Effective bounds of the variance of statistics on multisets of necklaces

Vėrinių multiaibių statistikos dispersijos efektyvūs įverčiai

Let (P, ǁ · ǁ) be an initial set of weighted objects and

for every j = 1, 2, . . . . Examine the set G with the extended weight function ǁ · ǁ of multisets comprised of p ∈ P. Namely, a ∈ G if a = {p₁, . . . , p_r} and ǁaǁ = ǁp₁ǁ + · · · + ǁp_rǁ including the empty multiset ∅ of weight 0. Then

where l(k¯) = 1k₁ + + nk_n if k¯ = (k₁, . . . , k_n) $∊N_0^n$ and n $∊{\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.

In the present paper, we deal with the multisets for which m(n) = qⁿ, where q ≥ 2 is an arbitrary natural number. If q is a prime power, then may be interpreted as ${\mathrm{F}}_q^{*}\left[t\right]$, the set of monic polynomials over a finite field F_q. Then P is the subset of irre- ducible polynomials. For an arbitrary such q, there exist combinatorial constructions, called multisets of necklaces satisfying m( n) = qⁿ (see, [1, Example 2.12, p. 43]). For multisets, we have the following relations

where in the summations, d runs over natural divisors of n and µ(d) stands for the Möbius function. The equalities are equivalent to the formal power series relation

Take an a ∈ G_n uniformly at random, that is, sample it with probabilityν_n({a}) = q⁻ⁿ, n ∈ N and ν₀({∅ }) = 1. If k_j(a) ≥ 0 is the number of elements pi in a ∈ G_n of weight j, then k(a) = k₁(a), . . . , k_n(a) is the structure vector of a ∈ G_n satisfying l(k(a)) = n. Its distribution is

where $\bar{s}=(s_1,...,s_{\mathrm{n}})\in {\mathbb{N}}_0^n$ and 1.{.} stands for the indicator function.

We are interested in the distribution with respect to ν_n of the linear statistics

The number of components in a is such a function, namely, it equals k₁(a)+ +k_n(a). We refer to [1] for more sophisticated examples.

The present paper is devoted to the variance of $h\left(\bar{c}\right)$ which is a sum of dependent random variables (r.vs) as the relation $h(\bar{j},a)=\ell \left(\bar{k}\right(a\left)\right)=n$ for each a $\in G_n$ shows. Estimating it, we propose an approach to overcome technical obstacles stemming from dependence.

In the sequel, the expectations and variances with respect to νn will be denoted by E_n and V_n while, when the probability space (Ω, F , P ) is not specified, we will respectively use the notation E and V. The summation indexes i, j, l, k, m, m¹ and m² will be natural numbers.

Theorem 1 If $\bar{c}\in {ℝ}^n$ and $n\in {\mathbb{N}}_0$, then

The sketch of the proof is given at the beginning of Section 2.

t is known [1] that, for a fixed j, the r.v. k_j(a) converges in distribution to the r.v. γj distributed according the negative binomial law NB(π(j), q^−j). If {γ1, γ2, . .} . are mutually independent, define the statistics Y_n = c₁γ₁ + + nγ_n. We shall see that the first sum on the right-hand side in (4) is close to VY_n; therefore, estimating V_nh(c¯), we use the following quadratic forms:

Theorem 2 If n ≥ 2, then

The inequality becomes an equality for

Corollary 1 If n ≥ 2 and$\bar{c}\ne \bar{0},$, then

The inequalities are trivial for functions proportional to $h(\bar{j},a)=n$ if $a∊G_n$ because of $V_{n}h\left(j\right)=0$ then. A shift of $\bar{c}$ eliminates this inconvenience. Observe that either of $B_n(\bar{c}-t\bar{j})$ and $R_n(\bar{c}-t\bar{j})$ attain their minimums in $t∊ℝ$ at

Theorem 3 If n ≥ 3, then

Both inequalities are sharp.

Corollary 2If n ≥ 3 and $\bar{c}\ne \alpha \bar{j}$ for every $\alpha \in ℝ$

The proofs of the last two theorems presented in Section 2 are built upon the ideas and auxiliary results obtained in [4], [2] and [5].

We firstly recall known facts about random multisets which can be found in [3] and [1, Section 2.3]. Let ${\bar{y}}^{\left(x\right)}=(y_1^{\left(x\right)},y_2^{\left(x\right)},...)$ be the infinite dimensional vector of independent r.vs having the negative binomial distributions NB(π(j), x^j), namely,

where 0 < x ≤ q⁻¹. Then γ_j^{(q−1 )} = γ_jwhich has been introduced in Introduction. For convenience, we extend k(a) to k(a) := (k₁(a), . . . , k_n(a), 0, . . . ) and use infinite dimensional vectors.$Set{\theta }^{\left(x\right)}=1y_1^{\left(x\right)}+···+n,y_{\mathrm{n}}^{\left(x\right)}+(n+1)y_{\mathrm{n}+1}^{\left(x\right)}+,...$The latter r.v. is well defined if 0 < x < q⁻¹, since the condition of the Boreli–Cantelli lemma is satisfied:

Lemma 1 $If\bar{s}=(s_1,...,s_{\mathrm{j}},s_{\mathrm{j}}+1,...)∊{\mathbb{N}}_0^{\mathrm{\infty }}$ and 0 < x < q⁻¹, then

Proof. Actually, this is Lemma 2.2 in [3] stated there for F_q[t]. The details remain the same in the more general case.

Lemma 2 For a functional $\mathrm{\Psi }:{\mathbb{N}}_0^{\mathrm{\infty }}\to ℝ$ such that $E\left|\mathrm{\Psi }\right({\bar{\mathrm{\gamma }}}^{\left(x\right)}\left)\right|<\mathrm{\infty },$, we have

Proof. Apply Lemma 1 in the double averaging as follows:

Proof of Theorem 1. It is straightforward. Applying the last lemma for the relevant Ψ , one can easily find the needed mixed moments of k_j(a), 1 ≤ j ≤ n, and further, the variance of the linear combination h(a).

To prove Theorems 2 and 3, we will apply the following lemmas concerning par- ticular matrices and quadratic forms.

Lemma 3LetU = ((u_ij)), i, j ≤ n, be the symmetric matrix with the entries

The spectrum of U is the set{1, 1/2, 1/3, . . . , ( 1)ⁿ⁻¹/n} . The eigenvectors corresponding to the first three eigenvalues areproportional to ${\bar{c}}_r=c_{r1},...,c_{r\mathrm{n}}),$ wherer = 1, 2, 3and, for j ≤ n,

Proof. This is the byproduct of works [4] and [2].

Afterwards, let ${\bar{e}}_r,1⩽m⩽n,$ be the orthogonal basis of ${ℝ}^n$ comprised of the eigenvectors of U and ${\bar{x}}^{\prime }$ means the transposed vector $\bar{x}$.

Lemma 4 If $b_m\in ℝ$ and 1 ≤ m ≤ n and n ≥ 2, then

If n ≥ 3and

then

Moreover, each bound in (9) and (11) are achieved, respectively, for $b_m=e_{rm}/\sqrt{m},$, wherer = 2, 1, 3and $e_{rm}$ have been defined in Lemma 3.

Proof. Inequalities (9) are seen from Lemma 3 after the substitution $b_m=x_m/\sqrt{m},$,m<n since the extreme eigenvalues are 1 and 1/2.

After the same substitution, we further examine the quadratic form with the matrix U . Condition (10) reckons the subspace of vectors $\bar{x}=(x_1,...,x_{\mathrm{n}})$ satisfying $x_1+···+x_{\mathrm{j}}\sqrt{j}+···+x_{\mathrm{n}}\sqrt{n}=\bar{x}·{\bar{e}}_1^{\prime }=0.$ This subspace is spanned over the first eigenvector. In other words, under (10), only the form values obtained in the subspace L ⊂ Rⁿ spanned over the vectors ${\bar{e}}_2,...,{\bar{e}}_{\mathrm{n}}$ count. Hence

Returning to b_m, from this we obtain inequality (11).

Proof of Theorem 2. After grouping the summands, expression (4) can be rewritten as follows:

Now evidently estimate (5) follows from Lemma 4 with

Moreover, it becomes an equality if we take $c_j=c_j^{*}$ satisfying

which by the Möbius inversion formula and (1) may be rewritten as (6).

To prove the first assertion of Corollary 1, it suffices to estimate the inner sum in $R_n\left(\bar{c}\right),$, namely,

Further, using the expression of VY_n, we just estimate the remainder:

Plugging both estimates into (5), we obtain the first inequality in Corollary 1 with ≤ instead of <. In fact, we obtained the strict inequality since Cauchy’s inequality applied in the last step is strict if $\bar{c}$ is not proportional to$\bar{j}$, and in this exceptional case,$Vh\left(\bar{c}\right)=0.$ .

Proof of Theorem 3. Observe that $V_{n}h\left(\bar{c}\right)=V_{n}h\left(\bar{c}\right)-tn)-V_{n}h(\bar{c}-t\bar{j})$ for every $t∊ℝ$. Hence the right-hand inequality follows from (5) applied for the shifted statistics.

To get the lower bound of variance, we combine (4) and (11). We start with

where

and m ≤ n. By the definition of tc the latter sequence satisfies condition (10). Hence by (11),

This and (4) imply the lower bound. Moreover, the latter is sharp since Lemma 4 assures this by a choice of a particular sequence ${\bar{b}}_m,m⩽n.$.