as N → ∞ the random processes X_n,N^{r1} converge in distribution in Skorokhod space to nF_n.