Distribution of the combinatorial multisets component vectors

. We explore a class of random combinatorial structures called weighted multisets. Their components are taken from an initial set satisfying general boundedness conditions posed on the number of elements with a given weight. The component vector of a multiset of weight n taken with equal probability has dependent coordinates, nevertheless, up to r = o ( n ) of them as n → ∞ , we approximate by an appropriate vector comprised from independent negative binomial random variables. The main result is an estimate of the total variation distance. For illustration, we present a central limit theorem for a sequence of additive functions.


Introduction
We examine weighted combinatorial multisets. They are comprised from components belonging to an initial class P of elements having weights in N. The repetitions are allowed while the order is irrelevant. The weight of a multiset is the sum of weights of its components. The empty multiset has the zero weight.
Let us denote by P j ⊂ P the subset of elements of weight j ∈ N and let π(j) = |P j | < ∞ be its cardinality. For an n ∈ N, sets := (s 1 , . . . , s n ) ∈ Z n + and ℓ(s) := 1s 1 + · · · + ns n . Let M n be the class of multisets σ of weight n and denote by k j (σ) 0 the number of components of weight j, 1 j n, in σ ∈ M n . The vector k(σ) := k 1 (σ), . . . , k n (σ) is called the component vector of σ. Note that ℓ(k(σ)) = n if σ ∈ M n . All quantitative information about the introduced class of multisets lays in the following formal relation satisfied by the generating function: If the uniform probability measure ν n is introduced in the set M n , then the distribution of component vector satisfies the conditioning relation ν n (k(σ) =s) = P (γ = s|ℓ(γ) = n), whereγ = (γ 1 , . . . , γ n ), and γ j = NB (π(j), x j ), 1 j n, are mutually independent negative binomial random variables (i.r.vs) defined on some probability space (Ω, F , P ) with parameters (π(j), x), where 0 < x < 1 is arbitrary. An extensive list of instances and the historical survey on investigations of random multisets can be found in [2] and [1]. In the present note, we discuss only the results concerning the total variation approximations of the truncated component vectors k r (σ) = (k 1 (σ), . . . , k r (σ)) by appropriate vectors with independent coordinates if r = r(n) and r = o(n) as n → ∞.
Let ρ TV denote the total variation distance and L(·) be the distribution under the relevant probability measure. For brevity, we will use ≪ as an analog of O(·). As it has been proved by D. Stark [7] (see also [1]), the regularity condition π(j) ∼ θq j j −1 , j → ∞, where θ > 0 and q > 1 are constants, and some other extra technical requirements imply Here and afterwardsγ r = (γ 1 , . . . , γ r ) and γ j = NB(π(j), q −j ), 1 j r n, are mutually independent negative binomial r.vs. The positive quantity υ depends on the constants in the conditions. A similar problem for the so-called additive arithmetical semigroups has been dealt with by J. Knopfmacher and W.-B. Zhang [3]. Putting regularity conditions on the number of semigroup elements of a given degree, they actually exploited some regularity of the number of prime elements. We generalize the estimates obtained in [1] and the most interesting part of that from [3].
In the sequel, the hidden constants, if not indicated otherwise, will depend only on c 0 , c 1 and q.

Theorem 1. Let the class of multisets be generated by a set P such that
for all j 1, where 0 < c 0 c 1 < ∞ and q > 1 are constants. Then there is a positive constant υ = υ(c 0 , c 1 ) such that (1) holds for 1 r n.
Theorem 1 will be proved using the analytical method proposed in 2002 by E. Manstavičius [5] and applied by him for other combinatorial structures called assemblies (see [4]). In Section 1, we present the main steps of the proof, the detailed exposition can be found in our master thesis [6]. In the last section, we prove a central limit theorem for a sequence of additive functions defined on the discussed class of multisets.
The next claim is crucial in the applied approach. Instead of integrating the remaining integral J 0 , we change its integrand and return to D n .

Central limit theorem
As an application of Theorem 1, we now present an analog of the well-known Feller-Lindeberg theorem. As previously, let condition (2) be satisfied and γ j = NB (π(j), q −j ), where q > 1 and 1 j n, are i.r.vs. Let a nj ∈ R, X nj = a nj γ j if 1 j n, and X n = X n1 + · · · + X nn . Set Φ(x) for the standard normal distribution function, u * := min{|u|, 1}sgnu, and , 0 y n.
Assume that n → ∞ in the limit relations.
Lemma 7. In the notation above, let a nj = o(1) for each fixed j ∈ N. The relation P (X n − b n < x) = Φ(x) + o(1) with some b n ∈ R uniformly in x ∈ R holds if and only if, for every ε > 0, and b n = α(n) + o(1).
Proof. The i.r.vs X nj , 1 j n, are infinitesimal. Hence the claim is just a special case of the mentioned Feller-Lindeberg theorem.
Let h nj (k) be a three-dimensional real sequence such that h nj (0) ≡ 0 for j n. Define the sequence of additive functions h n : M n → R by setting h n (σ) = j n h nj (k j (σ)). Theorem 2. Let the class of multisets M n satisfy condition (2). Assume that h nj (k) = o(1) for every fixed j, k ∈ N. If conditions (7) and (8) are satisfied for a nj := h nj (1), then for every 0 < δ < 1, then convergence (9) with some b n implies relations (7) and (8).
Proof. We indicate the main steps only. First, we verify that convergence (9) can hold only simultaneously with that for the sequence of functions h n (σ) defined via h nj (k) satisfying the condition h nj (k) = kh nj (1) =: ka nj for 1 j n. Next, we split the latter into two parts: h n (σ) = ( j r + r<j n )a nj k j (σ) =: h (r) n (σ) + f n (σ) As in [4], one can check that condition (7) yields a sequence r = r(n) → ∞ such that r = o(n) and ν n f n (σ) − α(n) − α(r) ε = o(1) for every ε > 0. Moreover, by Theorem 1 and Lemma 7, uniformly in x ∈ R. The last two relations furnish the proof of the sufficiency part.
In the necessity part, we can again use Theorem 1 and Lemma 7 because of condition (10) also implies (11). So we arrive at the last relation. Consequently, the necessity in Theorem 2 is assured by that in Lemma 7. The theorem is proved.