The central limit theorem for Riesz-Raikov sums II

TL;DR

This paper proves a central limit theorem for sums of periodic functions evaluated along sequences generated by expanding matrices, extending understanding of their probabilistic behavior.

Contribution

It establishes the CLT for Riesz-Raikov sums involving expanding matrices, advancing the theory for multidimensional dynamical systems.

Findings

Proves CLT for sums of periodic functions along matrix orbits

Extends previous results to multidimensional expanding matrices

Provides conditions under which the CLT holds

Abstract

For a $d\times d$ expanding matrix $A$, we investigate randomness of the sequence $\{A^k x\}$ and prove the central limit theorem for $\sum f(A^k x)$ where $f$ is a periodic function with a mild regularity condition.

Equations

$$\Vert A\mathbf{x}{\Vert }_2\ge q\Vert \mathbf{x}{\Vert }_{2}for\mathbf{x}\in {\mathbf{R}}^d,$$

$$\Vert \mathbf{\xi }A{\Vert }_2\ge q\Vert \mathbf{\xi }{\Vert }_{2}for\mathbf{\xi }\in {\hat{\mathbf{R}}}^d.$$

$$f(\mathbf{x}+{\mathbf{e}}_i)=f(\mathbf{x})(i=1,\dots ,d)and{\int }_{[0,1{)}^d}f(\mathbf{x})d\mathbf{x}=0,$$

$$f(\mathbf{x})\sim \sum _{\mathbf{\xi }\in {\mathbf{Z}}^d}\hat{f}(\mathbf{\xi })\exp (2\pi \sqrt{-1}\mathbf{\xi }\mathbf{x}),$$

$$f_a(\mathbf{x})=\sum _{\mathbf{\xi }\in R_a}\hat{f}(\mathbf{\xi })\exp (2\pi \sqrt{-1}\mathbf{\xi }\mathbf{x}),$$

$$\sum _{m=1}^{\infty }\Vert f-f_{2^m}{\Vert }_{L^2[0,1{)}^d}<\infty .$$

$${\int }_0^1\frac{{\omega }^{(2)}(H,f)}{H}dH<\infty ,$$

$$\omega^{(2)}(H,f)=\sup\biggl{\{}\,\|f({}\cdot{}+\boldsymbol{h})-f({}\cdot{})\|_{L^{2}[0,1]^{d}}\biggm{|}{\boldsymbol{h}=(h_{1},\dots,h_{d})\in{\bf R}^{d},\atop|h_{1}|,\dots,|h_{d}|\leq H}\,\biggr{\}}.$$

$${\sigma }^2(f)=\sum _{l\ge 0}(2-{\delta }_{0,l})\sum _{\mathbf{\xi },{\mathbf{\xi }}^{\prime }\in {\mathbf{Z}}^d}\hat{f}(\mathbf{\xi })\hat{f}({\mathbf{\xi }}^{\prime })1(\mathbf{\xi }+{\mathbf{\xi }}^{\prime }{A}^l=0).$$

$$\lim_{N\to\infty}\int_{\Gamma}\biggl{(}\frac{1}{\sqrt{N}}\sum_{k=1}^{N}f(A^{k}\boldsymbol{x})\biggr{)}^{2}\,d\boldsymbol{x}=\operatorname{\rm Leb}(\Gamma)\sigma^{2}(f),$$

$$\lim_{N\to\infty}\operatorname{\rm Leb}\biggl{\{}\boldsymbol{x}\in\Gamma\biggm{|}\frac{1}{\sqrt{N}}\sum_{k=1}^{N}f(A^{k}\boldsymbol{x})\leq t\biggr{\}}=\operatorname{\rm Leb}(\Gamma)\Phi_{\sigma^{2}(f)}(t)$$

$$\Vert \mathbf{j}{A}_{n+k}{\Vert }_{\infty }\ge q^k\Vert A_n{\Vert }_{\infty }if\mathbf{j}\in {\mathbf{Z}}^{d}andk\ge {\log }_q\Vert \mathbf{j}\Vert$$

$$A_n={\left(\begin{array}{cc}{c}_1& 0\\ 0& c_2\end{array}\right)}^n.$$

$$A=\left(\begin{array}{cc}\text{root}3\text{of}2& 0\\ 0& \text{root}2\text{of}3\end{array}\right),$$

$$f(x)=g_1(A^6\mathbf{x})-g_1(\mathbf{x})+g_2(A^3\mathbf{x})-g_2(\mathbf{x})+g_3(A^2\mathbf{x})-g_3(\mathbf{x})$$

$$J_{\gamma }(\lambda )=\left(\begin{array}{cccc}\lambda & 1& & \text{raise}-4.0pt0\\ & \lambda & \ddots & \\ & & \ddots & 1\\ \text{raise}0.0pt0& & & \lambda \end{array}\right),C_{\gamma }(\lambda )=\left(\begin{array}{cccc}|\lambda |Z_{\theta }& Z_0& & \text{raise}-4.0pt0\\ & |\lambda |Z_{\theta }& \ddots & \\ & & \ddots & Z_0\\ \text{raise}0.0pt0& & & |\lambda |Z_{\theta }\end{array}\right),$$

$$B=\left(\begin{array}{cccccc}{J}_{d_1}({\lambda }_1)& & & & & \text{raise}-4.0pt0\\ & \ddots & & & & \\ & & J_{d_{\alpha }}({\lambda }_{\alpha })& & & \\ & & & C_{d_{\alpha +1}/2}({\lambda }_{\alpha +1})& & \\ & & & & \ddots & \\ \text{raise}0.0pt0& & & & & C_{d_{\beta }/2}({\lambda }_{\beta })\end{array}\right),$$

$$Q=({\mathbf{q}}_1^{(1)},\dots ,{\mathbf{q}}_{d_1}^{(1)},{\mathbf{q}}_1^{(2)},\dots ,{\mathbf{q}}_{d_2}^{(2)},\dots ,{\mathbf{q}}_1^{(\beta )},\dots ,{\mathbf{q}}_{d_{\beta }}^{(\beta )})$$

$$\frac{\Vert Q\mathbf{x}\Vert }{\Vert \mathbf{x}\Vert },\frac{\Vert Q^{-1}\mathbf{x}\Vert }{\Vert \mathbf{x}\Vert },\frac{\Vert \mathbf{\xi }Q\Vert }{\Vert \mathbf{\xi }\Vert },\frac{\Vert \mathbf{\xi }{Q}^{-1}\Vert }{\Vert \mathbf{\xi }\Vert }\in (D_{1,Q}^{-1},D_{1,Q})(\mathbf{x}\ne 0,\mathbf{\xi }\ne 0).$$

$${\mathbf{R}}^d=\underset{h=1}{\overset{\beta }{⨁}}{W}_h,\dim W_h=d_{h}and\sum _{h=1}^{\beta }{d}_h=d.$$

$$\Vert A^k{\mathbf{x}}_h\Vert \le D_{2,A}|{\lambda }_h{|}^k{k}^d\Vert {\mathbf{x}}_h\Vert ({\mathbf{x}}_h\in W_h).$$

$$m_{i(h(\mathbf{\xi }))}({\tilde{\mathbf{\xi }}}_{h(\mathbf{\xi })}{B}_{h(\mathbf{\xi })}^k)\ge D_{3,a}|{\lambda }_{h(\mathbf{\xi })}{|}^k\ge D_{3,a}{\Lambda }^k(\mathbf{\xi }\in R_a).$$

$$D_{4,a}(K)$$

$$f(\boldsymbol{x}+\boldsymbol{h})-f(\boldsymbol{x})=\sum_{\boldsymbol{\xi}\in{\bf Z}^{d}}\widehat{f}(\boldsymbol{\xi})\exp(2\pi\sqrt{-1}\,\boldsymbol{\xi}\boldsymbol{x})\bigl{(}\exp(2\pi\sqrt{-1}\,\boldsymbol{\xi}\boldsymbol{h})-1\bigr{)},$$

$$\displaystyle\bigl{(}\omega^{(2)}(H,f)\bigr{)}^{2}$$

$$\displaystyle=2\sum_{\boldsymbol{\xi}\in{\bf Z}^{d}}|\widehat{f}(\boldsymbol{\xi})|^{2}\bigl{(}1-{\rm Re}\exp(2\pi\sqrt{-1}\,\boldsymbol{\xi}\boldsymbol{h})\bigr{)}.$$

$$\bigl{(}\omega^{(2)}(H,f)\bigr{)}^{2}\geq 2\sum_{\boldsymbol{\xi}\in{\bf Z}^{d}}|\widehat{f}(\boldsymbol{\xi})|^{2}\biggl{(}1-\prod_{i=1}^{d}\frac{\sin(2\pi\xi_{i}H)}{2\pi\xi_{i}H}\biggr{)}.$$

$$\biggl{|}\prod_{i=1}^{d}\frac{\sin(2\pi\xi_{i}H)}{2\pi\xi_{i}H}\biggr{|}\leq\frac{1}{2\pi|\xi_{j_{0}}|H}\leq\frac{1}{2\pi},$$

$$\bigl{(}\omega^{(2)}(H,f)\bigr{)}^{2}\geq(2-1/\pi)\sum_{\boldsymbol{\xi}\in{\bf Z}^{d}:\|\boldsymbol{\xi}\|\geq 1/H}|\widehat{f}(\boldsymbol{\xi})|^{2}.$$

$$\rho_{0}(x)=\Bigl{(}\frac{\sin x}{x}\Bigr{)}^{2}\quad\hbox{and}\quad\rho_{1}(x)=\rho_{0}\bigl{(}x/\sqrt{d}\,\bigr{)}+\rho_{0}\bigl{(}x/\sqrt{2d}\,\bigr{)}.$$

$${\hat{\rho }}_0(\xi )={\int }_{\mathbf{R}}{\rho }_0(x)e^{2\pi \sqrt{-1}\xi x}dx=\pi (1-\pi |\xi |)\vee 0,$$

Full text

This research is partially supported by JSPS KAKENHI 16K05204 and 15KT0106. It was also partially supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.

1. Introduction

For $\theta>1$ and a real valued square integrable function $f$ on $\bf R$ with period 1 satisfying $\int_{0}^{1}f=0$, regarded as random variables on $[\,0,1\,]$ Riesz-Raikov sums $\sum_{k=1}^{N}f(\theta^{k}x)$ obey the central limit theorem. This fact was first proved by Fortet [5] and Kac [8] when $\theta$ is an integer, and was extended to general case by Petit [16] and by the author [6]. In [7] the case when $\theta$ is a complex number was investigated. In this note, we consider a multidimensional analogue of this problem.

Let $\mathbf{R}^{d}$ denote the vector space of $d$-dimensional real column vectors and $\widehat{\mathbf{R}}^{d}$ that of real row vectors.

Fan [4] assumed a very mild condition on $A$ and proved that the sequence $\{A^{n}\boldsymbol{x}\}$ is uniformly distributed mod 1 for a.e. $\boldsymbol{x}$, and Lesigne [11] extended this result. Since the result of Fan implies the law of large numbers for $\{f(A^{n}\boldsymbol{x})\}$ where $f$ is periodic and continuous, it is very natural to ask whether the central limit theorem holds.

To have the central limit theorem, we here assume that $A$ is expanding, i.e., there exists a $q>1$ such that

[TABLE]

or equivalently,

[TABLE]

Let $f$ be a real valued square integrable function on $\mathbf{R}^{d}$ which satisfies

[TABLE]

where $\boldsymbol{e}_{1}$, …, $\boldsymbol{e}_{d}$ is the canonical basis of $\mathbf{R}^{d}$. We denote the Fourier series of $f$ by

[TABLE]

where ${\bf Z}^{d}$ is regarded as a subset of $\widehat{\mathbf{R}}^{d}$, and define the subsum $f_{a}$ by

[TABLE]

where $R_{a}=\{\boldsymbol{\xi}=(\xi_{1},\dots,\xi_{d})\in{\bf Z}^{d}\setminus\{\boldsymbol{0}\}\mid|\xi_{1}|,\dots,|\xi_{d}|\leq a\}$. Note that we have $\widehat{f}(\boldsymbol{0})=0$. We assume that $f$ satisfies the condition

[TABLE]

Note that a function of bounded variation over $[\,0,1)^{d}$ in the sense of Hardy-Krause satisfies this condition (Cf. [21]). An easy sufficient condition for (1.5) is the $L^{2}$-Dini condition

[TABLE]

where

[TABLE]

This fact is stated in Lemma 2.1.

To state our result, we introduce a quantity $\sigma^{2}(f)$, the limiting variance of the central limit theorem, by

[TABLE]

The series on the right hand side is shown to be absolutely convergent under the conditions (1.1) and (1.5). We denote the Lebesgue measure on $\mathbf{R}^{d}$ by $\operatorname{\rm Leb}$.

Theorem 1.1**.**

Let $A$ be a $d\times d$ real matrix satisfying (1.1), and let $f$ be a real valued function on $\mathbf{R}^{d}$ satisfying (1.3) and (1.5). Then for every bounded measurable $\Gamma\subset\mathbf{R}^{d}$,

[TABLE]

and

[TABLE]

for any $t\not=0$, where $\Phi_{\sigma^{2}(f)}$ is the distribution function of $N(0,\sigma^{2}(f))$. If $\sigma^{2}(f)>0$, then (1.9) holds also for $t=0$.

In the case when every coefficient of $A$ is an integer, that is the case when $A^{n}$ are endomorphisms on ${\bf T}^{d}$, Leonov [10], Fan [3], Levin [12] and Conze, Le Borgne, and Roger [2] assumed so called the partially expanding condition and proved the central limit theorem.

Löbbe [13] proved the central limit theorem, the law of the iterated logarithm, and the metric discrepancy results for $\{A_{n}\boldsymbol{x}\}$, where $A_{n}$ is a $d\times d$ matrix with integer coefficients satisfying

[TABLE]

for some $q>1$.

By putting $\boldsymbol{j}=(1,0)$ and $1<c_{1}<c_{2}$, we see that this condition cannot be satisfied even by

[TABLE]

This example suggests the following inference. For a matrix $A$, the sequence of matrix $A_{n}=A^{n}$ satisfies this condition only if absolute values of characteristic values of $A$ are all equal.

Since the condition (1.1) implies that modulus of all eigenvalues of $A$ are greater than one, we expect that the central limit theorem is valid under this assumption. But we could not prove because of some technical reason.

In the case of endomorphisms on ${\bf T}^{d}$, if $\sigma(f)=0$, one can express $f(\boldsymbol{x})=g(A\boldsymbol{x})-g(\boldsymbol{x})$ by using some locally square integrable periodic $g$. We could not prove it in our case, since the situation is more complicated. For example if

[TABLE]

we have $\sigma(f)=0$ for the function $f$ of the form

[TABLE]

where $g_{1}$, $g_{2}$, and $g_{3}$ are locally square integrable and periodic, $g_{2}$ does not depend on $x_{2}$, and $g_{3}$ does not depend of $x_{1}$. Although by a similar argument as one-dimensional case, we can prove that $\sigma(f)=0$ implies this expression, we do not have the result for general case so far.

2. Real Jordan form and related estimates

In this section we state some preliminary facts.

Irrespectively of the value of $\gamma\in\mathbf{N}$, for vectors $\boldsymbol{x}={}^{T}(x_{1},\dots,x_{\gamma})\in\mathbf{R}^{\gamma}$ and $\boldsymbol{\xi}=(\xi_{1},\dots,\xi_{\gamma})\in\widehat{\mathbf{R}}^{\gamma}$, we define $\|\boldsymbol{x}\|=\max_{\delta\leq\gamma}|x_{\delta}|$ and $\|\boldsymbol{\xi}\|=\max_{\delta\leq\gamma}|\xi_{\delta}|$. Abusing the notation we denote by $\boldsymbol{0}$ zero vectors in ${\mathbf{R}}^{\gamma}$ and $\widehat{\mathbf{R}}^{\gamma}$ for any $\gamma$.

For $\lambda\in{\bf R}$ a standard Jordan block $J_{\gamma}(\lambda)$ is a $\gamma\times\gamma$ matrix, and for $\lambda\in{\bf C}\setminus{\bf R}$ a real Jordan block $C_{\gamma}(\lambda)$ is a $2\gamma\times 2\gamma$ matrix defined as follows.

[TABLE]

where $Z_{\theta}=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\hfill\cos\theta\end{pmatrix}$ and $\theta=\arg\lambda$.

It is known (See e.g. Theorem 6.65 of [19]) that a real matrix is similar to the matrix of the form

[TABLE]

where $\lambda_{1}$, …, $\lambda_{\alpha}\in\mathbf{R}$, $\lambda_{\alpha+1}$, $\bar{\lambda}_{\alpha+1}$, …, $\lambda_{\beta}$, $\bar{\lambda}_{\beta}\in\mathbf{C}\setminus\mathbf{R}$ are characteristic values, i.e. there exists a real regular matrix

[TABLE]

such that $A=QBQ^{-1}$. We simply write $B=\begin{pmatrix}B_{1}&&\smash{\raise-4.0pt\hbox{\Large\bf 0}}\\ &\ddots\\ \smash{\raise 0.0pt\hbox{\Large\bf 0}}&&B_{\beta}\end{pmatrix}.$

Since $Q$ is regular, there exists a constant $1<D_{1,Q}<\infty$ such that

[TABLE]

Denoting the span of $\boldsymbol{q}_{1}^{(h)}$, …, $\boldsymbol{q}_{d_{h}}^{(h)}$ by $W_{h}$, we have

[TABLE]

By calculating $J_{\gamma}(\lambda)^{k}$ and $C_{\gamma}(\lambda)^{k}$ for $|\lambda|>1$, we have $\|J_{\gamma}(\lambda)^{k}\boldsymbol{x}\|\leq\gamma k^{\gamma}|\lambda|^{k}\|\boldsymbol{x}\|$ and $\|C_{\gamma}(\lambda)^{k}\boldsymbol{x}\|\leq 2\gamma k^{\gamma}|\lambda|^{k}\|\boldsymbol{x}\|$. Hence there exists a constant $D_{2,A}$ depending only on $A$ such that

[TABLE]

Denote the $\delta$-th component of $\boldsymbol{\xi}$ by $(\boldsymbol{\xi})_{\delta}$, and the 2-dimensional vector consisting of the $\delta$-th and the $(\delta+1)$-th components of $\boldsymbol{\xi}$ by $(\boldsymbol{\xi})_{\delta,\delta+1}$.

Put $m_{1}(\boldsymbol{0})=0$ and $m_{1}(\boldsymbol{\xi})=|\xi_{\delta}|$ if $\boldsymbol{\xi}=(\xi_{1},\dots,\xi_{\gamma})\in\widehat{\mathbf{R}}^{\gamma}$ satisfies $\xi_{1}=\cdots=\xi_{\delta-1}=0\not=\xi_{\delta}$. In the last case, we have $(\boldsymbol{\xi}J_{\gamma}(\lambda)^{k})_{1}=\cdots=(\boldsymbol{\xi}J_{\gamma}(\lambda)^{k})_{\delta-1}=0$, $(\boldsymbol{\xi}J_{\gamma}(\lambda)^{k})_{\delta}=\lambda^{k}\xi_{\delta}$, or $m_{1}(\boldsymbol{\xi}J_{\gamma}(\lambda)^{k})=|\lambda|^{k}m_{1}(\boldsymbol{\xi})$.

Put $m_{2}(\boldsymbol{0})=0$ and $m_{2}(\boldsymbol{\xi})=\|\boldsymbol{\eta}_{\delta}\|_{2}$ if $\boldsymbol{\xi}=(\boldsymbol{\eta}_{1},\dots,\boldsymbol{\eta}_{\gamma})\in\widehat{\mathbf{R}}^{2\gamma}$ ($\boldsymbol{\eta}_{1}$, …, $\boldsymbol{\eta}_{\gamma}\in\widehat{\mathbf{R}}^{2}$) satisfies $\boldsymbol{\eta}_{1}=\cdots=\boldsymbol{\eta}_{\delta-1}=\boldsymbol{0}\not=\boldsymbol{\eta}_{\delta}$. In the last case we have $(\boldsymbol{\xi}C_{\gamma}(\lambda)^{k})_{1,2}=\cdots=(\boldsymbol{\xi}C_{\gamma}(\lambda)^{k})_{2\delta-3,2\delta-2}=\boldsymbol{0}$ and $(\boldsymbol{\xi}C_{\gamma}(\lambda)^{k})_{2\delta-1,2\delta}=|\lambda|^{k}\boldsymbol{\eta}_{\delta}Z_{k\theta}$, or $m_{2}(\boldsymbol{\xi}C_{\gamma}(\lambda)^{k})=|\lambda|^{k}m_{2}(\boldsymbol{\xi})$.

For $\boldsymbol{\xi}\in{\bf Z}^{d}\setminus\{\boldsymbol{0}\}$, we write $\boldsymbol{\xi}Q=(\widetilde{\boldsymbol{\xi}}_{1},\dots,\widetilde{\boldsymbol{\xi}}_{\beta})$ where $\widetilde{\boldsymbol{\xi}}_{h}\in\widehat{\mathbf{R}}^{d_{h}}$. Put $i(h)=1$ for $h\in[\,1,\alpha\,]$ and $i(h)=2$ for $h\in[\,\alpha+1,\beta\,]$. The arguments above show that $m_{i(h)}(\widetilde{\boldsymbol{\xi}}_{h}B_{h}^{k})=|\lambda_{h}|^{k}m_{i(h)}(\widetilde{\boldsymbol{\xi}}_{h})$.

We denote $\Lambda=\min_{h\leq\beta}|\lambda_{h}|>1$. Since $Q$ is regular, there exists an $h\leq\beta$ such that $\widetilde{\boldsymbol{\xi}}_{h}\not=\boldsymbol{0}$. We denote the smallest such $h$ by $h({\boldsymbol{\xi}})$. Denote $D_{3,a}=\min_{\boldsymbol{\xi}\in R_{a}}m_{i(h({\boldsymbol{\xi}}))}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})})$. The arguments above show that

[TABLE]

We put

[TABLE]

and prove that $D_{4,a}:=D_{4,a}(\infty)$ is positive. Put $\ell_{1}=\max\{m_{i(h)}(\widetilde{\boldsymbol{\xi}}^{\prime}_{h})\mid\boldsymbol{\xi}^{\prime}\in R_{a},h\leq\beta\}$ and $\ell_{2}=\min\{m_{i(h)}(\widetilde{\boldsymbol{\xi}}^{\prime}_{h})\mid\boldsymbol{\xi}^{\prime}\in R_{a},h\leq\beta,m_{i(h)}(\widetilde{\boldsymbol{\xi}}^{\prime}_{h})\not=0\}>0$. By (2.3), there exists a $K_{0}$ such that $k\geq K_{0}$ implies $m_{i(h({\boldsymbol{\xi}}))}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})\geq 2\ell_{1}$.

For $k\geq K_{0}$, we can verify $m_{i(h({\boldsymbol{\xi}}))}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}^{\prime})\geq\ell_{1}\wedge\ell_{2}$. Here we prove it in the case $h({\boldsymbol{\xi}})=1$. The other case can be proved in the same way. Suppose that $(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})_{1}=\cdots=(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})_{\delta-1}=0$, $|(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})_{\delta}|=m_{1}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})$. If $\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}^{\prime}={\bf 0}$, we have $\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})}=\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}$ and $m_{1}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})=m_{1}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})\geq 2\ell_{1}$. If not, suppose that $(\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{1}=\dots=(\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta^{\prime}-1}=0$ and $|(\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta^{\prime}}|=m_{1}(\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta^{\prime}}$. If $\delta^{\prime}<\delta$, then we have $(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{1}=\cdots=(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta^{\prime}-1}=0$ and $|(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta^{\prime}}|=|(\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta^{\prime}}|=m_{1}(\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta^{\prime}}\geq\ell_{2}$. Hence we have $m_{1}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})\geq\ell_{2}$ in this case. If $\delta^{\prime}=\delta$, then we have $(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{1}=\cdots=(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta-1}=0$ and $|(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta}|\geq|(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})_{\delta}|-|(\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta}|\geq 2\ell_{1}-\ell_{1}$. Hence we have $m_{1}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})\geq\ell_{1}$ in this case. If $\delta^{\prime}>\delta$, then we have $(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{1}=\cdots=(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta-1}=0$ and $|(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})_{\delta}|=|(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k})_{\delta}|\geq 2\ell_{1}$. Hence we have $m_{1}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})\geq 2\ell_{1}$ in this case.

Since $D_{4,a}(K_{0})$ is positive, we have $D_{4,a}>0$.

Lemma 2.1**.**

The condition (1.6) implies the condition (1.5).

Proof.

Let $\|\boldsymbol{h}\|\leq H$. By

[TABLE]

we have

[TABLE]

By integrating both sides over $[\,-H,H\,]^{d}$ by $\boldsymbol{h}$ and dividing by $(2H)^{d}$, we have

[TABLE]

If $\|\boldsymbol{\xi}\|\geq 1/H$, then there exists a $j_{0}$ such that $|\xi_{j_{0}}|H\geq 1$. Since we have

[TABLE]

we have

[TABLE]

Hence we have $\omega^{(2)}(2^{-m},f)\geq\|f-f_{2^{m}}\|_{L^{2}[0,1]^{d}}$. Since the condition (1.6) is equivalent to $\sum_{m}\omega^{(2)}(2^{-m},f)<\infty$, it implies the condition (1.5). ∎

3. Fourth moment estimates

In this section, we assume the condition (1.1), Put

[TABLE]

By

[TABLE]

we obtain $\widehat{\rho}_{1}(\xi)=0$ if $|\xi|\geq 1/(\pi\sqrt{d}\,)$ and $\rho_{1}(x)>0$ for $x\in{\bf R}$. By putting

[TABLE]

by $\sqrt{d}\|\boldsymbol{\xi}\|\geq\|\boldsymbol{\xi}\|_{2}$, we obtain

[TABLE]

Lemma 3.1**.**

Assume that $A$ satisfies (1.1), For any bounded measurable set $\Gamma\subset\mathbf{R}^{d}$, and for any trigonometric polynomial $f_{a}$ satisfying (1.3), there exists a constant $D_{5,\Gamma,A,f_{a}}$ such that

[TABLE]

for any finite set $\Delta\subset\mathbf{N}$.

Proof.

Since we can take a constant $D_{6,\Gamma}<\infty$ such that ${\boldsymbol{1}}_{\Gamma}(\boldsymbol{x})\leq D_{6,\Gamma}\,\rho(\boldsymbol{x})$, it is sufficient to prove

[TABLE]

Komlós-Révész [9] proved the following: Suppose that $X$ is a non-empty set, $m$ is a $\sigma$-finite measure on $X$, and $\{\varphi_{i}\}$ is a sequence of real valued measurable functions satisfying

[TABLE]

for some $M<\infty$, then there exists an absolute constant $D_{8}$ such that

[TABLE]

Noting this estimate and by applying Erdős-Stečkin Theorem (See [15]), we can derive

[TABLE]

where $D_{9}$ is an absolute constant. Note that any subsequence of $\{\varphi_{k}\}$ satisfying (3.2) also satisfies (3.2) and our version

[TABLE]

follows.

We use the expression

[TABLE]

to have

[TABLE]

Because of $\sum_{\boldsymbol{\xi}\in R_{a}}|\widehat{f}(\boldsymbol{\xi})|<\infty$, if we have (3.3) for $\varphi_{k}({}\cdot{})=\cos(2\pi\boldsymbol{\xi}A^{k}{}\cdot{}+\gamma_{\boldsymbol{\xi}})$, then we have (3.3) for $\varphi_{k}({}\cdot{})=f_{a}(A^{k}{}\cdot{})$. Hence it is enough to prove that the sequence $\{\cos(2\pi\boldsymbol{\xi}A^{k}\boldsymbol{x}+\gamma_{\boldsymbol{\xi}})\}_{k\in\mathbf{N}}$ satisfies (3.2) under the measure $\rho(\boldsymbol{x})\,d\boldsymbol{x}$.

Take $p\in\mathbf{N}$ large enough to satisfy

[TABLE]

For $r=0$, $1$, …, $p-1$, we show that $\{\cos(2\pi\boldsymbol{\xi}A^{kp-r}\boldsymbol{x}+\gamma_{\boldsymbol{\xi}})\}_{k\in\mathbf{N}}$ satisfies (3.2) under the measure $\rho(\boldsymbol{x})\,d\boldsymbol{x}$. Then we can see that it satisfies (3.3), and then by using Minkowski’s inequality we see that $\{\cos(2\pi\boldsymbol{\xi}A^{k}\boldsymbol{x}+\gamma_{\boldsymbol{\xi}})\}_{k\in\mathbf{N}}$ itself satisfies (3.3).

We first note that for $h\leq\beta$ and $k_{1}<k_{2}<k_{3}<k_{4}$,

[TABLE]

By putting $\varsigma_{4}=1$, we have

[TABLE]

Suppose that $h(\boldsymbol{\xi})\leq\alpha$. By denoting $h(\boldsymbol{\xi})$ simply by $h$, denoting $\widetilde{\boldsymbol{\xi}}_{h}$ by$(\xi_{1},\dots,\xi_{d_{h}})$, and by taking a $\delta$ such that $\xi_{1}=\dots=\xi_{\delta-1}=0\not=\xi_{\delta}$, by $|\xi_{\delta}|=m_{1}(\widetilde{\boldsymbol{\xi}}_{h})\geq D_{3,a}$, we have

[TABLE]

Hence we have

[TABLE]

which implies

[TABLE]

Suppose that $h(\boldsymbol{\xi})>\alpha$. By denoting $h(\boldsymbol{\xi})$ simply by $h$, denoting $\widetilde{\boldsymbol{\xi}}_{h}$ by$(\boldsymbol{\eta}_{1},\dots,\boldsymbol{\eta}_{d_{h}/2})$, and by taking a $\delta$ such that $\boldsymbol{\eta}_{1}=\dots=\boldsymbol{\eta}_{\delta-1}=\boldsymbol{0}\not=\boldsymbol{\eta}_{\delta}$, we have

[TABLE]

Hence in the same way as before, we can verify (3.4). ∎

4. Martingale approximation

Take $L_{\delta}^{(h)}\in\mathbf{R}$ ($\delta=1$, …, $d_{h}$, $h=1$, …$\beta$) and $L>0$ arbitrarily and put

[TABLE]

Let $\mathcal{F}$ be the Borel $\sigma$-field on $\Omega$ and put

[TABLE]

We consider the sequence $\{f_{a}(A^{k}{}\cdot{})\}$ on the probability space $(\Omega,\mathcal{F},P_{\Omega})$. We state the almost sure invariance principle for the sequence. We denote the Lebesgue measure on $[\,0,1)$ by $\operatorname{\rm leb}$.

Proposition 4.1**.**

Let $A$ be a $d\times d$ real matrix satisfying (1.1), and let $f_{a}$ be a trigonometric polynomial on $\mathbf{R}^{d}$ satisfying (1.3). By taking the product probability space $(\Omega\times[\,0,1),\mathcal{F}\otimes\mathcal{B}([\,0,1)),P_{\Omega}\times\operatorname{\rm leb})$ and regard the sequence $\{f_{a}(A^{k}{}\cdot{})\}$ defined on this space. If $\sigma^{2}(f_{a})>0$, then we can define a sequence $\{Z_{i}\}$ of standard normal i.i.d. such that

[TABLE]

From this proposition, we can derive the central limit theorem.

Corollary 4.2**.**

Let $A$ be a $d\times d$ real matrix satisfying (1.1), and let $f_{a}$ be a trigonometric polynomial on $\mathbf{R}^{d}$ satisfying (1.3). Then for any probability measure $P$ on $\mathbf{R}^{d}$ which is absolutely continuous with respect to Lebesgue measure, on the probability space $(\mathbf{R}^{d},\mathcal{B}^{d},P)$ we have the convergence in law

[TABLE]

Proof of Proposition 4.1.

We divide the increasing sequence $\bf N$ of positive integers into consecutive blocks

[TABLE]

where

[TABLE]

Put $i^{-}=\min\Delta_{i}$ and $i^{+}=\max\Delta_{i}$. Clearly we have

[TABLE]

Put

[TABLE]

For $i\in\mathbf{N}$, $1\leq h\leq\beta$, $1\leq\delta\leq d_{h}$, and $j_{\delta}^{(h)}=0$, …, $2^{\mu_{h}{(i)}}-1$, we set

[TABLE]

and denote the collection of all such cubes by $\mathcal{J}(i)$. Let $\mathcal{F}_{i}$ be the $\sigma$-field on $\Omega$ generated by $\mathcal{J}(i)$. $\{\mathcal{F}_{i}\}$ forms a filtration on $(\Omega,\mathcal{F},P_{\Omega})$. Let

[TABLE]

Clearly $\{\widetilde{\mathcal{F}}_{i}\}$ forms a filtration on $(\Omega\times[\,0,1),\mathcal{F}\times\mathcal{B}[\,0,1),P_{\Omega}\times\operatorname{\rm leb})$.

For $(\boldsymbol{x},x)\in\Omega\times[\,0,1)$, we here put

[TABLE]

and prove

[TABLE]

where $\widetilde{E}({}\cdot{}\mid{}\cdot{})$ denotes the conditional expectation on $\Omega\times[\,0,1)$ and $E({}\cdot{}\mid{}\cdot{})$ that on $\Omega$. The first equality is trivial. Take $\boldsymbol{x}\in\Omega$ arbitrarily and take $J\in\mathcal{J}(i-1)$ such that $\boldsymbol{x}\in J$. We note that

[TABLE]

By putting

[TABLE]

where $E_{\gamma}$ is the unit matrix of size $\gamma\times\gamma$, we can write

[TABLE]

by using some $\boldsymbol{b}\in{\bf R}^{d}$. Changing variables by $\boldsymbol{y}=\boldsymbol{b}+QR_{i-1}\boldsymbol{t}$ and noting

[TABLE]

we have

[TABLE]

where $c=\boldsymbol{\xi}QB^{k}Q^{-1}\boldsymbol{b}$, and $\widetilde{\boldsymbol{\xi}}_{h}\in\widehat{\bf R}^{d_{h}}$ and $\boldsymbol{t}_{h}\in{\bf R}^{d_{h}}$ are given by $\boldsymbol{\xi}Q=(\widetilde{\boldsymbol{\xi}}_{1},\dots,\widetilde{\boldsymbol{\xi}}_{\beta})$ and $\boldsymbol{t}=\begin{pmatrix}\boldsymbol{t}_{1}\\ \vdots\\ \boldsymbol{t}_{\beta}\end{pmatrix}$. If we write $\widetilde{\boldsymbol{\xi}}_{h}B_{h}^{k}=(\zeta^{(h)}_{1},\dots,\zeta^{(h)}_{d_{h}})$, we have

[TABLE]

where $c^{\prime}=L2^{-\mu_{h}(i-1)}\zeta^{(h)}_{\delta}$, $\phi(x)=(\sin x)/x$ if $x\not=0$ and $\phi(0)=1$.

By (2.3), there exists a $\delta(\boldsymbol{\xi})$ such that $|\zeta^{(h(\boldsymbol{\xi}))}_{\delta(\boldsymbol{\xi})}|\geq{D_{3,a}}|\lambda_{h(\boldsymbol{\xi})}|^{k}/2$. Hence we have

[TABLE]

By

[TABLE]

we have (4.4).

Secondly, we prove

[TABLE]

Assume that $k\in\Delta_{i}$ and $\boldsymbol{x}\in J\in\mathcal{J}(i)$. Again the first equality is trivial. We have

[TABLE]

By $\boldsymbol{x}$, $\boldsymbol{y}\in J$, we have

[TABLE]

for some $-1<t^{(h)}_{\delta}<1$. By (2.2), we have

[TABLE]

and

[TABLE]

Lipschitz continuity of $f$ implies

[TABLE]

and thereby (4.5).

Put

[TABLE]

Clearly $\{Y_{i},\mathcal{F}_{i}\}$ and $\{\widetilde{Y}_{i},\widetilde{\mathcal{F}}_{i}\}$ are martingale differences and $\widetilde{Y}_{i}(\boldsymbol{x},x)=Y_{i}(\boldsymbol{x})$. By combining (4.4) and (4.5), we have

[TABLE]

By $\|T_{i}\|_{\infty}=O(i)$, we have $\|E(T_{i}\mid\mathcal{F}_{i})\|_{\infty}$, $\|E(T_{i}\mid\mathcal{F}_{i-1})\|_{\infty}=O(i)$, and $\|Y_{i}\|_{\infty}=O(i)$, which implies $\|Y_{i}+T_{i}\|_{\infty}=O(i)$ and

[TABLE]

By $\|Y_{i}^{2}+T_{i}^{2}\|_{\infty}=O(i^{2})$, we have

[TABLE]

By the last inequality and (3.1), we have

[TABLE]

We have

[TABLE]

Put

[TABLE]

There exists an $l_{0}$ such that for all $\boldsymbol{\xi}\in R_{a}$ and $\boldsymbol{\xi}^{\prime}\in R_{a}$, $\boldsymbol{\xi}A^{l}+\boldsymbol{\xi}^{\prime}\not=\boldsymbol{0}$ and $\boldsymbol{\xi}+\boldsymbol{\xi}^{\prime}A^{l}\not=\boldsymbol{0}$ hold for $l>l_{0}$. If $l\leq l_{0}$ and $\boldsymbol{\xi}+\boldsymbol{\xi}^{\prime}A^{l}=\boldsymbol{0}$, we have

[TABLE]

By noting

[TABLE]

we have

[TABLE]

For $\boldsymbol{x}\in\Omega$, take $J\in\mathcal{F}_{i-1}$ such that $\boldsymbol{x}\in J$. Under the condition $\boldsymbol{\xi}A^{k}+\boldsymbol{\xi}^{\prime}A^{k^{\prime}}\not=\boldsymbol{0}$, in the same way as before, we have

[TABLE]

and by $m_{i(h({\boldsymbol{\xi}}))}(\widetilde{\boldsymbol{\xi}}_{h({\boldsymbol{\xi}})}B_{h({\boldsymbol{\xi}})}^{k-k^{\prime}}+\widetilde{\boldsymbol{\xi}}^{\prime}_{h({\boldsymbol{\xi}})})\geq D_{4,a}$, as before we have

[TABLE]

Since the number of choices of $(\boldsymbol{\xi},\boldsymbol{\xi}^{\prime})$ is finite and the number of choices of $(k,k^{\prime})$ is at most $i$, we have

[TABLE]

By putting

[TABLE]

we have

[TABLE]

for some $D_{10,f_{a}}<\infty$. Since we have

[TABLE]

we have

[TABLE]

where

[TABLE]

Now we use the theorem which is a version of Strassen’s theorem (Theorem 4.4 of [18]).

Theorem 4.3** **(Monrad-Philipp [14] Theorem 7.

This version is Lemma A.4 in Philipp [17]).

Let $\{\widetilde{Y}_{i},\widetilde{\mathcal{F}}_{i}\}$ be a square integrable martingale difference satisfying

[TABLE]

for some non-decreasing $\psi$ such that $\psi(x)(\log x)^{\alpha}/x$ is non-increasing for some $\alpha>50$ and $\lim_{x\to\infty}\psi(x)=\infty$. If there exists a uniformly distributed random variable $\widetilde{U}$ which is independent of $\{\widetilde{Y}_{n}\}$, there exists a sequence $\{Z_{i}\}$ of standard normal i.i.d. such that

[TABLE]

where

[TABLE]

From now on, we regard $f_{a}(A^{k}\boldsymbol{x})$ as a random variable on $\Omega\times[\,0,1)$.

Recall that $\sigma^{2}(f_{a})>0$ and put $\psi(x)=x^{4/5}$. One can see that

[TABLE]

by (4.8) and (4.9).

Since

[TABLE]

holds for large $M$, we see by (4.7) that

[TABLE]

Because of $\beta_{M+1}-\beta_{M}=v_{M+1}\to\infty$, we obtain

[TABLE]

for large M, and $\widetilde{V}_{i}\leq\beta_{M}+D_{11}$ becomes equivalent to $i\leq M$. By putting $t=\beta_{M}+D_{11}$ in (4.10) and by noting (4.6) we have

[TABLE]

Put

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By ${}^{\#}\Delta_{M}^{\natural}=O(M\log M)$, we obtain

[TABLE]

Note that

[TABLE]

We can prove

[TABLE]

since (3.1) implies

[TABLE]

and is summable in $M$.

Hence for $N\in\Delta_{M}^{\prime}\cup\Delta_{M}$, we obtain

[TABLE]

Now we apply the following result on the fluctuation of the standard Wiener process $W(t)$ due to Csörgő-Révész ((1.2.4) in Theorem 1.2.1 of [1]). For non-decreasing $a_{T}$ such that $0<a_{T}\leq T$ and $T/a_{T}$ is non-decreasing, we have

[TABLE]

By putting

[TABLE]

we have

[TABLE]

By combining (4.11), (4.14) and (4.15) we have (4.2). ∎

5. Variance control

We first prove that the series in (1.7) is absolutely convergent. By using the convention $\widehat{f}(\boldsymbol{\xi})=0$ for $\boldsymbol{\xi}\notin{\bf Z}^{d}$, we have

[TABLE]

Hence by noting

[TABLE]

for $\boldsymbol{\xi}\in{\bf Z}^{d}\setminus\{\boldsymbol{0}\}$, we have

[TABLE]

Let $\Gamma\subset{\bf R}^{d}$ be a bounded measurable set. Since we have $L^{2}$ convergence $f_{a}\to f$ on $[\,0,1)^{d}$ and by periodicity, we have the convergence on any bounded set $\Gamma^{\prime}$.

By changing variable we have $L^{2}(\Gamma)$ convergence $f_{a}(A^{k}\boldsymbol{x})\to f(A^{k}\boldsymbol{x})$. Hence we have $L^{1}(\Gamma)$ convergence $f_{a}(A^{k}\boldsymbol{x})f_{a}(A^{k^{\prime}}\boldsymbol{x})\allowbreak\to f(A^{k}\boldsymbol{x})f(A^{k^{\prime}}\boldsymbol{x})$, and hence convergence in measure. Thus we have the convergence

[TABLE]

in measure on $\Gamma$ under the measure $\rho(\boldsymbol{x})\,d\boldsymbol{x}$. That is why we can apply Fatou’s Lemma and have

[TABLE]

By (1.2), we obtain

[TABLE]

For $\boldsymbol{\eta}\in\mathbf{R}^{d}$, there is at most one $\boldsymbol{\xi}\in{\bf Z}^{d}$ such that $\|\boldsymbol{\eta}-\boldsymbol{\xi}\|_{2}\leq 1/\pi$. In case such $\boldsymbol{\xi}$ exists, let $\chi(\boldsymbol{\eta})=\boldsymbol{\xi}$, and $\chi(\boldsymbol{\eta})=\boldsymbol{0}$ otherwise. If $\boldsymbol{\xi}\not=\boldsymbol{\xi}^{\prime}$, then $\|\boldsymbol{\xi}A^{m}-\boldsymbol{\xi}^{\prime}A^{m}\|_{2}\geq\|\boldsymbol{\xi}-\boldsymbol{\xi}^{\prime}\|_{2}\geq 1$, and $\chi(\boldsymbol{\xi}A^{m})\not=\chi(\boldsymbol{\xi}^{\prime}A^{m})$ if $\chi(\boldsymbol{\xi}A^{m})\not=\boldsymbol{0}$ and $\chi(\boldsymbol{\xi}^{\prime}A^{m})\not=\boldsymbol{0}$. If $\chi(\boldsymbol{\xi}A^{m})\not=\boldsymbol{0}$, then

[TABLE]

By

[TABLE]

we have

[TABLE]

We have proved

[TABLE]

where $\widehat{C}=D_{6,\Gamma}\sum_{m=0}^{\infty}\|f-f_{q^{m}/2\sqrt{d}}\|_{L^{2}[0,1)^{d}}$.

If $\boldsymbol{\xi}A^{k^{\prime}}+\boldsymbol{\xi}^{\prime}\not=0$, by Riemann-Lebesgue Lemma we have

[TABLE]

as $k\to\infty$. Hence for a trigonometric polynomial $f_{a}$, we have (1.8) as below:

[TABLE]

Because of the absolute convergence (5.1), we obtain

[TABLE]

Proof of Corollary 4.2.

Put

[TABLE]

First we assume $\sigma^{2}(f_{a})>0$ and prove the central limit theorem under the measure $P_{\Omega}$. By using Proposition 4.1 we have

[TABLE]

and the law of ${G_{N\sigma^{2}(f_{a})}}/{\sqrt{N}}$ is $N(0,\sigma^{2}(f_{a}))$, we see that the limit law under $P_{\Omega}\times\operatorname{\rm leb}$ of $X_{N}^{a}$ is $N(0,\sigma^{2}(f_{a}))$. Since the law of $X_{N}^{a}$ under $P_{\Omega}\times\operatorname{\rm leb}$ is identical with the law under $P_{\Omega}$, we see that $N(0,\sigma^{2}(f_{a}))$ is also the limit law under $P$ of $X_{N}^{a}$.

Now, denote by $\mathcal{G}$ the class of integrable functions $g$ on ${\bf R}^{d}$ satisfying

[TABLE]

and denote by $\mathcal{H}$ the collection of ${\bf 1}_{\Omega}\in\mathcal{G}$ given by (4.1) using arbitrary $L>0$ and $L^{(h)}_{\delta}\in{\bf R}$ ($\delta=1$, …, $d_{h}$, $h=1$, …, $\beta$). We can easily show

[TABLE]

By the above argument we have already proved $\mathcal{H}\subset\mathcal{G}$, and by (5.4) we can see that any simple function which is given as a linear combination of indicator functions with supports in $\mathcal{H}$ belongs to $\mathcal{G}$. Since any continuous function with compact support can be arbitrarily approximated in the sense of $L^{1}({\bf R}^{d})$ by such simple function, we see that it belongs to $\mathcal{G}$. Since any integrable function with compact support can be arbitrarily approximated in the sense of $L^{1}({\bf R}^{d})$ by a continuous function with compact support, we see that it belongs to $\mathcal{G}$. Hence we can see that (4.3) holds under any probability measure $P$ on ${\bf R}^{d}$ which is absolutely continuous with respect to the Lebesgue measure.

In case when $\sigma^{2}(f_{a})=0$, by (1.8) we see that the limit law of $X_{N}^{a}$ is the delta measure concentrated on $\boldsymbol{0}$, that is $N(0,0)$. It proves (4.3) for $\sigma^{2}(f_{a})=0$. ∎

Proof of Theorem 1.1.

Put

[TABLE]

We have proved

[TABLE]

which implies

[TABLE]

where $b_{a}=\|f-f_{a}\|_{L^{2}[0,1)^{d}}^{1/3}$. By

[TABLE]

we have

[TABLE]

By letting $a\to\infty$, we have (1.9).

By putting

[TABLE]

we have

[TABLE]

and hence by letting $N\to\infty$,

[TABLE]

By letting $a\to\infty$, we have (1.8). ∎

6. Acknowledgement

The author thank the referee for his or her valuable advices, especially for the information on the literature [3].