Gaussian Fluctuation for Smoothed Local Correlations in CUE

Abstract

Motivated by the Rudnick-Sarnak theorem we study limiting distribution of smoothed local correlations of the form

$$\begin{aligned} \sum _{j_1, j_2, \ldots , j_n} f(N(\theta _{j_2}-\theta _{j_1}), N(\theta _{j_3}-\theta _{j_1}), \ldots , N(\theta _{j_n}-\theta _{j_1})) \end{aligned}$$

for the Circular United Ensemble of random matrices for sufficiently smooth test functions.

1 Introduction and Formulation of the Main Result

The Circular $\beta$-Ensemble (C$\beta$E) is determined by the joint probability density

$$\begin{aligned} p_N^\beta (\theta _1, \ldots , \theta _N)=\frac{1}{Z_{N,\beta }}\prod _{1\le j< k\le N}\left| e^{i\theta _j}-e^{i\theta _k}\right| ^\beta , \ \ \ \end{aligned}$$

(1.1)

where $\theta =(\theta _1, \ldots , \theta _N) \in {\mathbb {T}}^N,$ i.e. $0\le \theta _1, \ldots , \theta _N <2 \pi , \ \beta >0,$ and the partition function $Z_{N,\beta }$ is given by

$$\begin{aligned} Z_{N,\beta }=\int _{{\mathbb {T}}^N} \prod _{1\le j< k\le N}\left| e^{i\theta _j}-e^{i\theta _k}\right| ^\beta d\theta = (2\pi )^N \frac{\Gamma \left( 1+\frac{\beta N}{2}\right) }{\Gamma \left( 1+ \frac{\beta }{2}\right) ^N}. \end{aligned}$$

(1.2)

The ensemble was introduced in Random Matrix Theory by Dyson in [8]. The Circular Unitary Ensemble (CUE) corresponds to $\beta =2.$ In this special case (1.1) describes the joint distribution of the eigenvalues of an $N\times N$ random unitary matrix U distributed according to the Haar measure.

In [1]–[3], we studied the limiting distribution of pair counting functions

$$\begin{aligned} S_N(f)=\sum _{1\le i\ne j\le N} f(L_N (\theta _i-\theta _j)_c), \end{aligned}$$

(1.3)

in the Circular $\beta$-Ensemble, where $1\le L_N\le N$ and $(\theta _i-\theta _j)_c$ is the phase difference on the unit circle, i.e.

$$\begin{aligned} (x-y)_c= {\left\{ \begin{array}{ll} x-y&{}\text {if } -\pi \le x-y<\pi ,\\ x-y -2 \pi &{}\text {if } \pi \le x-y<2 \pi ,\\ x-y +2 \pi &{}\text {if} -2 \pi<x-y<-\pi . \end{array}\right. } \end{aligned}$$

(1.4)

The case $\beta =2, \ L_N=N$ was motivated by a classical result of Montgomery on pair correlation of zeros of the Riemann zeta function [18]–[19]. Let us denote the “non-trivial” zeros as $\{ 1/2 \pm i \gamma _k, \ \gamma _k>0\}.$ Montgomery’s theorem suggested that the rescaled zeros

$$\begin{aligned} {\tilde{\gamma }}_k= \frac{\gamma _k}{2 \pi } \log (\gamma _k), \end{aligned}$$

asymptotically behave as the eigenvalues of a large random unitary (CUE) matrix. Namely, Montgomery studied the asymptotic behavior of the the statistic

$$\begin{aligned} F_T(\alpha )=T^{-1} \sum _{0<{\tilde{\gamma }}_j, \tilde{\gamma _k}\le T} \exp (i \alpha ({\tilde{\gamma }}_j-{\tilde{\gamma }}_k)) \frac{4}{4+({\tilde{\gamma }}_j-{\tilde{\gamma }}_k)^2/\log (T)^2}, \end{aligned}$$

for real $\alpha$ and large real T. Assuming the Riemann Hypothesis, Montgomery argued that

$$\begin{aligned} F(\alpha ):=\lim _{T\rightarrow \infty } F_T(\alpha )={\left\{ \begin{array}{ll} |\alpha |, &{}\text { if } \ 0<|\alpha |<1 \\ 1 , &{}\text {if} \ |\alpha |\ge 1. \end{array}\right. }. \end{aligned}$$

(1.5)

Montgomery rigorously proved (1.5) for $0<|\alpha |<1$ and provided heuristic arguments to support the formula for $|\alpha |\ge 1.$ He also proved that for $\alpha =0$ and large T the statistic behaves as

$$\begin{aligned} F_T(0)=\log (T)^{-1} (1+o(1)). \end{aligned}$$

Since $\alpha \mapsto \delta (\alpha )+\min (|\alpha |, 1)$ is the Fourier transform of $\delta (x)+ 1-\left( \frac{\sin (\pi x)}{\pi x}\right) ^2,$ the formula (1.5) implies that the asymptotic behavior of the two-point correlations of the rescaled zeros of the Riemann zeta functions and the eigenvalues of a large random unitary matrix coincide in the limit.

In 1994, Hejhal [13] extended the result to the case of three-point correlation functions, again under a technical condition on the support of the Fourier transform of a test function. In 1996, Rudnick and Sarnak [20] considered the l-level correlation sum, $l\ge 2,$

$$\begin{aligned} R_{T,l}(g)=\sum ^{*} g({\tilde{\gamma }}_{j_1}, \ldots , {\tilde{\gamma }}_{j_l}), \end{aligned}$$

(1.6)

where the sum is restricted to distinct rescaled zeros in the interval [0, T] for large T,  and g is a smooth test function satisfying:

• (i) $g(x_1, x_2, \ldots , x_l)$ is symmetric.

• (ii) $g(x_1+t, x_2+t, \ldots , x_l+t)=g(x_1, x_2, \ldots , x_l)$ for all $t\in {\mathbb {R}}.$

• (iii) $g(x_1, x_2, \ldots , x_l)\rightarrow 0$ rapidly as $|x|\rightarrow \infty$ in the hyperplane $\sum x_j=0.$

Assuming the Riemann Hypothesis and an additional important technical condition that ${\hat{g}}(\xi _1, \ldots , \xi _l)$, the Fourier transform of g, is supported in $\sum _j |\xi _j|<2,$ they proved that

$$\begin{aligned} R_{T,l}(g) \rightarrow \int _{{\mathbb {R}}^l} g(x_1, \ldots , x_l) \rho _l(x_1, \ldots , x_l) \delta (\frac{x_1+\ldots +x_l}{l}) dx_1\cdots dx_l, \end{aligned}$$

(1.7)

where $\delta (x)$ is the delta function and

$$\begin{aligned} \rho _l(x_1, \ldots , x_l)=det(K(x_i,x_j))_{1\le i,j\le l}, \ \ K(x,y)=\frac{sin(\pi (x-y))}{\pi (x-y)}, \end{aligned}$$

(1.8)

is the limiting l-point correlation function for the CUE. We refer the reader to [4, 5, 14, 17] and the references therein for additional information on exciting connections between Random Matrices and Number Theory.

Introducing $f(y_1, \ldots , y_{n})=g(0, y_1, \ldots , y_{l-1}), \ n=l-1, \$ we can rewrite (1.6) as

$$\begin{aligned} R_{T,l}(g)=\sum ^{*} f({\tilde{\gamma }}_{j_2}-{\tilde{\gamma }}_{j_1}, \ldots , {\tilde{\gamma }}_{j_l}-{\tilde{\gamma }}_{j_1}), \end{aligned}$$

(1.9)

where f is a smooth symmetric test function decaying fast at infinity and the sum is over all distinct rescaled zeros in the interval [0, T]. In this paper, we study the limiting fluctuation of the analogue of (1.9) for the CUE. Specifically, we consider

$$\begin{aligned} S_N(f)=\sum _{1\le j_1, j_2, \ldots , j_{n+1}\le N} f(N (\theta _{j_2}-\theta _{j_1})_c, \ldots , N (\theta _{j_{n+1}}-\theta _{j_1})_c), \end{aligned}$$

(1.10)

where $\theta =(\theta _1, \ldots , \theta _N) \in {\mathbb {T}}^N$ comes from the CUE and $f\in C^{\infty }_c({\mathbb {R}}^{n})$ is a smooth test function with compact support. Even though the number of terms in (1.10) is $N^{n+1},$ the number of non-zero terms in the sum is of order N.

When $n=1$ the following result was proven in [1]:

Theorem 1.1

[1]. Let $f \in C^{\infty }_c({\mathbb {R}})$ be an even, smooth, compactly supported function on the real line and $\theta =(\theta _1, \ldots , \theta _N) \in {\mathbb {T}}^N$ be CUE-distributed. Consider

$$\begin{aligned} S_N(f(N\cdot ))=\sum _{1\le i,j\le N} f(N (\theta _i-\theta _j)_c). \end{aligned}$$

(1.11)

Then

$$\begin{aligned} {\mathbb {E}}S_N(f)= \sum _{k\in {\mathbb {Z}}} \frac{1}{\sqrt{2\pi }}{\hat{f}}(k/N) \min \left( \frac{|k|}{N}, 1\right) + {\hat{f}}(0) N, \end{aligned}$$

and $(S_N(f(N\cdot )) -{\mathbb {E}}S_N(f(N\cdot ))) N^{-1/2}$ converges in distribution to the centered real Gaussian random variable with the variance

$$\begin{aligned}&\frac{1}{\pi } \int _{{\mathbb {R}}} |{\hat{f}}(t)|^2 \min (|t|,1)^2 dt -\frac{1}{\pi } \int _{|s-t|\le 1, |s|\vee |t|\ge 1}{\hat{f}}(t) {\hat{f}}(s) (1-|s-t|) ds dt\nonumber \\&-\frac{1}{\pi } \int _{0\le s,t\le 1, s+t>1}{\hat{f}}(s) {\hat{f}}(t) (s+t-1) ds dt. \end{aligned}$$

(1.12)

Remark 1.2

The limiting distribution of (1.11) does not change if one replaces the circular difference (1.4) in the argument of $f(N\cdot )$ by the regular one and studies instead

$$\begin{aligned} \sum _{1\le i,j\le N} f(N (\theta _i-\theta _j)), \end{aligned}$$

(1.13)

since the number of pairs of the eigenvalues in a $O(N^{-1})$ neighborhood of $\theta =0$ is bounded in probability.

The proof relied on the cumulant technique and had a strong combinatorial flavor. The purpose of this paper is to give a simpler proof that works for arbitrary $n>1.$ Below we formulate our main result.

Theorem 1.3

Let $f \in C^{\infty }_c({\mathbb {R}}^{n})$ be a smooth, symmetric, compactly supported function on ${\mathbb {R}}^{n}, \ n\ge 1,$ and $\theta =(\theta _1, \ldots , \theta _N) \in {\mathbb {T}}^N$ be a CUE-distributed random vector. Consider the l-tuple smoothed counting statistic $S_N(f), \ l=n+1,$ defined in (1.10). Then ${\mathbb {E}}S_N(f)$ satisfies (3.5–3.6) and the normalized random variable $(S_N(f) -{\mathbb {E}}S_N(f)) N^{-1/2}$ converges in distribution to the centered real Gaussian random variable $N(0, \sigma ^2(f)$ with the limiting variance $\sigma ^2(f)$ defined in (3.17).

The paper is organized as follows. Some preliminary facts are given in Sect. 2. Section 3 is devoted to the computation of mathematical expectation and variance of $S_N(f).$ Theorem 1.3 is proven in Sect. 4. Throughout the paper the notation $a_N=O(b_N)$ means that the ratio $a_N/b_N$ is bounded from above in absolute value. The notation $a_N=o(b_N)$ means that $a_n/b_N\rightarrow 0$ as $N\rightarrow \infty .$

2 Preliminary Facts

Let $f \in C^{\infty }_c({\mathbb {R}}^{n}).$ When N is sufficiently large, the support of $f(N x), \ x=(x_1, \ldots , x_{n} ),$ is contained in the cube $[-\pi ,\pi ]^{n}$ and one can write a Fourier series

$$\begin{aligned} f(N x)=f(N x_1, \ldots , N x_n)=\frac{1}{(2\pi )^{n/2} N^{n}} \sum _{k\in {\mathbb {Z}}^n} {\hat{f}}(k_1 N^{-1}, \ldots , k_n N^{-1})e^{i k\cdot x}, \end{aligned}$$

where

$$\begin{aligned} {\hat{f}}(\xi )=\frac{1}{(2\pi )^{n/2}} \int _{{\mathbb {R}}^n}f(x)e^{-i \xi \cdot x} dx. \end{aligned}$$

Then

$$\begin{aligned} S_N(f)&=\sum _{1\le j_1, j_2, \ldots , j_{n+1}\le N} f(N (\theta _{j_2}-\theta _{j_1})_c, \ldots , N (\theta _{j_{n+1}}-\theta _{j_1})_c) \nonumber \\&= \frac{1}{(2\pi )^{n/2} N^{n}} \sum _{k\in {\mathbb {Z}}^n} {\hat{f}}(k_1 N^{-1}, \ldots , k_n N^{-1}) \prod _{j=1}^{n+1} T_{N, k_j}, \end{aligned}$$

(2.1)

where

$$\begin{aligned} T_{N,s}=\sum _{m=1}^N e^{i s \theta _m}=Tr (U^s) \end{aligned}$$

(2.2)

is the trace of the s-th power of a random unitary (CUE) matrix, and $k_{n+1}=-\sum _{j=1}^n k_j.$

First, we evaluate ${\mathbb {E}}S_N(f).\$ One has

$$\begin{aligned} {\mathbb {E}}S_N(f)= \frac{1}{(2\pi )^{n/2} N^{n}} \sum _{k\in {\mathbb {Z}}^n} {\hat{f}}(k_1 N^{-1}, \ldots , k_n N^{-1}) \times {\mathbb {E}}[\prod _{j=1}^{n+1} T_{N, k_j}]. \end{aligned}$$

(2.3)

To compute ${\mathbb {E}}[\prod _{j=1}^{n+1} T_{N, k_j}],$ we study the joint cumulants of the traces of powers of a CUE matrix. We refer the reader to [15] (see also [1], Sect. 5) for the definition and basic properties of the joint cumulants. We will use the notation $\kappa _m^{(N)}(k_1, \ldots , k_m)$ for the joint cumulant of $T_{N, k_1}, \ldots , T_{N, k_m},$ i.e.

$$\begin{aligned} \kappa _m^{(N)}(k_1, \ldots , k_m)=\kappa (T_{N, k_1}, \ldots , T_{N, k_m}). \end{aligned}$$

Recall that

$$\begin{aligned} {\mathbb {E}}[\prod _{j=1}^{n+1} T_{N, k_j}]=\sum _{\pi } \prod _{B\in \pi } \kappa _{|B|}^{(N)}(k_i: i \in B), \end{aligned}$$

(2.4)

where the sum is over all partitions $\pi$ of $\{1, \ldots , n+1\},$ B runs through the list of all blocks of the partition $\pi , \$ and |B| is the cardinality of a block B. The following result was established in [22] (see also Sect. 5 of [1]):

Lemma 2.1

• (i) If $p>1$ and either $k_1+\ldots +k_p\ne 0$ or $\ \prod _i^p k_i=0, \$ or both, then

  $$\begin{aligned} \kappa _p^{(N)}(k_1, \ldots , k_p)=0. \end{aligned}$$

  (2.5)

• (ii) If $p>1, \ \ \prod _i^p k_i\ne 0,$ and $\sum _{i=1}^p k_i=0,$ then

  $$\begin{aligned} \kappa _p^{(N)}(k_1, \ldots , k_p)=\sum _{m=1}^p \frac{(-1)^{m}}{m} \sum _{\begin{array}{c} (p_1, \ldots , p_m):\\ p_1+\ldots +p_m=p, \ p_1, \ldots p_m\ge 1 \end{array}}\frac{1}{p_1!\cdots p_m!} \sum _{\sigma \in S_p} J_N(p_1, \ldots , p_m; k_{\sigma (1)},\ldots , k_{\sigma (p)}), \end{aligned}$$

  (2.6)

  where for positive integers $p_1, \ldots , p_m\ge 1, \ p_1+\ldots +p_m=p ,$ and integers $k_1, \ldots , k_p,$ satisfying $\sum _{i=1}^p k_i=0,$ we define

  $$\begin{aligned}&J_N(p_1, \ldots , p_m; k_1,\ldots , k_p):= \nonumber \\&\min \left( N, \ \max \left( 0, \sum _{i=1}^{p_1} k_i, \sum _{i=1}^{p_1+p_2} k_i, \ldots , \sum _{i=1}^{p_1+\ldots +p_{m-1}} k_i\right) \right. \nonumber \\&\quad \qquad \left. +\max \left( 0, \sum _{i=1}^{p_1} (-k_i), \ldots , \sum _{i=1}^{p_1+\ldots +p_{m-1}} (-k_i)\right) \right) . \end{aligned}$$

  (2.7)

• (iii) If $p=1$ then $\kappa _1^{(N)}(k)=N$ for $k=0$ and $\kappa _1^{(N)}(k)=0$ otherwise.

Remark 2.2

If $\sum _{i=1}^p k_i=0$ and $\sum _{i=1}^p |k_i|\le 2 N,$ then

$$\begin{aligned}&J_N(p_1, \ldots , p_m; k_1,\ldots , k_p)= \nonumber \\&\max \left( 0, \sum _{i=1}^{p_1} k_i, \sum _{i=1}^{p_1+p_2} k_i, \ldots , \sum _{i=1}^{p_1+\ldots +p_{m-1}} k_i\right) \quad +\max \left( 0, \sum _{i=1}^{p_1} (-k_i), \ldots , \sum _{i=1}^{p_1+\ldots +p_{m-1}} (-k_i)\right) . \end{aligned}$$

(2.8)

Using a combinatorial identity (Lemma 2 in [22]) one further obtains that

$$\begin{aligned} \kappa _p^{(N)}(k_1, \ldots , k_p)=0 \ \ {\text{ i }f} \ \ p>2 \ \ {\text{ a }nd} \ \ \sum _{i=1}^p |k_i|\le 2 N. \end{aligned}$$

(2.9)

Remark 2.3

It directly follows from (2.7) that

$$\begin{aligned} 0\le J_N(p_1, \ldots , p_m; k_1,\ldots , k_p)\le N. \end{aligned}$$

Therefore (2.5–2.6) imply

$$\begin{aligned} |\kappa _p^{(N)}(k_1, \ldots , k_p)|\le Const_p N, \end{aligned}$$

(2.10)

where $Const_p$ depends only on p.

Remark 2.4

Recall that $\kappa _p^{(N)}(k_1, \ldots , k_p)$ is a symmetric function of $k_1, \ldots , k_p$. Direct computations allow one to derive joint cumulants in several important special cases:

$$\begin{aligned}&\kappa _2^{(N)}(k_1, -k_1)=min(N, |k_1|), \end{aligned}$$

(2.11)

$$\begin{aligned}&\kappa _3^{(N)}(k_1, k_2, -(k_1+k_2))={\left\{ \begin{array}{ll} 0, &{}\text { if } k_1+k_2\le N, k_1, k_2\ge 0,\\ k_1+k_2-N, &{}\text { if } k_1+k_2>N, 0\le k_1, k_2 \le N, \\ k_2, &{}\text {if } k_1+k_2>N, k_1>N, 0\le k_2 \le N, \\ N, &{}\text { } K_1\ge N, k_2\ge N. \end{array}\right. }. \end{aligned}$$

(2.12)

$$\begin{aligned}&\kappa _4^{(N)}(k_1, k_2, -k_1, -k_2)={\left\{ \begin{array}{ll} 0, &{}\text { if } 1\le |k_1|=|k_2|\le N/2,\\ N -2|k_1|, &{}\text { if } N/2< |k_1|=|k_2|\le N,\\ -N , &{}\text {if } |k_1|=|k_2|\ge N,\\ ||k_1|-|k_2||-N, &{}\text { if } 1\le ||k_1|-|k_2||\le N-1,N\le \max (|k_1|,|k_2|),\\ N-|k_1|-|k_2|, &{}\text {if } 1\le |k_1|\ne |k_2|\le N-1, N+1\le |k_1|+|k_2|,\\ 0,&{}\text {else.} \end{array}\right. }. \end{aligned}$$

(2.13)

We note that (2.13) directly follows from Corollary 4.2 in [1]. (2.12) follows from (2.6) and (2.8). Since $\kappa _p^{(N)}(k_1, \ldots , k_p)$ is a symmetric function that satisfies $\kappa _p^{(N)}(k_1, \ldots , k_p)=\kappa _p^{(N)}(-k_1, \ldots , -k_p),$ the third cumulant formula (2.12) completely determines $\kappa _3^{(N)}(k_1, k_2, k_3)$ (recall that $\kappa _3^{(N)}$ vanishes if $k_3\ne -k_1-k_2$.) The second cumulant formula (2.11) immediately follows from (2.6) and (2.8).

The main contribution to (2.1) and (2.3) comes from $|k|=O(N).$ Define

$$\begin{aligned} t_i=\frac{k_i}{N}, \ \ 1\le i\le p. \end{aligned}$$

(2.14)

Then

$$\begin{aligned} c_p(t_1, \ldots , t_p):= \frac{1}{N} \kappa _p^{(N)}(t_1 N, \ldots , t_p N) \end{aligned}$$

(2.15)

does not depend on N and is a bounded function of $t_1, \ldots , t_p,$ which is identically zero on $t_1+\ldots +t_p\ne 0$ for $p>1$ and is piece-wise linear on $t_1+\ldots +t_p=0.$ To write down an explicit formula for $c_p(t_1, \ldots , t_p)$ we define functions $j(p_1, \ldots , p_m; t_1,\ldots , t_p)$ for positive integers $p_1, \ldots , p_m,$ satisfying $\ p_1+\ldots +p_m=p ,$ and real numbers $t_1, \ldots , t_p,$ as

$$\begin{aligned}&j(p_1, \ldots , p_m; t_1,\ldots , t_p):= \nonumber \\&\min \left( 1, \ \max \left( 0, \sum _{i=1}^{p_1} t_i, \sum _{i=1}^{p_1+p_2} t_i, \ldots , \sum _{i=1}^{p_1+\ldots +p_{m-1}} t_i\right) +\max \left( 0, \sum _{i=1}^{p_1} (-t_i), \ldots ,\sum _{i=1}^{p_1+\ldots +p_{m-1}} (-t_i)\right) \right) . \end{aligned}$$

(2.16)

Lemma 2.1 implies the following result.

Lemma 2.5

Rescaled joint cumulants $c_p$ defined in (2.15) can be written for $p>1$ and $\sum _{i=1}^p t_i=0$ as

$$\begin{aligned} c_p(t_1, \ldots , t_p)=\sum _{m=1}^p \frac{(-1)^{m}}{m} \sum _{\begin{array}{c} (p_1, \ldots , p_m): \\ p_1+\ldots +p_m=p, \ p_1, \ldots p_m\ge 1 \end{array}} \frac{1}{p_1!\cdots p_m!} \sum _{\sigma \in S_p} j(p_1, \ldots , p_m; t_{\sigma (1)},\ldots , t_{\sigma (p)}), \end{aligned}$$

(2.17)

where the functions $j(p_1, \ldots , p_m; t_1,\ldots , t_p)$ are defined in (2.16). Moreover, the following holds:

$$\begin{aligned}&(i) \ \ c_p(t_1, \ldots , t_p), \ p>1, \ \text { is a bounded symmetric piece-wise linear function on} \\&\quad \sum _{i=1}^p t_i=0. \\&(ii) \ \ c_p(t_1, \ldots , t_p)=0 \ \text {if} \ \ p>1 \ \ \text {and} \ \ \sum _{i=1}^p t_i\ne 0.\\&(iii) \ \ c_1(0)=1 \ \ \text {and} \ \ c_1(t)=0 \ \ \text {for} \ \ t\ne 0. \end{aligned}$$

Remark 2.6

We refer the reader to [9] and [10] for some important earlier results on the moments of $|T_{N,k}|^2$ for $k=O(N).$

3 Expectation and Variance

We start with a computation of ${\mathbb {E}}S_N(f).$ It follows from (2.3), (2.4), (2.15), and (2.12) that

$$\begin{aligned} {\mathbb {E}}S_N(f)&= \frac{1}{(2\pi )^{n/2} N^{n}} \sum _{k_1, \ldots , k_n\in {\mathbb {Z}}} {\hat{f}}(k_1/N, \ldots , k_n/N) \sum _{\pi } N^{|\pi |}\prod _{B\in \pi } c_{|B|}(k_i/N: i \in B) \nonumber \\&= \frac{1}{(2\pi )^{n/2}} \sum _{\pi } N^{|\pi |-n} \sum _{k_1, \ldots , k_n\in {\mathbb {Z}}} {\hat{f}}(k_1/N, \ldots , k_n/N) \prod _{B\in \pi } c_{|B|}(k_i/N: i \in B), \end{aligned}$$

(3.1)

where $k_{n+1}=-\sum _{i=1}^n k_i, \$ the sum is over all partitions $\pi$ of $\{1, \ldots , n+1\},$ B runs through the list of all blocks of the partition $\pi ,\ |\pi | \$ is the number of blocks in a partition $\pi ,$ and |B| is the cardinality of a block B.

Denote a linear subspace of ${\mathbb {R}}^{n+1}$ as

$$\begin{aligned} L_{\pi }:=\{t=(t_1, \ldots , t_{n+1})\in {\mathbb {R}}^{n+1} : \sum _{i=1}^{n+1} t_i=0; \ \ \sum _{i\in B} t_i=0 \ \forall B\in \pi \}. \end{aligned}$$

(3.2)

It follows from Lemma 2.5, parts (ii)–(iii), that for any fixed partition $\pi$ the summation in (3.1) is over $k=(k_1, \ldots , k_{n+1}) \in L_{\pi } \cap \frac{1}{N} {\mathbb {Z}}^{n+1}.$ Indeed, k satisfies the following system of linear equations of rank $|\pi |:$

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} \sum _{i=1}^{n+1} k_i=0, \\ &{} \sum _{i\in B} k_i=0, \ \ \forall B\in \pi , \end{array}\right. } \end{aligned}$$

(3.3)

Observe that the first linear equation in (3.3) follows from the remaining $|\pi |$ independent linear equations in (3.3). Therefore

$$\begin{aligned} \dim L_{\pi }=n+1-|\pi |. \end{aligned}$$

(3.4)

We note that the sums in (3.1) are Riemann sums $\left( {\text {up\,to\,a\,multiplicative\,factor}}\,\frac{N}{(2\pi )^{n/2}}\right)$ corresponding to smooth and fast decaying at infinity functions ${\hat{f}}(t_1, \ldots , t_n) \prod _{B\in \pi } c_{|B|}(t_i: i \in B)$ on $L_{\pi }$. Denote by m the number of blocks in $\pi$ and by $n_1, \ldots , n_m$ the cardinalities of the blocks. Using Lemma 2.5 we are ready to obtain asymptotics of the mean of $S_N(f).$

Lemma 3.1

$$\begin{aligned} {\mathbb {E}}S_N(f)={\mathcal {M}}(f) N +O(1), \end{aligned}$$

(3.5)

where

$$\begin{aligned}&{\mathcal {M}}(f)= \frac{1}{(2\pi )^{n/2}} \sum _{m=1}^n \frac{1}{m!} \sum _{\begin{array}{c} (n_1, \ldots , n_m):\\ n_1+\ldots +n_m=n+1, \ n_1, \ldots , n_m\ge 1 \end{array}} \frac{(n+1)!}{n_1!\cdots n_m!}\times \nonumber \\&\int _{L_{\pi }} {\hat{f}}(t_1, \ldots , t_n) \prod _{j=1}^m c_{n_j}(t_{M_{j-1}+1}, \ldots , t_{M_j-1}, -t_{M_{j-1}+1}-\ldots - t_{M_j-1}) d\lambda , \end{aligned}$$

(3.6)

$t_{n+1}=-\sum _{i=1}^n t_i, \ \ M_j=n_1+\ldots +n_j, \ 1\le j\le m,\ M_0=0, \$ $L(n_1, \ldots , n_m)\subset {\mathbb {R}}^{n+1}$ is defined as

$$\begin{aligned} L_{n_1, \ldots , n_m}:=\{t=(t_1, \ldots , t_{n+1})\in {\mathbb {R}}^{n+1} : \ \sum _{i=1}^{n+1} t_i=0; \ \ \sum _{M_{j-1}<i\le M_j} t_i=0, \ \ 1\le j\le m\}, \end{aligned}$$

(3.7)

$\ \lambda \$ is the Lebegsue measure on $L_{\pi },$ i.e.

$$\begin{aligned} \lambda =\prod _{j: n_j>1} dt_{M_{j-1}+1}\cdots dt_{M_j-1}, \end{aligned}$$

(3.8)

and rescaled joint cumulant functions $c_{n_j}$’s are defined in (2.11 and 2.12).

Remark 3.2

For singleton blocks $B: |B|=n_j=1, \$ we write $c_{n_j}(t_{M_{j-1}+1})$ in (3.6) and use $c_{n_j}(t_{M_{j-1}+1})=c_{n_j}(0)=1$ on $L_{\pi }.$

Next, we study $\text {Var}S_N(f).$ It follows from (2.1) that

$$\begin{aligned}&\text {Var}(S_N(f))= \frac{1}{(2\pi )^{n} N^{2n}} \sum _{k^{(1)}\in {\mathbb {Z}}^n} \sum _{k^{(2)}\in {\mathbb {Z}}^n} {\hat{f}}(k^{(1)}/N) {\hat{f}}(k^{(2)}/N) \times \nonumber \\&\left( {\mathbb {E}}\left[ \prod _{j=1}^{n+1} T_{N, k_j} \prod _{j=1}^{n+1} T_{N, k_{n+1+j}}\right] - {\mathbb {E}}\left[ \prod _{j=1}^{n+1} T_{N, k_j}\right] {\mathbb {E}}\left[ \prod _{j=1}^{n+1} T_{N, k_{n+1+j}}\right] \right) , \end{aligned}$$

(3.9)

where $k^{(1)}=(k_1, \ldots , k_{n}), \ \ k^{(2)}=(k_{n+2}, \ldots , k_{2n+1})\in {\mathbb {Z}}^{n}, \ \ k_{n+1}:=-\sum _{i=1}^n k_i,$ $k_{2n+2}:=-\sum _{i=1}^n k_{n+1+i},$ and $T_{N,s}$ is defined in (2.2).

To study the covariance of the products of traces in (3.9), we rewrite it using (2.4) as

$$\begin{aligned} {\mathbb {E}}\left[ \prod _{j=1}^{n+1} T_{N, k_j} \prod _{j=1}^{n+1} T_{N, k_{n+1+j}}\right] - {\mathbb {E}}\left[ \prod _{j=1}^{n+1} T_{N, k_j}\right] {\mathbb {E}}\left[ \prod _{j=1}^{n+1} T_{N, k_{n+1+j}}\right] = \sum _{\pi }^* \prod _{B\in \pi } \kappa _{|B|}^{(N)}(k_i: i \in B), \end{aligned}$$

(3.10)

where the sum is over all partitions $\pi$ of $\{1,2,\ldots , 2n+2\},$ satisfying the condition (3.11) below, B runs through the list of all blocks of the partition $\pi , \$ and |B| is the cardinality of a block $B.\$ The condition on $\pi$ is that it is not a union of partitions of $\{1, \dots , n+1 \}$ and $\{n+2, \ldots , 2n+2\},$ in other words:

$$\begin{aligned}&{\text{ T }he \ set \ }\{1,2,\ldots , n+1\} { is \ not \ a \ union \ of \ some \ blocks \ of \ } \pi . \end{aligned}$$

(3.11)

Similarly to the ${\mathbb {E}}S_N(f)$ computations, define

$$\begin{aligned} L_{\pi }:=\{t=(t_1, \ldots , t_{2n+2})\in {\mathbb {R}}^{2n+2} : \sum _{i=1}^{n+1} t_i=0, \ \sum _{i=1}^{n+1} t_{n+1+i}=0; \ \ \sum _{i\in B} t_i=0 \ \forall B\in \pi \}. \end{aligned}$$

(3.12)

Using (3.10) we obtain

$$\begin{aligned} \text {Var}(S_N(f))&= \frac{1}{(2\pi )^{n} N^{2n}} \sum _{k^{(1)}\in {\mathbb {Z}}^n} \sum _{k^{(2)}\in {\mathbb {Z}}^n} {\hat{f}}(k^{(1)}/N) {\hat{f}}(k^{(2)}/N) \sum _{\pi }^* \prod _{B\in \pi } \kappa _{|B|}^{(N)}(k_i: i \in B)\nonumber \\&= \frac{1}{(2\pi )^{n}} \sum _{\pi }^* N^{|\pi |-2 n} \sum _{k^{(1)}\in {\mathbb {Z}}^n} \sum _{k^{(2)}\in {\mathbb {Z}}^n} {\hat{f}}(k^{(1)}/N) {\hat{f}}(k^{(2)}/N) \prod _{B\in \pi } c_{|B|}(k_i/N: i \in B). \end{aligned}$$

(3.13)

Remark 3.3

As before, the sum $\sum _{\pi }^*$ in (3.13) is over all partitions $\pi$ of $\{1,2,\ldots , 2n+2\}$ satisfying the condition (3.11). The shorthand notation $c_{|B|}(t_i: i\in B)$ means that the arguments of $c_{|B|}$ are the $t_i$’s corresponding to $i\in B.$ Since the joint cumulant functions are symmetric the order of the variables is not important.

It follows from Lemma 2.5, (ii) and (iii), that for any fixed $\pi$ the summation in (3.13) is over $k=(k_1, \ldots , k_{2n+2}) \in L_{\pi } \cap \frac{1}{N} {\mathbb {Z}}^{2n+2},$ since k satisfies the following system of linear equations of rank $|\pi |+1:$

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} \sum _{i=1}^{n+1} k_i=0, \\ &{} \sum _{i=1}^{n+1+i} k_i=0, \\ &{} \sum _{i\in B} k_i=0, \ \ \forall B\in \pi . \end{array}\right. } \end{aligned}$$

(3.14)

The first linear equation $\sum _{i=1}^{n+1} k_i=0$ follows from the remaining $|\pi |+1$ independent linear equations in (3.14). Therefore

$$\begin{aligned} \dim L_{\pi }=2n+2-(|\pi |+1)=2n+1-|\pi |. \end{aligned}$$

(3.15)

Proceeding as in the case of the mathematical expectation above we arrive at

Lemma 3.4

$$\begin{aligned} \text {Var}(S_N(f))=\sigma ^2(f) N +O(1), \end{aligned}$$

(3.16)

where

$$\begin{aligned} \sigma ^2(f)= \frac{1}{(2\pi )^{n}} \sum _{\pi }^* \prod _{B\in \pi } \int _{L_{\pi }} \ f(t^{(1)}) f(t^{(2)}) \prod _{B\in \pi } c_{|B|}(t_i: i\in B) \ d\lambda , \end{aligned}$$

(3.17)

where the sum in (3.17) is over all partitions $\pi$ of $\{1,2,\ldots , 2n+2\},$ satisfying the condition (3.11), B runs through the list of all blocks of the partition $\pi , \ \ |B|$ is the cardinality of a block $B,\ \ t^{(1)}=(t_1, \ldots , t_n), \ t^{(2)}=(t_{n+2}, \ldots , t_{2n+1}),$ and $\lambda$ is the Lebesgue measure on $L_{\pi },$ i.e. it is the product of $dt_i$’s taken over all independent variables $t_i$ from the system of linear equations

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} \sum _{i=1}^{n+1} t_i=0, \\ &{} \sum _{i=1}^{n+1+i} t_i=0, \\ &{} \sum _{i\in B} t_i=0, \ \ \forall B\in \pi . \end{array}\right. } \end{aligned}$$

(3.18)

4 Proof of the Main Result

The section is devoted to the proof of Theorem 1.3. The proof uses the method of moments and is combinatorial in nature.

Proof

Fix a positive integer $m>2.$ We have

$$\begin{aligned}&{\mathbb {E}}(S_N(f)-{\mathbb {E}}S_N(f))^m= \frac{1}{(2\pi )^{\frac{n m}{2}} N^{m n}} \sum _{k^{(1)}\in {\mathbb {Z}}^n} \ldots \sum _{k^{(m)}\in {\mathbb {Z}}^n} {\hat{f}}(k^{(1)}/N)\cdots {\hat{f}}(k^{(m)}/N) \times \nonumber \\&{\mathbb {E}}\left[ \prod _{i=0}^{m-1} \left( \prod _{j=1}^{n+1} T_{N, k_{(n+1) i+j}}- {\mathbb {E}}\prod _{j=1}^{n+1} T_{N, k_{(n+1)i+j}}\right) \right] , \end{aligned}$$

(4.1)

$k^{(1)}=(k_1, \ldots , k_{n}), \ldots , k^{(m)}=(k_{(m-1) (n+1)+1}, \ldots , k_{m (n+1)-1})\in {\mathbb {Z}}^{n},$ $k_{n+1}:=-\sum _{j=1}^n k_j, \ \ k_{2(n+1)}=-\sum _{j=1}^{n} k_{n+1+j},\ldots , k_{m(n+1)}:=- \sum _{j=1}^n k_{(m-1) (n+1)+j}.$

The mathematical expectation on the r.h.s. of (4.1) can be written in terms of joint cumulants using the following lemma. $\square$

Lemma 4.1

For centered random variables $X_1, \ldots , X_{m l}$ with finite moments,

$$\begin{aligned} {\mathbb {E}}\left[ \prod _{i=0}^{m-1} \left( \prod _{j=1}^{l} X_{i l+j}- {\mathbb {E}}\prod _{j=1}^{l} X_{i l+j}\right) \right] = \sum ^*_{\pi } \prod _{B\in \pi } \kappa (X_i: i \in B), \end{aligned}$$

(4.2)

where the sum on the r.h.s. of (4.2) is over all partitions $\pi$ of $\{1, \ldots , m l\}$ that do not contain a partition of $\{ i l +1, \ldots , (i+1) l\}$ for any $0\le i\le m-1,$ i.e. none of the sets $\{ i l +1, \ldots , (i+1) l\}$ can be represented as a union of some blocks of $\pi .$

Proof

It follows from the formula expressing moments in terms of cumulants (see e.g. (2.4)) that the r.h.s. of (4.2) is equal to a linear combination of $\prod _{B\in \pi } \kappa (X_i: i \in B),$ where $\pi$ runs over the list of partitions of $\{1,2,\ldots , m l\}.$ Thus our goal is to show that a coefficient in the linear combination is either 1 or 0 depending on whether $\pi$ satisfies the condition in Lemma 4.1 or not.

If any sub-collection of $\pi$ is not a partition of $\{ i l +1, \ldots , (i+1) l\}$ for any $\ i=1, \ldots ,m,$ then the coefficient in front of the product $\prod _{B\in \pi } \kappa (X_i: i \in B)$ in the linear combination is 1 since the only contribution comes from ${\mathbb {E}}\left[ \prod _{i=0}^{m-1} \prod _{j=1}^{l} X_{i l+j}\right] .$

Finally, suppose $s, \ 1\le s\le m,$ of the sets $\{ i l +1, \ldots , (i+1) l\}, \ 1\le i \le m,$ can be represented as unions of some blocks of $\pi .$ Then the coefficient in front of $\prod _{B\in \pi } \kappa (X_i: i \in B)$ is equal to

$$\begin{aligned} \sum _{k=0}^s (-1)^k \frac{s!}{k! (s-k)!}=0. \end{aligned}$$

(4.3)

$\square$

Following Lemma 4.1 we can rewrite the m-th centered moment as

$$\begin{aligned}&{\mathbb {E}}(S_N(f)-{\mathbb {E}}S_N(f))^m= \nonumber \\&\frac{1}{(2\pi )^{\frac{n m}{2}} N^{m n}} \sum _{k^{(1)}\in {\mathbb {Z}}^n} \ldots \sum _{k^{(m)}\in {\mathbb {Z}}^n} {\hat{f}}(k^{(1)}/N)\cdots {\hat{f}}(k^{(m)}/N) \sum _{\pi }^* \prod _{B\in \pi } \kappa _{|B|}^{(N)}(k_i: i \in B), \end{aligned}$$

(4.4)

where, as before, $k^{(1)}=(k_1, \ldots , k_{n}), \ldots , k^{(m)}=(k_{(m-1) (n+1)+1}, \ldots , k_{m (n+1)-1})\in {\mathbb {Z}}^{n},$ $k_{n+1}=-\sum _{j=1}^n k_j, \ldots , k_{m(n+1)}=- \sum _{j=1}^n k_{(m-1) (n+1)+j}.$

As in Sect. 3, we consider a linear subspace

$$\begin{aligned} L_{\pi }:=\{t\in {\mathbb {R}}^{m (n+1)} : \sum _{j=1}^{n+1} t_{i(n+1)+j}=0, \ \forall \ 0\le i\le m-1; \ \ \sum _{j\in B} t_j=0 \ \forall B\in \pi \}. \end{aligned}$$

(4.5)

It follows from Lemma 2.5, (ii) and (iii), that for any fixed $\pi$ the summation in (4.4) is over $k=(k_1, \ldots , k_{m (n+1)}) \in L_{\pi } \cap \frac{1}{N} {\mathbb {Z}}^{m (n+1)},$ since k satisfies the following system of linear equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} \sum _{j=1}^{n+1} k_{i (n+1)+j}=0, \ \forall \ 0\le i\le m-1, \\ &{} \sum _{i\in B} k_i=0, \ \ \forall B\in \pi . \end{array}\right. } \end{aligned}$$

(4.6)

Using (2.15), we rewrite (4.4) as

$$\begin{aligned}&{\mathbb {E}}(S_N(f)-{\mathbb {E}}S_N(f))^m= \nonumber \\&\frac{1}{(2\pi )^{\frac{n m}{2}} N^{m n}} \sum _{k\in L^{\pi }\cap \frac{1}{N} {\mathbb {Z}}^{m (n+1)}} {\hat{f}}(k^{(1)}/N)\cdots {\hat{f}}(k^{(m)}/N) \sum _{\pi }^* \ N^{|\pi |} \prod _{B\in \pi } c_{|B|}(k_i/N: i \in B)=\nonumber \\&\frac{1}{(2\pi )^{\frac{n m}{2}}}\sum _{\pi }^* N^{|\pi |-m n} \sum _{k\in L^{\pi }\cap \frac{1}{N} {\mathbb {Z}}^{m (n+1)}} {\hat{f}}(k^{(1)}/N)\cdots {\hat{f}}(k^{(m)}/N) \prod _{B\in \pi } c_{|B|}(k_i/N: i \in B). \end{aligned}$$

(4.7)

The crucial question in the power counting analysis of the r.h.s. of (4.7) is the dimension of the vector subspace $L_{\pi }\subset {\mathbb {R}}^{m (n+1)},$ or, equivalently, the rank of the system of linear equations (4.6).

Denote

$$\begin{aligned}{}[1]:=\{1,\ldots , n+1\}, \ [2]:=\{n+2, \ldots , 2n+2\}, \ldots , [m]=\{(m-1) (n+1)+1, \ldots , m (n+1)\}. \end{aligned}$$

Definition 4.2

For a given partition $\pi$ of $\{1, \ldots , m n\}$ an equivalence relation $\sim _{\pi }$ is defined on the set $\{[1], [2], \ldots , [m]\}$ in the following way:

$$\begin{aligned}{}[i]\sim _{\pi } [j] \end{aligned}$$

(4.8)

if and only if there is a block B in the partition $\pi$ such that $B\cap [i]\ne \emptyset$ and $B\cap [j]\ne \emptyset .$

Remark 4.3

It follows from Lemma 4.1 that the cardinality of each equivalence class of the equivalence relation $\sim _{\pi }$ is at least 2.

Definition 4.4

We call a partition $\pi$ optimal if the cardinality of every equivalence class of the equivalence relation $\sim _{\pi }$ is 2. If $\pi$ is not optimal, it is called sub-optimal.

Clearly, optimal partitions exist if and only if m is even. Next lemma is the main ingredient of the proof of Theorem 1.3.

Lemma 4.5

• (i) Let m be an even positive integer and $\pi$ be an optimal partition of $\{1, \ldots , m n\}$. Then a linear subspace $L_{\pi } \subset {\mathbb {R}}^{m (n+1)}$ defined in (4.5) satisfies

  $$\begin{aligned} \dim L_{\pi }= m n +m/2 -|\pi |. \end{aligned}$$

  (4.9)

• (ii) Let $m>1$ be a positive integer and $\pi$ be an sub-optimal partition of $\{1, \ldots , m n\}$. Then

  $$\begin{aligned} \dim L_{\pi }<m n +m/2 -|\pi |. \end{aligned}$$

  (4.10)

Proof

• (i) If $\pi$ is optimal it can be viewed as a union of $\frac{m}{2}$ partitions $\pi _{i,j}$ of $[i]\cup [j], \ [i]\sim _{\pi }[j].$ Each partition $\pi _{i,j}$ corresponds to a vector subspace $L_{\pi _{i,j}}\subset {\mathbb {R}}^{2n+2}$ of dimension

  $$\begin{aligned} \dim L_{\pi _{i,j}}=2 n+1-|\pi _{i,j}| \end{aligned}$$

  (see (3.12) and (3.15).) Since $L_{\pi }$ is the Cartesian product of $L_{\pi _{i,j}}$’s we have

  $$\begin{aligned} \dim L_{\pi } =\sum \dim L_{\pi _{i,j}}, \end{aligned}$$

  (4.11)

  and (4.9) immediately follows.

• (ii) Now let us assume that $\pi$ is a sub-optimal partition. Recall that the subspace $L_{\pi }$ is determined by the following system of linear equations

  $$\begin{aligned} {\left\{ \begin{array}{ll} &{} \sum _{j=1}^{n+1} t_{i (n+1)+j}=0, \ \ \forall \ 0\le i\le m-1,\\ &{} \sum _{j\in B} t_j=0, \ \ \forall B\in \pi . \end{array}\right. } \end{aligned}$$

  (4.12)

To prove (4.10) we have to show that the rank of this system is bigger than $|\pi |+\frac{m}{2}.\$ As before, $L_{\pi }$ can be viewed as the Cartesian product of the subspaces corresponding to the equivalence classes. Using the additivity of dimension, we can assume without a loss of generality that $\pi$ in (4.12) has only one equivalence class. We claim that in this case the rank of (4.12) is $|\pi |+m-1>|\pi |+\frac{m}{2},$ since $m>2.$ To show this, consider $|\pi |+m-1$ vectors in ${\mathbb {R}}^{m (n+1)},\$ namely $\{ \chi _B, \ B\in \pi \} \cup \{ \chi _{[i]}, \ 0\le i<m-1\}:$

$$\begin{aligned}&\chi _B(j)={\left\{ \begin{array}{ll} &{} 1 \ \ j\in B, \\ &{} 0 \ \ j \notin B, \end{array}\right. } \end{aligned}$$

(4.13)

$$\begin{aligned}&\chi _{[i]}(j)={\left\{ \begin{array}{ll} &{} 1 \ \ j\in [i]=\{i (n+1) +1, \ldots , (i+1) (n+1)\}, \\ &{} 0 \ \ j \notin [i]. \end{array}\right. } \end{aligned}$$

(4.14)

Part (ii) of Lemma (4.5) follows from $\square$

Lemma 4.6

Let $\pi$ be a sub-optimal partition of $\{1, \ldots , m n\}, \ \ m\ge 1, \$ such that the equivalence relation $\sim _{\pi }$ on $\{[1], \ldots , [m]\}$ has only one equivalence class. Then the vectors

$$\begin{aligned} \{ \chi _B, \ B\in \pi \} \cup \{ \chi _{[i]}, \ 0\le i<m-1\} \end{aligned}$$

(4.15)

are linearly independent.

Proof

The vectors $\{ \chi _B, \ B\in \pi \}$ are linearly independent since their supports are disjoint. We show by induction that adding $\chi _{[i]}$ one by one preserves linear independence. Suppose that $\{ \chi _B, \ B\in \pi \} \cup \{ \chi _{[i]}, \ 0\le i<k-1\}, \ k<m, \$ are linearly independent and $\chi _{[k]}$ can be written as a linear combination of $\{ \chi _B, \ B\in \pi \} \cup \{ \chi _{[i]}, \ 0\le i<k-1\}.$ Then a non-trivial linear combination of $\{ \chi _{[i]}, \ 0\le i<k\} \$ can be written as a linear combination of some of the vectors $\chi _B.$ This implies that $[1]\cup \ldots \cup [k]$ contain an equivalence class of the equivalence relation $\sim _{\pi }, \$ which is a contradiction. Lemma 4.6 is proven. $\square$

Since $L_{\pi }$ is the orthogonal complement of the linear span of the vectors (4.15) this implies part (ii) of Lemma (4.5).

Now we are ready to finish the proof of Theorem 2.5. Let $m=2 k$ be even. Denote a subsum in (4.7) corresponding to a partition $\pi$ by $\Sigma _{\pi }.$ It follows from Lemma (4.5) that only the optimal partitions $\pi$ give leading contribution of order $N^{m/2}$ to the r.h.s. of (4.7). For each sub-optimal $\pi$ we have $\ \Sigma _{\pi }= O(N^{\frac{m-1}{2}}). \$

There are exactly $(2 k-1)!!$ ways to split the set $\{[1], \ldots , [2 k] \}$ into pairs. By repeating the variance computations, we note that the sum of $\Sigma _{\pi }$’s over all $\pi$ corresponding to any particular splitting of $\{[1], \ldots , [2 k] \}$ into pairs gives $\ \sigma ^{m}(f) N^{\frac{m}{2}} (1+o(1)). \$

We conclude that the 2k-th moment of $\frac{S_N(f)-{\mathbb {E}}S_N(f)}{\sqrt{\text {Var}S_N(f)}}$ converges to $(2k-1)!!$ in the limit $N\rightarrow \infty .$ The case of odd m is treated similarly. Specifically, one obtains that the odd moments of $\frac{S_N(f)-{\mathbb {E}}S_N(f)}{\sqrt{\text {Var}S_N(f)}}$ converges to 0 in the limit $N\rightarrow \infty .$

This finishes the proof of Theorem 2.5.