Asymptotic expansion of an estimator for the Hurst coefficient

Abstract

Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. We first derive the expansion formula of the principal term of the error of the estimator using a recently developed theory of asymptotic expansion of the distribution of Wiener functionals, and utilize the perturbation method on the obtained formula in order to calculate the expansion of the estimator. We also discuss some second-order modifications of the estimator. Numerical results show that the asymptotic expansion attains higher accuracy than the normal approximation.

1 Introduction

Let $B=(B_t)_{t\in [0,T]}$ be a fractional Brownian motion with Hurst coefficient $H\in (0,1)$. For a fixed positive number T, let ${t_j}=t_j^n=jT/n$ for $n\in {{\mathbb {N}}}$ and $j\in \{0,1,...,n\}$. We are interested in estimation of the Hurst parameter H from the sampled data $(B_{t_j})_{j=0,1,...,n}$. The second-order difference $d_{n,j}$ of B is denoted by

$$\begin{aligned} d_{n,j}= & {} B_{t_{j+1}}-2B_{t_j}+B_{t_{j-1}}. \end{aligned}$$

Note that $t_j$ depends on n. We make the sum of squares $V_{n,T}^{(2)}$ of $d_{n,j}$ by

$$\begin{aligned} V_{n,T}^{(2)}= & {} \sum _{j=1}^{n-1} d_{n,j}^2. \end{aligned}$$

In Kubilius et al. (2018) this estimator is denoted by $V_{n,T}^{(2)B}$. For estimation of H, Benassi et al. (1998) and Istas and Lang (1997) introduced the estimator

$$\begin{aligned} \widehat{H}_n^{(2)}= & {} 0\vee \bigg ( \frac{1}{2}-\frac{1}{2\log 2}\log \frac{V_{2n,T}^{(2)}}{V_{n,T}^{(2)}} \bigg )\wedge 1. \end{aligned}$$

(1.1)

The estimator $\widehat{H}_n^{(2)}$ is preferable since it is consistent and asymptotically normal as $n\rightarrow \infty$. On the other hand, it is known that normal approximation to the distribution of a statistic is not necessarily satisfactory. Section 4 shows that the error of the normal approximation to the histogram of the estimator is not negligible numerically. In this paper, we will consider a higher-order approximation of the distribution of the error of $\widehat{H}_n^{(2)}$ using the asymptotic expansion.

Asymptotic expansion is a standard concept in statistics and well developed for independent models. As for basic literature, we refer the reader to Cramér (1928), Cramér (2004), Gnedenko and Kolmogorov (1954), Bhattacharya (1971), Petrov (1975), Bhattacharya and Ranga Rao (1976), and to a recent textbook by Bhattacharya et al. (2016). The theory of asymptotic expansion has been extended to dependent models such as mixing Markov processes and martingales, and an amount of literature is available today. See Yoshida (2016) and references therein.

The theory of asymptotic expansion for Wiener functionals has been developed recently. In the central limit case, Tudor and Yoshida (2019a) derived the first-order expansion for vector-valued sequences of random variables, and Tudor and Yoshida (2019b) did the arbitrary order of asymptotic expansion for Wiener functionals. The latter has been applied to asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion by Tudor and Yoshida (2020). In the mixed Gaussian limit case, Nualart and Yoshida (2019) presented asymptotic expansion of Skorohod integrals. This method is used in Yoshida (2020) for Brownian functionals having anticipative weights. Yamagishi and Yoshida (2022) provided order estimates of functionals related to fractional Brownian motion, with weighted graphs, and asymptotic expansion of the quadratic variation of fractional diffusions. In this paper, we will apply the scheme by Tudor and Yoshida (2019a) combined with a perturbation method.

The organization of this paper is as follows. Section 2 gives the main results, and Sect. 3 proves them. The real performance of asymptotic expansion is numerically studied in Sect. 4, and we verify the improvement produced by the correction term of the asymptotic expansion formula of $\widehat{H}_n^{(2)}$. The technical part of this paper is Sect. 3. The technicalities are outlined at the beginning of Sect. 3.

2 Asymptotic expansion for $\widehat{H}_n^{(2)}$

To carry out the computation, we will work with the Malliavin calculus. Suppose that $H\in (0,1)$. Denote by $\mathcal{E}$ the set of step functions on [0, T]. Introduce an inner product in $\mathcal{E}$ by

$$\begin{aligned} \langle 1_{[0,t]},1_{[0,s]}\rangle _{{{\mathfrak {H}}}}= & {} \frac{1}{2}\big (t^{2H}+s^{2H}-|t-s|^{2H}\big ) \quad (t,s\in [0,T]) \end{aligned}$$

Let $\Vert h\Vert _{{\mathfrak {H}}}=\langle h,h\rangle _{{\mathfrak {H}}}^{1/2}$ for $h\in \mathcal{E}$. The closure of $\mathcal{E}$ with respect to $\Vert \cdot \Vert _{{\mathfrak {H}}}$ is denoted by ${{\mathfrak {H}}}$. It is a real separable Hilbert space with an extended inner product $\langle \cdot ,\cdot \rangle _{{\mathfrak {H}}}$ and the corresponding norm denoted by $\Vert \cdot \Vert _{{\mathfrak {H}}}$. With an isonormal Gaussian process ${{\mathbb {W}}}=({{\mathbb {W}}}(h))_{h\in {{\mathfrak {H}}}}$ on some probability space $(\Omega ,\mathcal{F},P)$, it is possible to realize a fractional Brownian motion B by $B_t={{\mathbb {W}}}(1_{[0,t]})$. Denote by $({{\mathbb {D}}}_{s,p},\Vert \cdot \Vert _{s,p})$ the Sobolev space of Wiener functionals with differentiability index $s\in {{\mathbb {R}}}$ and integrability index $p>1$. We write ${{\mathbb {D}}}_\infty =\cap _{s,p}{{\mathbb {D}}}_{s,p}$. The Malliavin derivative is denoted by D, and the divergence operator (Skorohod integral) by $\delta$. The k-fold Wiener integral of $h^{\otimes k}$ is denoted by $I_k(h^{\otimes k})=\delta ^k(h^{\otimes k})$ for $h\in {{\mathfrak {H}}}$. For basic elements in the Malliavin calculus, we refer the reader to Watanabe (1984), Ikeda and Watanabe (1989), Nualart (2006) and Nourdin and Peccati (2012).

Let $1_j^n=1_{(t_{j-1}^n,t_j^n]}$. The function $1_j^n$ will simply be denoted by $1_j$. Then

$$\begin{aligned} d_{n,j} = \delta (1_{j+1}-1_j)={{\mathbb {W}}}(1_{j+1}-1_j)=I_1(1_{j+1}-1_j). \end{aligned}$$

By the product formula, we have

$$\begin{aligned} d_{n,j}^2= & {} I_1(1_{j+1}-1_j)^2 = I_2\big ((1_{j+1}-1_j)^{\otimes 2}\big )+\Vert 1_{j+1}-1_j\Vert _{{\mathfrak {H}}}^2. \end{aligned}$$

By definition,

$$\begin{aligned} \Vert 1_{j+1}-1_j\Vert _{{\mathfrak {H}}}^2= & {} 2\Vert 1_j\Vert _{{\mathfrak {H}}}^2-2\langle 1_{j+1},1_j\rangle \\ {}= & {} 2(T/n)^{2H}-\big ((2T/n)^{2H}-2(T/n)^{2H}\big ) \\ {}= & {} \big (4-2^{2H}\big )(T/n)^{2H}. \end{aligned}$$

Thus,

$$\begin{aligned} E[d_{n,j}^2]= & {} \big (4-2^{2H}\big )(T/n)^{2H} \end{aligned}$$

for $j=1,...,n-1$. In particular,

$$\begin{aligned} E\big [V_{n,T}^{(2)}\big ]= & {} \big (4-2^{2H}\big )(T/n)^{2H}(n-1). \end{aligned}$$

(2.1)

The use of $V_{n,T}^{(2)}$ for the estimator $\widehat{H}_n^{(2)}$ of (1.1) is accounted by (2.1), that entails

$$\begin{aligned} \log \frac{E\big [V_{2n,T}^{(2)}\big ]}{E\big [V_{n,T}^{(2)}\big ]}= & {} \log \frac{\big (4-2^{2H}\big )(T/(2n))^{2H}(2n-1)}{\big (4-2^{2H}\big )(T/n)^{2H}(n-1)} \\ {}= & {} -2H\log 2+\log \frac{2(1-1/(2n))}{1-1/n} \\ {}= & {} -2H\log 2+\log 2+\frac{1}{2n}+O(n^{-2}) \end{aligned}$$

as $n\rightarrow \infty$.

We say $X_n=o_M(n^\nu )$ [resp. $X_n=O_M(n^\nu )$] for a sequence $(X_n)_{n\in {{\mathbb {N}}}}$ of random variables and $\nu \in {{\mathbb {R}}}$ if $X_n\in {{\mathbb {D}}}_\infty$ and $\Vert X_n\Vert _{s,p}=o(n^\nu )$ [resp. $\Vert X_n\Vert _{s,p}=O(n^\nu )$] as $n\rightarrow \infty$ for any $s\in {{\mathbb {R}}}$ and $p\in (1,\infty )$. We shall need a high-order of stochastic expansion for $\widehat{H}_n^{(2)}$ to go up to the asymptotic expansion from the central limit theorem.

Proposition 2.1

There exists a sequence $(\psi _n)_{n\in {{\mathbb {N}}}}$ of Wiener functionals satisfying the following conditions.

• (i) $\psi _n:\Omega \rightarrow [0,1]$, $\psi _n\in {{\mathbb {D}}}_\infty$ for $n\in {{\mathbb {N}}}$, and $\Vert \psi _n-1\Vert _{s,p}=O(n^{-L})$ as $n\rightarrow \infty$ for every $(s,p,L)\in {{\mathbb {R}}}\times (1,\infty )\times (0,\infty )$.

• (ii) The sequence $(Z_n)_{n\in {{\mathbb {N}}}}$ of random variables defined by

  $$\begin{aligned} Z_n= & {} (2\log 2)\sqrt{n}\big (\widehat{H}_n^{(2)}-H\big )\psi _n \end{aligned}$$

  (2.2)

  for $n\in {{\mathbb {N}}}$ admits a stochastic expansion

  $$\begin{aligned} Z_n= & {} M_n+n^{-1/2}N_n, \end{aligned}$$

  (2.3)

  where

  $$\begin{aligned} M_n= & {} -n^{1/2}\bigg \{\bigg [\frac{V^{(2)}_{2n,T}}{E[V^{(2)}_{2n,T}]}-1\bigg ] -\bigg [\frac{V^{(2)}_{n,T}}{E[V^{(2)}_{n,T}]}-1\bigg ]\bigg \} \end{aligned}$$

  (2.4)

  and

  $$\begin{aligned} N_n= & {} n\bigg \{ \frac{1}{2}\bigg [\frac{V^{(2)}_{2n,T}}{E[V^{(2)}_{2n,T}]}-1\bigg ]^2 {-}\frac{1}{2}\bigg [\frac{V^{(2)}_{n,T}}{E[V^{(2)}_{n,T}]}-1\bigg ]^2 \bigg \}-\frac{1}{2}+O_M(n^{-1/2}) \end{aligned}$$

  (2.5)

  as $n\rightarrow \infty$. In particular, $Z_n\in {{\mathbb {D}}}_\infty$ for $n\in {{\mathbb {N}}}$.

See Sect. 3.3 for the proof of Proposition 2.1.

From now on, our approach will be as follows. We will first validate asymptotic expansion of the distribution of $M_n$, and next apply the perturbation method to the stochastic expansion (2.3) with (2.4) and (2.5) in order to derive an Edgeworth expansion formula for $\widehat{H}_n^{(2)}$. For positive numbers K and $\gamma$, $\mathcal{E}(K,\gamma )$ denotes the set of measurable functions on ${{\mathbb {R}}}$ such that $|f(z)|\le K(1+|z|^\gamma )$ for all $z\in {{\mathbb {R}}}$. The density function of the normal distribution with mean $\mu$ and variance C is denoted by $\phi (z;\mu ,C)$. The main interest of this paper is in the following theorem.

Theorem 2.2

There exist a positive constant $v_H$ and an odd polynomial q(z) of degree 3 such that the function

$$\begin{aligned} p_n(z)= & {} \big (1+n^{-1/2}q(z)\big )\phi (z;0,v_H) \end{aligned}$$

(2.6)

satisfies

$$\begin{aligned} \sup _{f\in \mathcal{E}(K,\gamma )} \bigg |E\big [f\big (\sqrt{n}(\widehat{H}_n^{(2)}-H)\big )\big ]-\int _{{\mathbb {R}}}f(z)p_n(z)dz\bigg |= & {} o(n^{-1/2})\quad (n\rightarrow \infty ) \end{aligned}$$

(2.7)

for every positive numbers K and $\gamma$. See (3.38) of Sect. 3.5 for the definition of q(z) and $v_H$.

The second-order modification of $\widehat{H}_n^{(2)}$ is given by

$$\begin{aligned} \widehat{H}_n^{(b)}= & {} { 0\vee \bigg \{\widehat{H}_n^{(2)}-\frac{1}{n}b\big (\widehat{H}_n^{(2)}\big )\bigg \}\wedge 1,} \end{aligned}$$

(2.8)

where the function $b:[0,1]\rightarrow {{\mathbb {R}}}$ is assumed to be smooth on (0, 1), and the symbols $\wedge$ and $\vee$ stand for $\min$ and $\max$, respectively. Asymptotic expansion of the distribution of $\widehat{H}_n^{(b)}$ can be obtained easily by a trivial modification of that of $\widehat{H}_n^{(2)}$. Define $p_n^{(b)}(z)$ by

$$\begin{aligned} p_n^{(b)}(z)= & {} \big (1+n^{-1/2}q^{(b)}(z)\big )\phi (z;0,v_H) \end{aligned}$$

(2.9)

with

$$\begin{aligned} q^{(b)}(z)= & {} q(z) - \frac{b(H)}{v_H}z \end{aligned}$$

Theorem 2.3

For any positive numbers K and $\gamma$,

$$\begin{aligned} \sup _{f\in \mathcal{E}(K,\gamma )} \bigg |E\big [f\big (\sqrt{n}(\widehat{H}_n^{(b)}-H)\big )\big ]-\int _{{\mathbb {R}}}f(z)p_n^{(b)}(z)dz\bigg |= & {} o(n^{-1/2}) \end{aligned}$$

(2.10)

as $n\rightarrow \infty$.

As an application of Theorem 2.3, we obtain a second-order unbiased estimator $\widehat{H}_n^{(*)}$ by choosing the function

$$\begin{aligned} b^*(H)= & {} \int _{{\mathbb {R}}}zq(z)\phi (z;0,v_H)dz \end{aligned}$$

as b(H), namely, $\widehat{H}_n^{(*)}=\widehat{H}_n^{(b^*)}$. Then

$$\begin{aligned} E\big [\sqrt{n}(\widehat{H}_n^{(*)}-H)\big ]= & {} o(n^{-1/2}) \end{aligned}$$

as $n\rightarrow \infty$. If to take the function

$$\begin{aligned} b^{**}(H)= & {} \int _0^\infty q(z)e^{-\frac{z^2}{2v_H}}dz \end{aligned}$$

as b(H), then the estimator $\widehat{H}_n^{(**)}=\widehat{H}_n^{(b^{**})}$ satisfies

$$\begin{aligned} P\big [\sqrt{n}\big (\widehat{H}_n^{(**)}-H\big )\le 0\big ]= & {} \frac{1}{2}+o(n^{-1/2}) \end{aligned}$$

and

$$\begin{aligned} P\big [\sqrt{n}\big (\widehat{H}_n^{(**)}-H\big )\ge 0\big ]= & {} \frac{1}{2}+o(n^{-1/2}) \end{aligned}$$

as $n\rightarrow \infty$. That is, the estimator $\widehat{H}_n^{(**)}$ is second-order median unbiased. See Remark 3.15 about regularity of the functions $b^*$ and $b^{**}$.

The rest of this paper will be devoted to the proof of Theorems 2.2 and 2.3.

3 Proofs

This section proves Theorems 2.2 and 2.3. We first derive the asymptotic expansion formula of $M_n$ in Sect. 3.4 (Theorem 3.14). In Sect. 3.5, using the perturbation method with the asymptotic expansion of $M_n$ and the stochastic expansion of $Z_n$ (Proposition 2.1, justified in Sect. 3.3), we obtain the asymptotic expansion formula of $Z_n$, and hence of $\widehat{H}_n^{(2)}$. To obtain the asymptotic expansion of $M_n$ and to use the perturbation method, we need to know the asymptotic behavior of $U_n=(M_n^{[1]},M_n^{[2]},n^{1/2}\widetilde{G}_n)$, which is defined by (3.20) below. Indeed, the fourth moment theorem will be applied to $U_n$ in Proposition 3.12 (Sect. 3.2.3). In order to calculate the asymptotic covariance matrix of $U_n$, we give technical lemmas in Sects. 3.1 and 3.2. The proof of Theorem 2.3 is given in Sect. 3.6.

The estimator $\widehat{H}_n^{(2)}$ includes the second-order differences of the fBm B of two different time steps, namely $d_{n,j}$ and $d_{2n,j}$. The variance and covariance of $d_{n,j}$ and $d_{2n,j'}$ are represented by $\widehat{\rho }$ and $\widetilde{\rho }$ in Lemma 3.5(a) and (b), and 3.7(a). (See (3.5) and (3.13) for the definition of $\widehat{\rho }$ and $\widetilde{\rho }$.) Since the tail of $\widehat{\rho }(j)$ and $\widetilde{\rho }(j)$ (as $j\rightarrow \infty$) decay fast enough (Lemma 3.4 and 3.6), we can estimate the asymptotic behavior of the cumulants of $(M_n^{[1]},M_n^{[2]})$: Lemmas 3.5 and 3.7 for the second cumulants, Lemmas 3.8 and 3.9 for the third, and Lemmas 3.10 and 3.11 for the fourth. The basic idea of the estimates above are presented at Lemma 3.1. (See also Remark 3.2.)

3.1 Preliminary lemmas

Let $k\in {{\mathbb {N}}}$ with $k\ge 2$. Consider $k+1$ sequences of positive integers

$$\begin{aligned} \big (\nu (n)\big )_{n\in {{\mathbb {N}}}},\> \big (\nu _1(n)\big )_{n\in {{\mathbb {N}}}},...,\big (\nu _k(n)\big )_{n\in {{\mathbb {N}}}} \end{aligned}$$

such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\nu (n)=\infty \quad \text {and}\quad \lim _{n\rightarrow \infty }\frac{\nu _\alpha (n)}{\nu (n)} \>=\>p_\alpha \end{aligned}$$

(3.1)

for some constant $p_\alpha \in (0,\infty )$ for every $\alpha =1,...,k$.

For functions $\varrho _1,...,\varrho _k:{{\mathbb {Z}}}\rightarrow {{\mathbb {R}}}$, let

$$\begin{aligned} A_n(\varrho _1,...,\varrho _k)= & {} \nu (n)^{-1}\sum _{j_1=1}^{\nu _1(n)}\cdots \sum _{j_k=1}^{\nu _k(n)} \varrho _1(j_1-j_2)\varrho _2(j_2-j_3) \\{} & {} \cdots \varrho _{k-1}(j_{k-1}-j_k)\varrho _k(j_k-j_1) \end{aligned}$$

and also let

$$\begin{aligned} {{\mathbb {V}}}(\varrho _1,...,\varrho _k)= & {} \bigg |\sum _{i_1,...,i_{k-1}\in {{\mathbb {Z}}}} \varrho _1(i_1)\varrho _2(i_1-i_2)\varrho _3(i_2-i_3)\cdots \varrho _{k-1}(i_{k-2}-i_{k-1})\varrho _k(i_{k-1}) \bigg |. \end{aligned}$$

For ${{\mathbb {I}}}=\{1,...,k-1\}$, let

$$\begin{aligned} p_n(i_1,...,i_{k-1})= & {} \bigg [ \bigg \{\bigg (\bigwedge _{\alpha \in {{\mathbb {I}}}}\frac{\nu _{\alpha +1}(n)-i_\alpha }{\nu (n)}\bigg ) \wedge \frac{\nu _1(n)}{\nu (n)}\bigg \} - \bigg \{\bigg (\bigvee _{\alpha \in {{\mathbb {I}}}}\frac{-i_\alpha }{\nu (n)}\bigg )\vee 0\bigg \} \bigg ] \\{} & {} \times 1_{\{1-\nu _1(n)\le i_\alpha \le \nu _{\alpha +1}(n)\> (\alpha \in {{\mathbb {I}}}) \}}. \end{aligned}$$

Lemma 3.1

• (a) $\displaystyle {{\mathbb {V}}}(\varrho _1,...,\varrho _k) \>\le \>{{\mathbb {V}}}(|\varrho _1|,...,|\varrho _k|) \>\le \>\big \Vert \varrho _1\big \Vert _{\ell _{\frac{k}{k-1}}}\cdots \big \Vert \varrho _k\big \Vert _{\ell _{\frac{k}{k-1}}}$.

• (b)
  $$\begin{aligned} A_n(\varrho _1,...,\varrho _k)= & {} \sum _{i_1,...,i_{k-1}\in {{\mathbb {Z}}}} \varrho _1(-i_1)\varrho _2(i_1-i_2)\varrho _{3}(i_2-i_3)\cdots \\{} & {} \cdots \varrho _{k-1}(i_{k-2}-i_{k-1})\varrho _k(i_{k-1}) p_n(i_1,...,i_{k-1}). \end{aligned}$$

• (c) Suppose that the condition (3.1) is satisfied. If ${{\mathbb {V}}}(|\varrho _1(-\cdot )|,|\varrho _2|,...,|\varrho _k|) <\infty$, then the limit

  $$\begin{aligned} \lim _{n\rightarrow \infty }A_n(\varrho _1,...,\varrho _k)= & {} \bigg (\bigwedge _{\alpha =1,...,k} p_\alpha \bigg ) \sum _{i_1,...,i_{k-1}\in {{\mathbb {Z}}}} \varrho _1(-i_1)\varrho _2(i_1-i_2)\varrho _3(i_2-i_3)\cdots \nonumber \\{} & {} \times \varrho _{k-1}(i_{k-2}-i_{k-1})\varrho _k(i_{k-1}) \nonumber \\ \end{aligned}$$

  (3.2)

  exists, that is, the sum on the right-hand side of (3.2) is finite. In particular, if $\big \Vert \varrho _\alpha \big \Vert _{\ell _{\frac{k}{k-1}}}<\infty$ for all $\alpha \in \{1,...,k\}$, then the convergence (3.2) holds under (3.1).

• (d) Suppose that the functions $\varrho _{1},...,\varrho _{k}:{{\mathbb {Z}}}\rightarrow {{\mathbb {R}}}$ satisfy

  $$\begin{aligned} |\varrho _{\alpha }(j)|\le & {} C(1+|j|)^{-\gamma }\quad (j\in {{\mathbb {Z}}};\>\alpha =1,...,k) \end{aligned}$$

  for some constant $\gamma >\frac{k}{k-1}$. Suppose that the condition (3.1) is satisfied. Then

  $$\begin{aligned} {{\mathbb {V}}}(\varrho _{1},...,\varrho _{k}) \le {{\mathbb {V}}}(|\varrho _{1}|,...,|\varrho _{k}|) < \infty \end{aligned}$$

  In particular,

  $$\begin{aligned} \sup _{n\in {{\mathbb {N}}}}\big |A_n(\varrho _{1},...,\varrho _{k})\big | \le \sup _{n\in {{\mathbb {N}}}}\big |A_n(|\varrho _{1}|,...,|\varrho _{k}|)\big |<\infty , \end{aligned}$$

  and the convergence (3.2) holds.

Proof

(a) The system of linear equations

$$\begin{aligned} \left\{ \begin{array}{l} \frac{k-1}{p}\>=\>\frac{1}{q}+(k-2) \\ \frac{1}{p}+\frac{1}{q}\>=\>1 \end{array}\right. \end{aligned}$$

is solved by $p=\frac{k}{k-1}$ and $q=k$. Therefore, Young’s inequality yields

$$\begin{aligned} {{\mathbb {V}}}(|\varrho _1|,...,|\varrho _k|)\le & {} \big \Vert |\varrho |_1*\cdots *|\varrho _{k-1}|\big \Vert _{\ell _k}\>\big \Vert \varrho _k\big \Vert _{\ell _{\frac{k}{k-1}}} \\ {}\le & {} \big \Vert \varrho _1\big \Vert _{\ell _{\frac{k}{k-1}}}\cdots \big \Vert \varrho _k\big \Vert _{\ell _{\frac{k}{k-1}}}. \end{aligned}$$

(b) By the change of variables

$$\begin{aligned} i_\alpha \>=\>j_{\alpha +1} - j_1\quad (\alpha =1,...,k-1) \end{aligned}$$

for given $j_1$, we obtain

$$\begin{aligned}{} & {} A_n(\varrho _1,...,\varrho _k) \\{} & {} \quad = \nu (n)^{-1} \sum _{j_1=1}^{\nu _1(n)}\sum _{i_1,...,i_{k-1}\in {{\mathbb {Z}}}} \varrho _1(-i_1)\varrho _2(i_1-i_2)\varrho (i_2-i_3) \\{} & {} \qquad \cdots \varrho _{k-1}(i_{k-2}-i_{k-1})\varrho _k(i_{k-1}) \\{} & {} \qquad \times 1_{\{1-i_\alpha \le j_1\le \nu _\alpha (n)-i_\alpha \>(\alpha =1,...,k-1)\}} \\{} & {} \qquad \times 1_{\{1-\nu _1(n)\le i_\alpha \le \nu _{\alpha +1}(n)\>(\alpha =1,...,k-1)\}}\\{} & {} \quad = \sum _{i_1,...,i_{k-1}\in {{\mathbb {Z}}}} \varrho _1(-i_1)\varrho _2(i_1-i_2)\varrho (i_2-i_3)\cdots \varrho _{k-1}(i_{k-2}-i_{k-1})\varrho _k(i_{k-1})\\{} & {} \qquad \times p_n(i_1,...,i_{k-1}) \end{aligned}$$

(c) Under the condition (3.1),

$$\begin{aligned} \sup _{n\in {{\mathbb {N}}}}\sup _{i_1,...,i_{k-1}\in {{\mathbb {Z}}}}|p_n(i_1,...,i_{k-1})|<\infty \quad \text {and}\quad \lim _{n\rightarrow \infty }p_n(i_1,...,i_{k-1})\>=\>\bigwedge _{\alpha =1,...,k} p_\alpha . \end{aligned}$$

Then Lebesgue’s theorem implies the convergence (3.2).

(d) Use (a) and (c) to prove (d). $\square$

Remark 3.2

(i) We have ${{\mathbb {V}}}\big (\overbrace{|\rho _1(-\cdot )|,...,|\rho _k|}^{k}\big ) < \infty$ for $(\rho _1,...,\rho _k)\in \{\widehat{\rho },\widetilde{\rho }\}^k$ since $4-2H>2>\frac{k-1}{k}$ for every $k\ge 2$, thanks to the estimates (3.6) and (3.14) below. (ii) It is possible to strengthen the result (3.2) to a representation of the form $A_n(\varrho _1,...,\varrho _k)\>=\>\text {constant}+O(n^{-1})$ by putting a more restrictive condition on $\nu (n)$ and $\nu _\alpha (n)$ than (3.1). In this paper, such a convergence is necessary only in the case $k=2$, and we will give estimate for the error term directly without the help of Lemma 3.1.

Recall that $1^n_j=1_{(t_{j-1}^n,t_j^n]}$. Let

$$\begin{aligned} f^{[\alpha ]}_n= & {} \frac{\sqrt{n}\sum _{j=1}^{\alpha n-1}(1^{\alpha n}_{j+1}-1^{\alpha n}_j)^{\otimes 2}}{E[V^{(2)}_{\alpha n,T}]} \end{aligned}$$

(3.3)

for $\alpha \in \{1,2\}$. Then $M_n=I_2(f_n^{[1]})-I_2(f_n^{[2]})$ for $M_n$ defined by (2.4). The operator $L=-\delta D$ is the Malliavin operator (Ornstein-Uhlenbeck operator). It is a numeric operator such that $LF=(-q)F$ for elements F of the q-th Wiener chaos. Since $(-L)^{-1}M_n \>=\>2^{-1}I_2(f_n^{[1]})-2^{-1}I_2(f_n^{[2]})$ and $D(-L)^{-1}M_n \>=\>I_1(f_n^{[1]})-I_1(f_n^{[2]})$, the second-order $\Gamma$-factor $\Gamma _n^{(2)}(M_n)$ of $M_n$ is given by

$$\begin{aligned} G_n= & {} \Gamma _n^{(2)}(M_n) \>=\>\langle DM_n,D(-L)^{-1}M_n\rangle _{{\mathfrak {H}}}\\= & {} \big \langle 2I_1(f_n^{[1]})-2I_1(f_n^{[2]}),\>I_1(f_n^{[1]})-I_1(f_n^{[2]})\big \rangle _{{\mathfrak {H}}}\end{aligned}$$

for $n\in {{\mathbb {N}}}$. The product formula gives

$$\begin{aligned} G_n= & {} 2I_2\big (f_n^{[2]}\otimes _1f_n^{[2]}\big )+2\Vert f_n^{[2]}\Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2 -4I_2\big (f_n^{[2]}\odot _1f_n^{[1]}\big )-4\langle f_n^{[2]},f_n^{[1]}\rangle _{{{\mathfrak {H}}}^{\otimes 2}} \nonumber \\{} & {} +2I_2\big (f_n^{[1]}\otimes _1f_n^{[1]}\big )+2\Vert f_n^{[1]}\Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2, \end{aligned}$$

(3.4)

where $f_n^{[2]}\odot _1f_n^{[1]}$ is the symmetrized 1-contraction of $f_n^{[2]}\otimes f_n^{[1]}$.

Backward shift operator $\mathsf{{B}}$ is defined by $\mathsf{{B}}\theta (x)=\theta (x-1)$ for $x\in {{\mathbb {R}}}$ and a sequence $\theta =(\theta (x+t))_{t\in {{\mathbb {Z}}}}$ of numbers. The following lemma is an exercise.

Lemma 3.3

Let $x\in {{\mathbb {R}}}$, $I\in {{\mathbb {N}}}$ and $k_1,...,k_I\in {{\mathbb {N}}}$. Then

$$\begin{aligned} \prod _{i=1}^{I}(1-\mathsf{{B}}^{k_i})f(x)= & {} (-1)^I\int _0^{k_1}\cdots \int _0^{k_I}f^{(I)}\big (x-s_1-\cdots -s_I)ds_1\cdots ds_I \end{aligned}$$

for any function $f\in C^I\big ([x-\sum _{i=1}^Ik_i,x]\big )$, the set of functions of class $C^I$ on $[x-\sum _{i=1}^Ik_i,x]$.

Let

$$\begin{aligned} \widehat{\rho }(j)= & {} \frac{1}{2}\big \{-|j-2|^{2H}+4|j-1|^{2H}-6|j|^{2H}+4|j+1|^{2H}-|j+2|^{2H}\big \} \end{aligned}$$

(3.5)

Lemma 3.4

$$\begin{aligned} \widehat{\rho }(i)\sim & {} H(2H-1)(2H-2)(2H-3)|i|^{2H-4}\quad (|i|\rightarrow \infty ) \end{aligned}$$

(3.6)

In particular, $\sum _{i=1}^\infty |i|^3\widehat{\rho }(i)^2<\infty$.

Proof

With the backward shift operator $\mathsf{{B}}$, we have

$$\begin{aligned} (1-\mathsf{{B}})^4= & {} 1-4\mathsf{{B}}+6\mathsf{{B}}^2-4\textsf{B}^3+\mathsf{{B}}^4. \end{aligned}$$

Therefore, with $f(x)=|x+2|^{2H}$ we have

$$\begin{aligned} \widehat{\rho }(j)= & {} -\frac{1}{2}\big ((1-\mathsf{{B}})^4 f \big )(j) \end{aligned}$$

and hence (3.6) follows from Lemma 3.3. $\square$

Let

$$\begin{aligned} \Sigma _{11}= & {} 2\bigg [1+\frac{2}{\big (4-2^{2H}\big )^2}\sum _{i=1}^\infty \widehat{\rho }(i)^2\bigg ], \end{aligned}$$

and let

$$\begin{aligned} \Sigma _{22}=\frac{1}{2}\Sigma _{11}. \end{aligned}$$

(3.7)

Lemma 3.5

• (a) For $j,j'\in \{1,...,n-1\}$,

  $$\begin{aligned} \big \langle 1_{j+1}^n-1_j^n, 1_{j'+1}^n-1_{j'}^n\big \rangle _{{\mathfrak {H}}}= & {} \frac{T^{2H}}{n^{2H}}\widehat{\rho }(j-j'). \end{aligned}$$

  (3.8)

• (b) For $k,k'\in \{1,...,2n-1\}$,

  $$\begin{aligned} \big \langle 1_{k+1}^{2n}-1_k^{2n}, 1_{k'+1}^{2n}-1_{k'}^{2n}\big \rangle _{{\mathfrak {H}}}= & {} \frac{T^{2H}}{(2n)^{2H}}\widehat{\rho }(k-k'). \end{aligned}$$

  (3.9)

• (c) As $n\rightarrow \infty$,

  $$\begin{aligned} \Vert f_n^{[1]}\Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2= & {} \frac{1}{2}\Sigma _{11}+O(n^{-1}). \end{aligned}$$

  (3.10)

• (d) As $n\rightarrow \infty$,

  $$\begin{aligned} \Vert f_n^{[2]}\Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2= & {} \frac{1}{2}\Sigma _{22}+O(n^{-1}). \end{aligned}$$

  (3.11)

Proof

(a) and (b): We will use an elementary formula

$$\begin{aligned} E\big [(B_d-B_c)(B_b-B_a)\big ]= & {} \frac{1}{2}\big (|d-a|^{2H}+|c-b|^{2H}-|d-b|^{2H}-|c-a|^{2H}\big )\nonumber \\ \end{aligned}$$

(3.12)

for $a,b,c,d\in [0,T]$. We obtain (3.8) and(3.9) since

$$\begin{aligned} \big \langle 1_{j+1}^n-1_j^n, 1_{j'+1}^n-1_{j'}^n\big \rangle _{{\mathfrak {H}}}= & {} \langle 1_{j+1}^n,1_{j'+1}^n\rangle _{{\mathfrak {H}}}+\langle 1_j^n,1_{j'}^n\rangle _{{\mathfrak {H}}}-\langle 1_{j+1}^n,1_{j'}^n\rangle _{{\mathfrak {H}}}-\langle 1_j^n,1_{j'+1}^n\rangle _{{\mathfrak {H}}}\\ {}= & {} 2\times \frac{T^{2H}}{2n^{2H}}\big \{|j'-j+1|^{2H}+|j'-j-1|^{2H}-2|j'-j|^{2H}\big \} \\ {}{} & {} -\frac{T^{2H}}{2n^{2H}}\big \{|j'-j|^{2H}+|j'-j-2|^{2H}-2|j'-j-1|^{2H}\big \} \\ {}{} & {} -\frac{T^{2H}}{2n^{2H}}\big \{|j'-j|^{2H}+|j'-j+2|^{2H}-2|j'-j+1|^{2H}\big \} \\ {}= & {} \frac{T^{2H}}{n^{2H}}\widehat{\rho }(j'-j) \end{aligned}$$

from (3.5). (3.9) is now trivial.

(c) and (d): For $j,j'\in \{1,...,n-1\}$,

$$\begin{aligned} \sum _{j,j'=1}^{n-1}\big \langle (1_{j+1}^n-1_j^n)^{\otimes 2}, (1_{j'+1}^n-1_{j'}^n)^{\otimes 2} \big \rangle _{{{\mathfrak {H}}}^{\otimes 2}}= & {} \left( \frac{T^{2H}}{n^{2H}}\right) ^2 {\sum _{j,j'=1}^{n-1}}\widehat{\rho }(j-j')^2. \end{aligned}$$

Therefore

$$\begin{aligned} \Vert f_n^{[1]}\Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2= & {} {\bigg \Vert \frac{\sqrt{n}\sum _{j=1}^{n-1}(1^{n}_{j+1}-1^{n}_j)^{\otimes 2}}{E[V^{(2)}_{n,T}]} \bigg \Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2} \\ {}= & {} {\frac{n}{\big (4-2^{2H}\big )^2 (n-1)^2}\sum _{j,k=1}^{n-1}\widehat{\rho }(k-j)^2} \\ {}= & {} \frac{n^2}{\big (4-2^{2H}\big )^2 (n-1)^2} \bigg [\big (4-2^{2H}\big )^2\>\frac{n-1}{n}+2\sum _{i=1}^{n-2} \left( 1-\frac{i{+}1}{n}\right) \widehat{\rho }(i)^2\bigg ] \\ {}= & {} 1+\frac{2}{\big (4-2^{2H}\big )^2}\sum _{i=1}^\infty \widehat{\rho }(i)^2+O(n^{-1}) \\ {}= & {} \frac{1}{2}\Sigma _{11}+O(n^{-1}). \end{aligned}$$

With this result, we also obtain

$$\begin{aligned} \Vert f_n^{[2]}\Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2= & {} \bigg \Vert \frac{\sqrt{2^{-1}\cdot 2n}\sum _{j=1}^{2n-1}(1^{2n}_{j+1}-1^{2n}_j)^{\otimes 2}}{E[V^{(2)}_{2n,T}]} \bigg \Vert _{{{\mathfrak {H}}}^{\otimes 2}}^2 \\ {}= & {} \frac{1}{2}\bigg [\frac{1}{2}\Sigma _{11}+O(n^{-1})\bigg ] \>=\>\frac{1}{2}\Sigma _{22}+O(n^{-1}). \end{aligned}$$

$\square$

Let

$$\begin{aligned} \widetilde{\rho }(j)= & {} \frac{1}{2^{2H+1}}\bigg \{ -|j-3|^{2H}+2|j-2|^{2H}+|j-1|^{2H}-4|j|^{2H}+|j+1|^{2H} \nonumber \\{} & {} +2|j+2|^{2H}-|j+3|^{2H}\bigg \}. \end{aligned}$$

(3.13)

for $j\in {{\mathbb {Z}}}$. We will write

$$\begin{aligned} \widetilde{\rho }(k,j)= & {} \frac{1}{2^{2H+1}}\bigg \{ -|k-2j-3|^{2H}+2|k-2j-2|^{2H}+|k-2j-1|^{2H}-4|k-2j|^{2H} \\{} & {} +|k-2j+1|^{2H} +2|k-2j+2|^{2H}-|k-2j+3|^{2H}\bigg \}. \end{aligned}$$

Then $\widetilde{\rho }(k,j)=\widetilde{\rho }(k-2j)$.

Lemma 3.6

$$\begin{aligned} \widetilde{\rho }(j)\sim & {} 2^{2-2H}H(2H-1)(2H-2)(2H-3)|j|^{2H-4}\quad (|j|\rightarrow \infty ). \end{aligned}$$

(3.14)

In particular, $\sum _{j\in {{\mathbb {Z}}}}|i|^3\widetilde{\rho }(j)^2<\infty$.

Proof

We have

$$\begin{aligned} (1-\textsf{B})^2(1-\textsf{B}^2)^2= & {} 1-2\textsf{B}-\textsf{B}^2+4\textsf{B}^3-\textsf{B}^4-2\textsf{B}^5+\textsf{B}^6 \end{aligned}$$

for the backward shift operator $\textsf{B}$. Therefore we obtain (3.14). $\square$

Let

$$\begin{aligned} \Sigma _{12}= & {} \frac{{ 2^{2H}}}{ (4-2^{2H})^2 } \>\sum _{\ell =-\infty }^{\infty } \widetilde{\rho }(\ell )^2. \end{aligned}$$

Lemma 3.7

• (a) For $j\in \{1,...,n-1\}$ and $k\in \{1,...,2n-1\}$,

  $$\begin{aligned} \big \langle 1_{j+1}^{n}-1_j^{n}, 1_{k+1}^{2n}-1_k^{2n}\big \rangle _{{\mathfrak {H}}}= & {} \frac{T^{2H}}{n^{2H}}\widetilde{\rho }(k,j). \end{aligned}$$

  (3.15)

• (b) As $n\rightarrow \infty$,

  $$\begin{aligned} \langle f_n^{[2]},f_n^{[1]}\rangle _{{{\mathfrak {H}}}^{\otimes 2}}= & {} \frac{1}{2}\Sigma _{12}+O(n^{-1}). \end{aligned}$$

  (3.16)

Proof

From (3.12), we have

$$\begin{aligned}{} & {} \big \langle 1_{k+1}^{2n}-1_k^{2n}, 1_{j+1}^n-1_j^n\big \rangle _{{\mathfrak {H}}}\\{} & {} \quad = \langle 1_{k+1}^{2n},1_{j+1}^n\rangle _{{\mathfrak {H}}}+\langle 1_k^{2n},1_j^n\rangle _{{\mathfrak {H}}}-\langle 1_{k+1}^{2n},1_j^n\rangle _{{\mathfrak {H}}}-\langle 1_k^{2n},1_{j+1}^n\rangle _{{\mathfrak {H}}}\\{} & {} \quad = \frac{T^{2H}}{2(2n)^{2H}}\bigg \{ |k-2j-2|^{2H}+|k-2j+1|^{2H}-|k-2j|^{2H}-|k-2j-1|^{2H}\\{} & {} \qquad +|k-2j-1|^{2H}+|k-2j+2|^{2H}-|k-2j+1|^{2H}-|k-2j|^{2H} \\{} & {} \qquad -|k-2j|^{2H}-|k-2j+3|^{2H}+|k-2j+1|^{2H}+|k-2j+2|^{2H} \\{} & {} \qquad -|k-2j-3|^{2H}-|k-2j|^{2H}+|k-2j-2|^{2H}+|k-2j-1|^{2H}\bigg \} \\{} & {} \quad = \frac{T^{2H}}{2(2n)^{2H}}\bigg \{ -|k-2j-3|^{2H}+2|k-2j-2|^{2H}+|k-2j-1|^{2H}-4|k-2j|^{2H} \\{} & {} \qquad +|k-2j+1|^{2H} +2|k-2j+2|^{2H}-|k-2j+3|^{2H}\bigg \} \\{} & {} \quad = \frac{T^{2H}}{n^{2H}}\widetilde{\rho }(k,j) \end{aligned}$$

This shows (3.15). By using (2.1) and (3.15), we obtain

$$\begin{aligned} \langle f_n^{[2]},f_n^{[1]}\rangle _{{{\mathfrak {H}}}^{\otimes 2}}= & {} { \frac{n}{E[V^{(2)}_{2n,T}]E[V^{(2)}_{n,T}]} \sum _{k=1}^{2n-1} \sum _{j=1}^{n-1} \big \langle 1^{2n}_{k+1}-1^{2n}_k,\>1^{n}_{j+1}-1^{n}_j\big \rangle _{{{\mathfrak {H}}}}^2 } \\ {}= & {} \frac{n}{E[V^{(2)}_{2n,T}]E[V^{(2)}_{n,T}]}\left( \frac{T^{2H}}{n^{2H}}\right) ^2 \>\sum _{k=1}^{2n-1} \sum _{j=1}^{n-1}\widetilde{\rho }(k,j)^2 \end{aligned}$$

and hence

$$\begin{aligned} \langle f_n^{[2]},f_n^{[1]}\rangle _{{{\mathfrak {H}}}^{\otimes 2}}= & {} \frac{2^{2H}n}{ (4-2^{2H})^2(n-1)(2n-1) } \>\sum _{k=1}^{2n-1} \sum _{j=1}^{n-1}\widetilde{\rho }(k-2j)^2 \\ {}= & {} \frac{2^{2H}n}{ (4-2^{2H})^2(n-1)(2n-1) } \>\sum _{\ell =-2n+3}^{2n-3} {{\left( n-1-\lfloor {\left| \ell \right| }/2\rfloor \right) }} \widetilde{\rho }(\ell )^2 \\ {}= & {} \frac{2^{2H}}{ 2(4-2^{2H})^2 } \>\sum _{\ell =-\infty }^{\infty } \widetilde{\rho }(\ell )^2 +O(n^{-1}) \>=\>\frac{1}{2}\Sigma _{12}+O(n^{-1}) \end{aligned}$$

with the help of Lemma 3.6. $\square$

3.2 A central limit theorem toward the asymptotic expansion

Let

$$\begin{aligned} G_\infty \>=\>\Sigma _{22}-2\Sigma _{12}+\Sigma _{11} \>=\>\frac{3}{2}\Sigma _{11}-2\Sigma _{12}. \end{aligned}$$

(3.17)

From (3.4), Lemma 3.5 (c), (d) and Lemma 3.7 (b), we have

$$\begin{aligned} G_n= & {} \widetilde{G}_n+G_\infty +O(n^{-1}) \end{aligned}$$

(3.18)

where

$$\begin{aligned} \widetilde{G}_n= & {} 2I_2\big (f_n^{[2]}\otimes _1f_n^{[2]}\big ) -4I_2\big (f_n^{[2]}\odot _1f_n^{[1]}\big ) +2I_2\big (f_n^{[1]}\otimes _1f_n^{[1]}\big ). \end{aligned}$$

(3.19)

Write $M_n^{[i]}=I_2(f_n^{[i]})$ for $i=1,2$. By the definition of $f_n^{[i]}$ ($i=1,2$) given at (3.3), we have $M_n=I_2(f_n^{[1]})-I_2(f_n^{[2]}) =M_n^{[1]}-M_n^{[2]}.$ We will derive a central limit theorem for the sequence

$$\begin{aligned} U_n= & {} \big (M_n^{[1]},M_n^{[2]},n^{1/2}\widetilde{G}_n\big ). \end{aligned}$$

(3.20)

For $f,g\in {{\mathfrak {H}}}^{\odot 2}$, symmetric tensors, we have

$$\begin{aligned} E\big [I_2(f)I_2(g)\big ]= & {} 2f\odot _2g \>=\>2\langle f,g\rangle _{{{\mathfrak {H}}}^{\otimes 2}} \end{aligned}$$

Thus,

$$\begin{aligned} E\big [M_n^{[\alpha ]}\widetilde{G}_n\big ]= & {} E\big [I_2(f_n^{[\alpha ]})\big \{ 2I_2\big (f_n^{[2]}\otimes _1f_n^{[2]}\big ) -4I_2\big (f_n^{[2]}\odot _1f_n^{[1]}\big ) +2I_2\big (f_n^{[1]}\otimes _1f_n^{[1]}\big )\big \} \big ] \nonumber \\= & {} { 4}\big \langle f_n^{[\alpha ]},f_n^{[2]}\otimes _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} -{ 8}\big \langle f_n^{[\alpha ]},f_n^{[2]}\odot _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \nonumber \\{} & {} \quad +{ 4}\big \langle f_n^{[\alpha ]},f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \end{aligned}$$

(3.21)

for $\alpha =1,2$.

3.2.1 Cubic formulas

Define $\kappa (\alpha _1;\alpha _2,\alpha _2)$ ($\alpha _1,\alpha _2\in \{1,2\}$) as follows.

$$\begin{aligned} \kappa (1;1,1)= & {} \frac{1}{\big (4-2^{2H}\big )^3} \sum _{i_1,i_2\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widehat{\rho }(i_1-i_2)\widehat{\rho }(i_2), \\ \kappa (2;1,1)= & {} \frac{2^{2H-1}}{\big (4-2^{2H}\big )^3} \sum _{i_1,i_2\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widetilde{\rho }(i_2-2i_1)\widetilde{\rho }(i_2), \\ \kappa (1;2,2)= & {} {\frac{2^{2H-2}}{\big (4-2^{2H}\big )^3}\sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(i_1)\widetilde{\rho }(i_1-i_2)\widetilde{\rho }(i_2)} \end{aligned}$$

and

$$\begin{aligned} \kappa (2;2,2)\>=\>\frac{1}{4}\kappa (1;1,1). \end{aligned}$$

Lemma 3.8

As n tends to $\infty$, the following estimates hold:

• (a) $\displaystyle \big \langle f_n^{[1]},n^{1/2}f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1;1,1)+o(1)$.

• (b) $\displaystyle \big \langle f_n^{[2]},n^{1/2}f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (2;1,1)+o(1)$.

• (c) $\displaystyle \big \langle f_n^{[1]},n^{1/2}f_n^{[2]}\otimes _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1;2,2)+o(1)$.

• (d) $\displaystyle \big \langle f_n^{[2]},n^{1/2}f_n^{[2]}\otimes _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (2;2,2)+o(1)$.

Proof

(a): By (3.3) and Lemma 3.5 (a), (b), we have

$$\begin{aligned} f_n^{[\alpha ]}\otimes _1f_n^{[\alpha ]}= & {} \frac{n\sum _{j=1}^{\alpha n-1}\sum _{j'=1}^{\alpha n-1} (1^{\alpha n}_{j+1}-1^{\alpha n}_j)\odot (1^{\alpha n}_{j'+1}-1^{\alpha n}_{j'}) \langle 1^{\alpha n}_{j+1}-1^{\alpha n}_j,1^{\alpha n}_{j'+1}-1^{\alpha n}_{j'}\rangle _{{\mathfrak {H}}}}{E[V^{(2)}_{\alpha n,T}]^2} \nonumber \\= & {} \frac{n}{E[V^{(2)}_{\alpha n,T}]^2}\frac{T^{2H}}{(\alpha n)^{2H}} \sum _{j=1}^{\alpha n-1}\sum _{j'=1}^{\alpha n-1} (1^{\alpha n}_{j+1}-1^{\alpha n}_j)\odot (1^{\alpha n}_{j'+1}-1^{\alpha n}_{j'}) \widehat{\rho }(j-j')\nonumber \\ \end{aligned}$$

(3.22)

for $\alpha =1,2$. Therefore, from (3.3), (3.22) and (2.1), we obtain

$$\begin{aligned}{} & {} \big \langle f_n^{[1]},n^{1/2}f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \\{} & {} \quad = \frac{n^2}{E[V^{(2)}_{n,T}]^3}\frac{T^{2H}}{n^{2H}} \sum _{j,j',j''=1}^{n-1} \langle 1^{n}_{j''+1}-1^{n}_{j''}, 1^{n}_{j+1}-1^{n}_j\rangle _{{\mathfrak {H}}}\langle 1^{n}_{j''+1}-1^{n}_{j''}, 1^{n}_{j'+1}-1^{n}_{j'}\rangle _{{\mathfrak {H}}}\widehat{\rho }(j-j')\\{} & {} \quad = \frac{n^{2}}{\big (4-2^{2H}\big )^3(n-1)^3} \sum _{j=1}^{n-1}\sum _{j'=1}^{n-1}\sum _{j''=1}^{n-1} \widehat{\rho }(j-j')\widehat{\rho }(j'-j'')\widehat{\rho }(j''-j)\\{} & {} \quad = \kappa (1;1,1)+o(1) \end{aligned}$$

thanks to Lemma 3.1.

(b): Similarly to the proof of (a),

$$\begin{aligned}{} & {} \big \langle f_n^{[2]},n^{1/2}f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \nonumber \\{} & {} \quad = \frac{n^{2}}{E[V^{(2)}_{n,T}]^2E[V^{(2)}_{2n,T}]} {\frac{T^{2H}}{n^{2H}}} \sum _{k=1}^{2n-1}\sum _{j=1}^{n-1}\sum _{j'=1}^{n-1} \langle 1^{2n}_{k+1}-1^{2n}_k,1^{n}_{j+1}-1^{n}_{j}\rangle _{{\mathfrak {H}}}\nonumber \\{} & {} \qquad \times \langle 1^{2n}_{k+1}-1^{2n}_k,1^{n}_{j'+1}-1^{n}_{j'}\rangle _{{\mathfrak {H}}}\widehat{\rho }(j-j') \nonumber \\{} & {} \quad = \frac{2^{2H}n^{2}}{\big (4-2^{2H}\big )^3 (n-1)^2(2n-1)} \sum _{k=1}^{2n-1}\sum _{j=1}^{n-1}\sum _{j'=1}^{n-1} \widehat{\rho }(j-j')\widetilde{\rho }(2j'-k)\widetilde{\rho }(k-2j) \nonumber \\{} & {} \quad = \frac{2^{2H-1}}{\big (4-2^{2H}\big )^3} \sum _{i\in {{\mathbb {Z}}}}\sum _{j\in {{\mathbb {Z}}}} \widehat{\rho }(i)\widetilde{\rho }(j)\widetilde{\rho }(j-2i)+o(1) \nonumber \\{} & {} \quad = \kappa (2;1,1)+o(1). \end{aligned}$$

(3.23)

Indeed,

$$\begin{aligned}{} & {} n^{-1}\sum _{k=1}^{2n-1}\sum _{j=1}^{n-1}\sum _{j'=1}^{n-1} \widehat{\rho }(j-j')\widetilde{\rho }(2j'-k)\widetilde{\rho }(k-2j)\\{} & {} \quad = n^{-1}\sum _{k=1}^{n-1}\sum _{j=1}^{n-1}\sum _{j'=1}^{n-1} \widehat{\rho }(j-j')\widetilde{\rho }(2j'-2k)\widetilde{\rho }(2k-2j)\\{} & {} \qquad + n^{-1}\sum _{k=1}^{n}\sum _{j=1}^{n-1}\sum _{j'=1}^{n-1} \widehat{\rho }(j-j')\widetilde{\rho }(2j'-2k+1)\widetilde{\rho }(2k-2j-1) \\{} & {} \quad = \sum _{i\in {{\mathbb {Z}}}}\sum _{j\in {{\mathbb {Z}}}} \widehat{\rho }(i)\widetilde{\rho }(2i-2j)\widetilde{\rho }(2j) + \sum _{i\in {{\mathbb {Z}}}}\sum _{j\in {{\mathbb {Z}}}} \widehat{\rho }(i)\widetilde{\rho }(2i-2j+1)\widetilde{\rho }(2j-1)+o(1)\\{} & {} \quad = \sum _{i\in {{\mathbb {Z}}}}\sum _{j\in {{\mathbb {Z}}}} \widehat{\rho }(i)\widetilde{\rho }(2i-j)\widetilde{\rho }(j)+o(1) \end{aligned}$$

by Lemma 3.1.

(c): By using (3.3), (3.22) and (2.1), we obtain

$$\begin{aligned}{} & {} \big \langle f_n^{[1]},n^{1/2}f_n^{[2]}\otimes _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}}\nonumber \\{} & {} \quad = \frac{n^2}{E[V^{(2)}_{n,T}]E[V^{(2)}_{2n,T}]^2}\frac{T^{2H}}{(2n)^{2H}} \nonumber \\{} & {} \qquad \times \sum _{j=1}^{n-1}\sum _{k,k'=1}^{2n-1} \langle 1^{n}_{j+1}-1^{n}_{j}, 1^{2n}_{k+1}-1^{2n}_k\rangle _{{\mathfrak {H}}}\langle 1^{n}_{j+1}-1^{n}_{j}, 1^{2n}_{k'+1}-1^{2n}_{k'}\rangle _{{\mathfrak {H}}}\widehat{\rho }(k-k') \nonumber \\{} & {} \quad = \frac{n^{2}}{\big (4-2^{2H}\big )^3(T/n)^{2H}(T/(2n))^{4H}(n-1)(2n-1)^2} \frac{T^{2H}}{(2n)^{2H}}\left( \frac{T^{2H}}{n^{2H}}\right) ^2 \nonumber \\{} & {} \qquad \times \sum _{j=1}^{n-1}\sum _{k,k'=1}^{2n-1} \widehat{\rho }(k-k')\widetilde{\rho }(k'-2j)\widetilde{\rho }(2j-k) \nonumber \\{} & {} \quad = \frac{2^{2H-2}}{\big (4-2^{2H}\big )^3n} \sum _{j=1}^{n-1}\sum _{k,k'=1}^{2n-1} \widehat{\rho }(k-k')\widetilde{\rho }(k'-2j)\widetilde{\rho }(2j-k)+o(1) \nonumber \\{} & {} \quad = \kappa (1;2,2)+o(1). \end{aligned}$$

(3.24)

In fact, thanks to Lemma 3.1,

$$\begin{aligned}{} & {} n^{-1}\sum _{j=1}^{n-1}\sum _{k,k'=1}^{2n-1} \widehat{\rho }(k-k')\widetilde{\rho }(k'-2j)\widetilde{\rho }(2j-k)\\{} & {} \quad = n^{-1}\sum _{j=1}^{n-1}\sum _{k=1}^{n-1}\sum _{k'=1}^{n-1} \widehat{\rho }(2k-2k')\widetilde{\rho }(2k'-2j)\widetilde{\rho }(2j-2k)\\{} & {} \qquad +n^{-1}\sum _{j=1}^{n-1}\sum _{k=1}^{n-1}\sum _{k'=1}^{n} \widehat{\rho }(2k-2k'+1)\widetilde{\rho }(2k'-2j-1)\widetilde{\rho }(2j-2k)\\{} & {} \qquad +n^{-1}\sum _{j=1}^{n-1}\sum _{k=1}^{n}\sum _{k'=1}^{n-1} \widehat{\rho }(2k-2k'-1)\widetilde{\rho }(2k'-2j)\widetilde{\rho }(2j-2k+1)\\{} & {} \qquad +n^{-1}\sum _{j=1}^{n-1}\sum _{k=1}^{n}\sum _{k'=1}^{n} \widehat{\rho }(2k-2k')\widetilde{\rho }(2k'-2j-1)\widetilde{\rho }(2j-2k+1) \end{aligned}$$

and the last expression is equal to

$$\begin{aligned}{} & {} \sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(2i_1)\widetilde{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2) +\sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(2i_1-1)\widetilde{\rho }((2i_1-1)-2i_2)\widetilde{\rho }(2i_2)\\{} & {} \qquad +\sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(2i_1+1)\widetilde{\rho }((2i_1+1)-2i_2-1)\widetilde{\rho }(2i_2+1) \\{} & {} \qquad +\sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(2i_1)\widetilde{\rho }(2i_1-2i_2-1)\widetilde{\rho }(2i_2+1) +o(1)\\{} & {} \quad = \sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(i_1)\widetilde{\rho }(i_1-2i_2)\widetilde{\rho }(2i_2) +\sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(i_1)\widetilde{\rho }(i_1-2i_2-1)\widetilde{\rho }(2i_2+1) +o(1)\\{} & {} \quad = \sum _{i_1,i_2\in {{\mathbb {Z}}}}\widehat{\rho }(i_1)\widetilde{\rho }(i_1-i_2)\widetilde{\rho }(i_2) +o(1). \end{aligned}$$

(d): The proof of (d) is similar to that of (a). $\square$

Lemma 3.9

• (a) $\displaystyle n^{1/2}\big \langle f_n^{[1]},f_n^{[2]}\odot _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (2;1,1)+o(1)$.

• (b) $\displaystyle n^{1/2}\big \langle f_n^{[2]},f_n^{[2]}\odot _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1;2,2)+o(1)$.

Proof

We have

$$\begin{aligned} \big \langle f_n^{[1]},f_n^{[2]}\odot _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}}&= \big \langle f_n^{[1]},f_n^{[2]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} = \big \langle f_n^{[2]},f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}}. \end{aligned}$$

The first equality stands since $f_n^{[1]}$ is symmetric (i.e. $\in \mathcal{H}^{\odot 2}$). Hence we have (a) by Lemma 3.8 (b). Similar arguments and Lemma 3.8 (c) prove (b). $\square$

3.2.2 Fourth power formulas

Recall that

$$\begin{aligned} f_n^{[1]}\otimes _1f_n^{[1]}= & {} \frac{n}{E[V^{(2)}_{n,T}]^2}\frac{T^{2H}}{n^{2H}} \sum _{j=1}^{n-1}\sum _{j'=1}^{n-1} (1^{n}_{j+1}-1^{n}_j)\odot (1^{n}_{j'+1}-1^{n}_{j'}) \widehat{\rho }(j-j'), \\ f_n^{[2]}\otimes _1f_n^{[1]}= & {} \frac{n}{E[V^{(2)}_{n,T}]E[V^{(2)}_{2n,T}]}\frac{T^{2H}}{n^{2H}} \sum _{j=1}^{n-1}\sum _{k=1}^{2n-1} (1_{j+1}^n-1_j^n)\otimes (1_{k+1}^{2n}-1_k^{2n}) \widetilde{\rho }(k,j) \end{aligned}$$

and

$$\begin{aligned} f_n^{[2]}\otimes _1f_n^{[2]}= & {} \frac{n}{E[V^{(2)}_{2n,T}]^2}\frac{T^{2H}}{(2n)^{2H}} \sum _{j=1}^{2n-1}\sum _{k=1}^{2n-1} (1_{k+1}^{2n}-1_k^{2n})\otimes (1_{k'+1}^{2n}-1_{k'}^{2n}) \widehat{\rho }(k-k'). \end{aligned}$$

Let

$$\begin{aligned} \kappa (1,1;1,1)= & {} \frac{1}{\big (4-2^{2H}\big )^4} \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widehat{\rho }(i_1-i_2)\widehat{\rho }(i_2-i_3)\widehat{\rho }(i_3), \nonumber \\ \kappa (2,2;2,2)= & {} \frac{1}{8}\kappa (1,1;1,1), \nonumber \\ \kappa (1,1;2,2)= & {} \frac{2^{2H-2}}{\big (4-2^{2H}\big )^4} \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widetilde{\rho }(2i_1-i_2)\widehat{\rho }(i_2-i_3)\widetilde{\rho }(i_3). \end{aligned}$$

(3.25)

Lemma 3.10

• (a) $\displaystyle n\big \langle f_n^{[1]}\otimes _1f_n^{[1]},\> f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1,1;1,1)+o(1)$.

• (b) $\displaystyle n\big \langle f_n^{[1]}\otimes _1f_n^{[1]},\> f_n^{[2]}\otimes _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1,1;2,2)+o(1)$.

• (c) $\displaystyle n\big \langle f_n^{[2]}\otimes _1f_n^{[2]},\> f_n^{[2]}\otimes _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (2,2;2,2)+o(1)$.

Proof

It follows from Lemma 3.1 that

$$\begin{aligned}{} & {} n\big \langle f_n^{[1]}\otimes _1f_n^{[1]},\> f_n^{[1]}\otimes _1f_n^{[1]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}}\\{} & {} \quad = \frac{n^3}{E[V^{(2)}_{n,T}]^4}\left( \frac{T^{2H}}{n^{2H}}\right) ^2 \\{} & {} \qquad \times \sum _{j_1,j_2,j_3,j_4=1}^{n-1} \big \langle (1^{n}_{j_1+1}-1^{n}_{j_1})\otimes (1^{n}_{j_4+1}-1^{n}_{j_4}),\> (1^{n}_{j_2+1}-1^{n}_{j_2})\otimes (1^{n}_{j_3+1}-1^{n}_{j_3})\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \\{} & {} \qquad \times \widehat{\rho }(j_1-j_4)\widehat{\rho }(j_2-j_3) \\{} & {} \quad = \frac{n^3}{\big (4-2^{2H}\big )^4(T/n)^{8H}(n-1)^4}\left( \frac{T^{2H}}{n^{2H}}\right) ^4\\{} & {} \qquad \times \sum _{j_1,j_2,j_3,j_4=1}^{n-1} \widehat{\rho }(j_1-j_2)\widehat{\rho }(j_2-j_3)\widehat{\rho }(j_3-j_4)\widehat{\rho }(j_4-j_1) \\{} & {} \quad = \kappa (1,1;1,1)+o(1). \end{aligned}$$

Thus, (a) has been obtained. Similarly, we can show (c).

Now we have

$$\begin{aligned}{} & {} n\big \langle f_n^{[1]}\otimes _1f_n^{[1]},\> f_n^{[2]}\otimes _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \nonumber \\{} & {} \quad = \frac{n^3}{E[V^{(2)}_{n,T}]^2E[V^{(2)}_{2n,T}]^2} \left( \frac{T^{2H}}{n^{2H}}\right) ^3\frac{T^{2H}}{(2n)^{2H}}\nonumber \\{} & {} \quad \sum _{j_1,j_2=1}^{n-1} \sum _{k_1,k_2=1}^{2n-1} \widetilde{\rho }(k_1-2j_1)\widetilde{\rho }(k_2-2j_2)\widehat{\rho }(j_1-j_2)\widehat{\rho }(k_1-k_2) \nonumber \\{} & {} \quad = \frac{2^{2H}n^3}{\big (4-2^{2H}\big )^4(n-1)^2(2n-1)^2}\nonumber \\{} & {} \quad \sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-k_2)\widehat{\rho }(k_2-k_1)\widetilde{\rho }(k_1-2j_1)\\{} & {} \quad = \kappa (1,1;2,2)+o(1) \nonumber \end{aligned}$$

(3.26)

by the following argument:

$$\begin{aligned}{} & {} n^{-1}\sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-k_2)\widehat{\rho }(k_2-k_1)\widetilde{\rho }(k_1-2j_1) \\{} & {} \quad = n^{-1}\sum _{j_1,j_2=1}^{n-1}\sum _{m_1,m_2=1}^{n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-2m_2)\widehat{\rho }(2m_2-2m_1)\widetilde{\rho }(2m_1-2j_1) \\{} & {} \qquad +n^{-1}\sum _{j_1,j_2=1}^{n-1}\sum _{m_1=1}^{n-1}\sum _{m_2=1}^{n} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-(2m_2-1))\widehat{\rho }((2m_2-1)-2m_1)\widetilde{\rho }(2m_1-2j_1) \\{} & {} \qquad +n^{-1}\sum _{j_1,j_2=1}^{n-1}\sum _{m_1=1}^{n}\sum _{m_2=1}^{n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-2m_2)\widehat{\rho }(2m_2-(2m_1-1))\widetilde{\rho }((2m_1-1)-2j_1) \\{} & {} \qquad +n^{-1}\sum _{j_1,j_2=1}^{n-1}\sum _{m_1,m_2=1}^{n} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-(2m_2-1))\widehat{\rho }((2m_2-1)-(2m_1-1))\\{} & {} \qquad \times \widetilde{\rho }((2m_1-1)-2j_1) \end{aligned}$$

and with Lemma 3.1, we see the gap between the above expression and the one below is of o(1):

$$\begin{aligned}&\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} {\widehat{\rho }(i_1)} \widetilde{\rho }(2i_1-2i_2)\widehat{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3) \\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} {\widehat{\rho }(i_1)} \widetilde{\rho }(2i_1-2i_2+1)\widehat{\rho }(2i_2-2i_3-1)\widetilde{\rho }(2i_3) \\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} {\widehat{\rho }(i_1)} \widetilde{\rho }(2i_1-2i_2)\widehat{\rho }(2i_2-2i_3+1)\widetilde{\rho }(2i_3-1)\\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} {\widehat{\rho }(i_1)} \widetilde{\rho }(2i_1-2i_2+1)\widehat{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3-1)+o(1) \\&\quad = \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} {\widehat{\rho }(i_1)} \widetilde{\rho }(2i_1-i_2)\widehat{\rho }(i_2-2i_3)\widetilde{\rho }(2i_3) \qquad (\text {combined the 1st and 2nd}) \\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} {\widehat{\rho }(i_1)} \widetilde{\rho }(2i_1-i_2)\widehat{\rho }(i_2-2i_3+1)\widetilde{\rho }(2i_3-1) \qquad (\text {combined the 3rd and 4th}) \\&\qquad +o(1) \\&\quad = {\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widetilde{\rho }(2i_1-i_2)\widehat{\rho }(i_2-i_3)\widetilde{\rho }(i_3) +o(1).} \end{aligned}$$

Therefore we obtained (b). $\square$

Let

$$\begin{aligned} \kappa (1,1;1,2)= & {} \frac{2^{2H-1}}{\big (4-2^{2H}\big )^4} \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widetilde{\rho }(2i_1-i_2)\widetilde{\rho }(i_2-2i_3)\widehat{\rho }(i_3), \\ \kappa (1,2;2,2)= & {} {\frac{2^{2H-3}}{(4-2^{2H})^4} \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widehat{\rho }(i_1-i_2)\widetilde{\rho }(i_2-i_3)\widetilde{\rho }(i_3)} \end{aligned}$$

and

$$\begin{aligned} \kappa (1,2;1,2)= & {} \frac{1}{2}\kappa (1,1;2,2)+\kappa (1,2;1,2)_2 \end{aligned}$$

where $\kappa (1,1;2,2)$ is defined at (3.25) and

$$\begin{aligned} \kappa (1,2;1,2)_2= & {} \frac{2^{4H-3}}{(4-2^{2H})^4} \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(i_1)\widetilde{\rho }(i_1-2i_2)\widetilde{\rho }(2i_2-i_3)\widetilde{\rho }(i_3). \end{aligned}$$

Lemma 3.11

• (a) $\displaystyle n\big \langle f_n^{[1]}\otimes _1f_n^{[1]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1,1;1,2)+o(1)$.

• (b) $\displaystyle n\big \langle f_n^{[2]}\otimes _1f_n^{[2]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1,2;2,2)+o(1)$.

• (c) $\displaystyle n\big \langle f_n^{[1]}\odot _1f_n^{[2]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \>=\>\kappa (1,2;1,2)+o(1)$.

Proof

(a): We have

$$\begin{aligned}{} & {} n\big \langle f_n^{[1]}\otimes _1f_n^{[1]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \\{} & {} \quad = \frac{n^3}{E[V^{(2)}_{n,T}]^3E[V^{(2)}_{2n,T}]}\left( \frac{T^{2H}}{n^{2H}}\right) ^4 \sum _{j_1,j_2,j_3=1}^{n-1}\sum _{k=1}^{2n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-k)\widetilde{\rho }(k-2j_3)\widehat{\rho }(j_3-j_1)\\{} & {} \quad = \frac{2^{2H}n^3}{\big (4-2^{2H}\big )^4(n-1)^3(2n-1)} \sum _{j_1,j_2,j_3=1}^{n-1}\sum _{k=1}^{2n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-k)\widetilde{\rho }(k-2j_3)\widehat{\rho }(j_3-j_1)\\{} & {} \quad = \kappa (1,1;1,2)+o(1). \end{aligned}$$

The last equality can be verified by

$$\begin{aligned}{} & {} n^{-1}\sum _{j_1,j_2,j_3=1}^{n-1}\sum _{k=1}^{2n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-k)\widetilde{\rho }(k-2j_3)\widehat{\rho }(j_3-j_1)\\{} & {} \quad = n^{-1}\sum _{j_1,j_2,j_3=1}^{n-1}\sum _{k=1}^{n-1} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-2k)\widetilde{\rho }(2k-2j_3)\widehat{\rho }(j_3-j_1)\\{} & {} \qquad +n^{-1}\sum _{j_1,j_2,j_3=1}^{n-1}\sum _{k=1}^{n} \widehat{\rho }(j_1-j_2)\widetilde{\rho }(2j_2-2k+1)\widetilde{\rho }(2k-1-2j_3)\widehat{\rho }(j_3-j_1)\\{} & {} \quad = \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widetilde{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2-2i_3)\widehat{\rho }(i_3)\\{} & {} \qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widetilde{\rho }(2i_1-2i_2+1)\widetilde{\rho }(2i_2-1-2i_3)\widehat{\rho }(i_3)+o(1) \\{} & {} \quad = \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widetilde{\rho }(2i_1-i_2)\widetilde{\rho }(i_2-2i_3)\widehat{\rho }(i_3)+o(1). \end{aligned}$$

(b): We will consider the product

$$\begin{aligned}{} & {} n\big \langle f_n^{[2]}\otimes _1f_n^{[2]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \\{} & {} \quad = \frac{n^3}{E[V^{(2)}_{n,T}]E[V^{(2)}_{2n,T}]^3} \left( \frac{T^{2H}}{(2n)^{2H}}\right) ^2\left( \frac{T^{2H}}{n^{2H}}\right) ^2\\{} & {} \qquad \times \sum _{k_1,k_2,k_3=1}^{2n-1}\sum _{j=1}^{n-1} \widehat{\rho }(k_1-k_2)\widehat{\rho }(k_2-k_3)\widetilde{\rho }(k_3-2j)\widetilde{\rho }(2j-k_1) \\{} & {} \quad = \frac{2^{2H}n^3}{(4-2^{2H})^4(n-1)(2n-1)^3} \sum _{k_1,k_2,k_3=1}^{2n-1}\sum _{j=1}^{n-1} \widehat{\rho }(k_1-k_2)\widehat{\rho }(k_2-k_3)\widetilde{\rho }(k_3-2j)\widetilde{\rho }(2j-k_1). \end{aligned}$$

Define $k^0$ and $k^1$ by

$$\begin{aligned} k^0=2k,\quad k^1=2k-1, \end{aligned}$$

respectively for an integer k. The sum $\widetilde{\sum }_{k_1,k_2,k_3}$ should read according to a proper configuration of $k^0_i$ and $k^1_i$, $i=1,2,3$. To proceed,

$$\begin{aligned}{} & {} n^{-1}\sum _{k_1,k_2,k_3=1}^{2n-1}\sum _{j=1}^{n-1} \widehat{\rho }(k_1-k_2)\widehat{\rho }(k_2-k_3)\widetilde{\rho }(k_3-2j)\widetilde{\rho }(2j-k_1) \\{} & {} \quad = n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^0-k_2^0)\widehat{\rho }(k_2^0-k_3^0)\widetilde{\rho }(k_3^0-2j)\widetilde{\rho }(2j-k_1^0) \\{} & {} \qquad +n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^0-k_2^0)\widehat{\rho }(k_2^0-k_3^1)\widetilde{\rho }(k_3^1-2j)\widetilde{\rho }(2j-k_1^0) \\{} & {} \qquad +n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^0-k_2^1)\widehat{\rho }(k_2^1-k_3^0)\widetilde{\rho }(k_3^0-2j)\widetilde{\rho }(2j-k_1^0)\\{} & {} \qquad +n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^0-k_2^1)\widehat{\rho }(k_2^1-k_3^1)\widetilde{\rho }(k_3^1-2j)\widetilde{\rho }(2j-k_1^0) \\{} & {} \qquad +n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^1-k_2^0)\widehat{\rho }(k_2^0-k_3^0)\widetilde{\rho }(k_3^0-2j)\widetilde{\rho }(2j-k_1^1) \\{} & {} \qquad +n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^1-k_2^0)\widehat{\rho }(k_2^0-k_3^1)\widetilde{\rho }(k_3^1-2j)\widetilde{\rho }(2j-k_1^1) \\{} & {} \qquad +n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^1-k_2^1)\widehat{\rho }(k_2^1-k_3^0)\widetilde{\rho }(k_3^0-2j)\widetilde{\rho }(2j-k_1^1) \\{} & {} \qquad +n^{-1}\widetilde{\sum }_{k_1,k_2,k_3}\sum _{j=1}^{n-1} \widehat{\rho }(k_1^1-k_2^1)\widehat{\rho }(k_2^1-k_3^1)\widetilde{\rho }(k_3^1-2j)\widetilde{\rho }(2j-k_1^1) \end{aligned}$$

and by Lemma 3.1, the above sum can be written as follows:

$$\begin{aligned}{} & {} \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1)\widehat{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3) \\{} & {} \quad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1)\widehat{\rho }(2i_1-2i_2+1)\widetilde{\rho }(2i_2-2i_3-1)\widetilde{\rho }(2i_3) \\{} & {} \quad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1-1)\widehat{\rho }(2i_1-2i_2-1)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3) \\{} & {} \quad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1-1)\widehat{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2-2i_3-1)\widetilde{\rho }(2i_3) \\{} & {} \quad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1+1)\widehat{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3+1) \\{} & {} \quad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1+1)\widehat{\rho }(2i_1-2i_2+1)\widetilde{\rho }(2i_2-2i_3-1)\widetilde{\rho }(2i_3+1) \\{} & {} \quad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1)\widehat{\rho }(2i_1-2i_2-1)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3+1) \\{} & {} \quad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(2i_1)\widehat{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2-2i_3-1)\widetilde{\rho }(2i_3+1) +o(1) \end{aligned}$$

$$\begin{aligned}{} & {} = \sum _{k_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(k_1)\widehat{\rho }(k_1-2i_2)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3) \qquad (\text {combined the 1st and 3rd}) \\{} & {} \quad +\sum _{k_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(k_1)\widehat{\rho }(k_1-2i_2+1)\widetilde{\rho }(2i_2-2i_3-1)\widetilde{\rho }(2i_3) \qquad (\text {combined the 2nd and 4th}) \\{} & {} \quad +\sum _{k_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(k_1)\widehat{\rho }(k_1-2i_2-1)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3+1) \qquad (\text {combined the 5th and 7th}) \\{} & {} \quad +\sum _{k_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(k_1)\widehat{\rho }(k_1-2i_2)\widetilde{\rho }(2i_2-2i_3-1)\widetilde{\rho }(2i_3+1) \qquad (\text {combined the 6th and 8th}) \\{} & {} \quad +o(1) \\{} & {} = \sum _{k_1,k_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(k_1)\widehat{\rho }(k_1-k_2)\widetilde{\rho }(k_2-2i_3)\widetilde{\rho }(2i_3) \qquad (\text {combined the 1st and 2nd}) \\{} & {} \quad +\sum _{k_1,k_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(k_1)\widehat{\rho }(k_1-k_2)\widetilde{\rho }(k_2-2i_3-1)\widetilde{\rho }(2i_3+1) \qquad (\text {combined the 3rd and 4th}) \\{} & {} \quad +o(1) \\{} & {} = \sum _{k_1,k_2,k_3\in {{\mathbb {Z}}}} \widehat{\rho }(k_1)\widehat{\rho }(k_1-k_2)\widetilde{\rho }(k_2-k_3)\widetilde{\rho }(k_3) +o(1). \end{aligned}$$

Therefore

$$\begin{aligned}{} & {} n\big \langle f_n^{[2]}\otimes _1f_n^{[2]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}}\\{} & {} \quad = {\frac{2^{2H-3}}{(4-2^{2H})^4} \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}} \widehat{\rho }(i_1)\widehat{\rho }(i_1-i_2)\widetilde{\rho }(i_2-i_3)\widetilde{\rho }(i_3)}+o(1)\\{} & {} \quad = \kappa (1,2;2,2)+o(1). \end{aligned}$$

(c): For the product $n\big \langle f_n^{[1]}\odot _1f_n^{[2]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}}$, we have

$$\begin{aligned}{} & {} n\big \langle f_n^{[1]}\odot _1f_n^{[2]},\> f_n^{[1]}\odot _1f_n^{[2]}\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \\{} & {} \quad = \frac{n^3}{E[V^{(2)}_{n,T}]^2E[V^{(2)}_{2n,T}]^2}\frac{T^{4H}}{n^{4H}} \\{} & {} \qquad \times \sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1} \big \langle (1_{j_1+1}^n-1_{j_1}^n)\odot (1_{k_1+1}^{2n}-1_{k_1}^{2n}),\> (1_{j_2+1}^n-1_{j_2}^n)\odot (1_{k_2+1}^{2n}-1_{k_2}^{2n})\big \rangle _{{{\mathfrak {H}}}^{\otimes 2}} \\{} & {} \qquad \times \widetilde{\rho }(k_1,j_1)\widetilde{\rho }(k_2,j_2) \\{} & {} \quad = \frac{n^3}{E[V^{(2)}_{n,T}]^2E[V^{(2)}_{2n,T}]^2}\frac{T^{4H}}{n^{4H}} \sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1}\frac{1}{2}\widetilde{\rho }(k_1,j_1)\widetilde{\rho }(k_2,j_2) \\{} & {} \qquad \times \bigg \{ \big \langle 1_{j_1+1}^n-1_{j_1}^n,\>1_{j_2+1}^n-1_{j_2}^n\big \rangle _{{{\mathfrak {H}}}} \big \langle 1_{k_1+1}^{2n}-1_{k_1}^{2n},\>1_{k_2+1}^{2n}-1_{k_2}^{2n}\big \rangle _{{{\mathfrak {H}}}} \\{} & {} \qquad + \big \langle 1_{j_2+1}^n-1_{j_2}^n,\>1_{k_1+1}^{2n}-1_{k_1}^{2n}\big \rangle _{{{\mathfrak {H}}}} \big \langle 1_{j_1+1}^n-1_{j_1}^n,\>1_{k_2+1}^{2n}-1_{k_2}^{2n}\big \rangle _{{{\mathfrak {H}}}} \bigg \} \\{} & {} \quad = \frac{n^3}{E[V^{(2)}_{n,T}]^2E[V^{(2)}_{2n,T}]^2}\frac{T^{4H}}{n^{4H}} \sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1}\frac{1}{2}\widetilde{\rho }(k_1-2j_1)\widetilde{\rho }(k_2-2j_2)\\{} & {} \qquad \times \bigg \{ \frac{T^{2H}}{n^{2H}}\frac{T^{2H}}{(2n)^{2H}}\widehat{\rho }(j_1-j_2)\widehat{\rho }(k_1-k_2) +\frac{T^{2H}}{n^{2H}}\frac{T^{2H}}{n^{2H}}\widetilde{\rho }(k_2-2j_1)\widetilde{\rho }(k_1-2j_2) \bigg \}\\{} & {} \quad = \frac{2^{2H}n^3}{(4-2^{2H})^4(n-1)^2(2n-1)^2}\frac{1}{2}\\{} & {} \qquad \times \sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1}\widetilde{\rho }(k_1-2j_1)\widetilde{\rho }(k_2-2j_2) \widehat{\rho }(j_1-j_2)\widehat{\rho }(k_1-k_2) \\{} & {} \qquad +\frac{2^{4H}n^3}{(4-2^{2H})^4(n-1)^2(2n-1)^2}\frac{1}{2}\\{} & {} \qquad \times \sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1}\widetilde{\rho }(k_1-2j_1)\widetilde{\rho }(k_2-2j_2) \widetilde{\rho }(k_2-2j_1)\widetilde{\rho }(k_1-2j_2). \end{aligned}$$

Now the convergence in (c) is verified as follows. The first sum is the half of the sum (3.26). Hence it converges to $\frac{1}{2}\kappa (1,1;2,2)$. For the second sum,

$$\begin{aligned}&n^{-1}\sum _{j_1,j_2=1}^{n-1}\sum _{k_1,k_2=1}^{2n-1} \widetilde{\rho }(2j_1-k_2)\widetilde{\rho }(k_2-2j_2)\widetilde{\rho }(2j_2-k_1)\widetilde{\rho }(k_1-2j_1)\\&\quad = n^{-1}\widetilde{\sum }_{j_1,j_2,k_1,k_2} \widetilde{\rho }(2j_1-k_2^0)\widetilde{\rho }(k_2^0-2j_2)\widetilde{\rho }(2j_2-k_1^0)\widetilde{\rho }(k_1^0-2j_1) \\&\qquad +n^{-1}\widetilde{\sum }_{j_1,j_2,k_1,k_2} \widetilde{\rho }(2j_1-k_2^1)\widetilde{\rho }(k_2^1-2j_2)\widetilde{\rho }(2j_2-k_1^0)\widetilde{\rho }(k_1^0-2j_1) \\&\qquad +n^{-1}\widetilde{\sum }_{j_1,j_2,k_1,k_2} \widetilde{\rho }(2j_1-k_2^0)\widetilde{\rho }(k_2^0-2j_2)\widetilde{\rho }(2j_2-k_1^1)\widetilde{\rho }(k_1^1-2j_1) \\&\qquad +n^{-1}\widetilde{\sum }_{j_1,j_2,k_1,k_2} \widetilde{\rho }(2j_1-k_2^1)\widetilde{\rho }(k_2^1-2j_2)\widetilde{\rho }(2j_2-k_1^1)\widetilde{\rho }(k_1^1-2j_1) \\&\quad = \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(2i_1)\widetilde{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3) \\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(2i_1-1)\widetilde{\rho }(2i_1-2i_2{-}1)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3) \\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(2i_1)\widetilde{\rho }(2i_1-2i_2)\widetilde{\rho }(2i_2-2i_3+1)\widetilde{\rho }(2i_3-1) \\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(2i_1-1)\widetilde{\rho }(2i_1-2i_2{-}1)\widetilde{\rho }(2i_2-2i_3{+}1)\widetilde{\rho }(2i_3{-}1) +o(1)\\&\quad = \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(i_1)\widetilde{\rho }(i_1-2i_2)\widetilde{\rho }(2i_2-2i_3)\widetilde{\rho }(2i_3) \qquad (\text {combined the 1st and 2nd}) \\&\qquad +\sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(i_1)\widetilde{\rho }(i_1-2i_2)\widetilde{\rho }(2i_2-2i_3+1)\widetilde{\rho }(2i_3{-}1) \qquad (\text {combined the 3rd and 4th}) \\&\qquad +o(1)\\&\quad = \sum _{i_1,i_2,i_3\in {{\mathbb {Z}}}}\widetilde{\rho }(i_1)\widetilde{\rho }(i_1-2i_2)\widetilde{\rho }(2i_2-i_3)\widetilde{\rho }(i_3) +o(1). \end{aligned}$$

$\square$

3.2.3 Central limit theorems

We are now on the point of getting a central limit theorem for $U_n$. Recall the notation

$$\begin{aligned} U_n= & {} \big (M_n^{[1]},M_n^{[2]},n^{1/2}\widetilde{G}_n\big ). \end{aligned}$$

Denote by $N_k(0,C)$ the k-dimensional centered normal distribution with covariance matrix C.

Proposition 3.12

The random vector $U_n$ is asymptotically normal, that is,

$$\begin{aligned} U_n&\rightarrow ^d&N_3(0,{{\mathfrak {U}}}) \end{aligned}$$

as $n\rightarrow \infty$, where ${{\mathfrak {U}}}=({{\mathfrak {U}}}_{ij})_{i,j=1}^3$ is a symmetric matrix with components

$$\begin{aligned} {{\mathfrak {U}}}_{11}= & {} \Sigma _{11},\quad {{\mathfrak {U}}}_{12}\>=\>\Sigma _{12},\quad {{\mathfrak {U}}}_{22}\>=\>\Sigma _{22},\\ {{\mathfrak {U}}}_{13}= & {} { 4}\kappa (1;2,2)-{ 8}\kappa (2;1,1)+{ 4}\kappa (1;1,1),\\ {{\mathfrak {U}}}_{23}= & {} { 4}\kappa (2;2,2)-{ 8}\kappa (1;2,2)+{ 4}\kappa (2;1,1),\\ {{\mathfrak {U}}}_{33}= & {} { 8}\kappa (2,2;2,2)+{ 32}\kappa (1,2;1,2)+{ 8}\kappa (1,1;1,1)\\{} & {} -{ 32}\kappa (1,2;2,2)+{ 16}\kappa (1,1;2,2)-{ 32}\kappa (1,1;1,2). \end{aligned}$$

Proof

The Cramér-Wold device is used to show the three-dimensional central limit theorem. It follows from Lemmas 3.8, 3.9, 3.10 and 3.11 with the representations (3.21) and (3.19) that the asymptotic covariance matrix of $v\cdot U_n$ is $^tv{{\mathfrak {U}}}v$ for every $v\in {{\mathbb {R}}}^3$. To apply the fourth moment theorem (Nualart and Peccati 2005), what we need to show is $\text {Var}\big [|Dv\cdot U_n|_{{\mathfrak {H}}}^2\big ]\rightarrow 0$ as $n\rightarrow \infty$. We will demonstrate $\text {Var}[|DF_n|_{{\mathfrak {H}}}^2]\rightarrow 0$ only for the component $F_n=2^{-1}n^{1/2}I_2\big (f_n^{[1]}\otimes _1f_n^{[1]}\big )$, because the proof of the convergence of the other components is quite similar. For example, we consider a component $F_n=2^{-1}n^{1/2}I_2\big (f_n^{[1]}\otimes _1f_n^{[1]}\big )$. By definition of $F_n$, we have

$$\begin{aligned} DF_n= & {} n^{1/2}I_1\big (f_n^{[1]}\otimes _1f_n^{[1]}\big ) \\ {}= & {} \frac{n^{1.5}}{E[V^{(2)}_{n,T}]^2}\frac{T^{2H}}{n^{2H}} \sum _{j=1}^{n-1}\sum _{j'=1}^{n-1} I_1\big (1^{n}_{j+1}-1^{n}_j\big )(1^{n}_{j'+1}-1^{n}_{j'}) \widehat{\rho }(j-j') \end{aligned}$$

and

$$\begin{aligned} |DF_n |_{{\mathfrak {H}}}^2= & {} {\frac{n^3}{(4-2^{2H})^4(T/n)^{8H}(n-1)^4}\left( \frac{T^{2H}}{n^{2H}}\right) ^3} \\ {}{} & {} \times \sum _{j_1,j_2,j_3,j_4=1}^{n-1} I_1\big (1_{j_1+1}^n-1_{j_1}^n\big )I_1\big (1_{j_4+1}^n-1_{j_4}^n\big ) \widehat{\rho }(j_2-j_3) \widehat{\rho }(j_1-j_2)\widehat{\rho }(j_3-j_4) \\ {}= & {} {\frac{n^3}{(4-2^{2H})^4(T/n)^{8H}(n-1)^4}\left( \frac{T^{2H}}{n^{2H}}\right) ^3} \\ {}{} & {} \times \sum _{j_1,j_2,j_3,j_4=1}^{n-1} I_2\big ((1_{j_1+1}^n-1_{j_1}^n)\odot (1_{j_4+1}^n-1_{j_4}^n)\big ) \widehat{\rho }(j_2-j_3)\widehat{\rho }(j_1-j_2)\widehat{\rho }(j_3-j_4) \\ {}{} & {} +E[|DF_n |_{{\mathfrak {H}}}^2]. \end{aligned}$$

Therefore

$$\begin{aligned}{} & {} \text {Var}\big [|DF_n |_{{\mathfrak {H}}}^2\big ] \\{} & {} \quad = {\frac{n^6}{(4-2^{2H})^8(T/n)^{16H}(n-1)^8}\left( \frac{T^{2H}}{n^{2H}}\right) ^6} \sum _{j_1,...,j_8=1}^{n-1} \\{} & {} \qquad \times E\bigg [ I_2\big ((1_{j_1+1}^n-1_{j_1}^n)\odot (1_{j_4+1}^n-1_{j_4}^n)\big ) I_2\big ((1_{j_8+1}^n-1_{j_8}^n)\odot (1_{j_5+1}^n-1_{j_5}^n)\big )\bigg ] \\{} & {} \qquad \times \widehat{\rho }(j_2-j_3)\widehat{\rho }(j_1-j_2)\widehat{\rho }(j_3-j_4) \widehat{\rho }(j_6-j_7)\widehat{\rho }(j_5-j_6)\widehat{\rho }(j_7-j_8) \\{} & {} \quad = {\frac{n^6}{(4-2^{2H})^8(T/n)^{16H}(n-1)^8}\left( \frac{T^{2H}}{n^{2H}}\right) ^8} \\{} & {} \qquad \times \sum _{j_1,...,j_8=1}^{n-1} \big [\widehat{\rho }(j_1-j_8)\widehat{\rho }(j_4-j_5)+\widehat{\rho }(j_4-j_8)\widehat{\rho }(j_1-j_5) \big ]\\{} & {} \qquad \times \widehat{\rho }(j_2-j_3)\widehat{\rho }(j_1-j_2)\widehat{\rho }(j_3-j_4) \widehat{\rho }(j_6-j_7)\widehat{\rho }(j_5-j_6)\widehat{\rho }(j_7-j_8)\\{} & {} \quad = {\frac{2n^6}{(4-2^{2H})^8(n-1)^8}} \sum _{j_1,...,j_8=1}^{n-1} \widehat{\rho }(j_1-j_2)\widehat{\rho }(j_2-j_3)\widehat{\rho }(j_3-j_4)\widehat{\rho }(j_4-j_5) \\{} & {} \qquad \times \widehat{\rho }(j_5-j_6)\widehat{\rho }(j_6-j_7)\widehat{\rho }(j_7-j_8)\widehat{\rho }(j_8-j_1). \end{aligned}$$

We have ${{\mathbb {V}}}\big (\overbrace{\widehat{\rho },...,\widehat{\rho }}^{k}\big ) < \infty$ since $4-2H>2>\frac{k-1}{k}$ for every $k\ge 2$. Therefore

$$\begin{aligned} \text {Var}\big [|DF_n |_{{\mathfrak {H}}}^2\big ]= & {} O(n^{-1}) \end{aligned}$$

by Lemma 3.1. $\square$

A central limit theorem for $T_n\>=\>\big (M_n,n^{1/2}\widetilde{G}_n\big )$ follows immediately from Proposition 3.12 since $M_n=M_n^{[1]}-M_n^{[2]}$.

Corollary 3.13

$T_n \rightarrow ^d N_2(0,{{\mathfrak {T}}})$ as $n\rightarrow \infty$, where ${{\mathfrak {T}}}=({{\mathfrak {T}}}_{ij})_{i,j=1}^2$ is a symmetric matrix with components

$$\begin{aligned} {{\mathfrak {T}}}_{11}= & {} {{\mathfrak {U}}}_{11}-2 {{\mathfrak {U}}}_{12}+ {{\mathfrak {U}}}_{22},\quad {{{\mathfrak {T}}}_{12}\>=\>{{\mathfrak {U}}}_{13}-{{\mathfrak {U}}}_{23},}\quad {{\mathfrak {T}}}_{22}\>=\>{{\mathfrak {U}}}_{33}. \end{aligned}$$

We remark that

$$\begin{aligned} G_\infty \>=\>{{\mathfrak {T}}}_{11} \>=\>\Sigma _{22}-2\Sigma _{12}+\Sigma _{11} \>=\>\frac{3}{2}\Sigma _{11}-2\Sigma _{12} \>\ge \>\bigg (\frac{3}{2}-\sqrt{2}\bigg )\Sigma _{11} \>>\>0,\nonumber \\ \end{aligned}$$

(3.27)

where $G_\infty$ was defined by (3.17). Indeed, from Proposiion 3.12, $(M_n^{[1]},M_n^{[2]})\rightarrow ^d(M_\infty ^{[1]},M_\infty ^{[2]})$, a two-dimensional Gaussian random vector with covariance matrix $(\Sigma _{ij})_{i,j=1,2}$.

We applied the Schwarz inequality to obtain (3.27).

3.3 Proof of Proposition 2.1

Let $\psi :{{\mathbb {R}}}\rightarrow [0,1]$ be a smooth function satisfying $\psi (x)=1$ on $\{x;|x|\le 1/2\}$, and $\psi (x)=0$ on $\{x;|x|\ge 1\}$. We define $\psi _n$ by $\psi _1=0$ and

$$\begin{aligned} \psi _n= & {} \psi \bigg (\frac{4|V_{n,T}^{(2)}-E[V_{n,T}^{(2)}]|^2}{ {\eta _0^2}E[V_{n,T}^{(2)}]^2}\bigg ) \psi \bigg (\frac{4|V_{2n,T}^{(2)}-E[V_{2n,T}^{(2)}]|^2}{ {\eta _0^2}E[V_{2n,T}^{(2)}]^2}\bigg ) \end{aligned}$$

for $n\ge 2$. Whenever $\psi _n>0$,

$$\begin{aligned} \bigg |\frac{V^{(2)}_{n,T}}{E[V^{(2)}_{n,T}]}-1\bigg |<\frac{1}{2}\eta _0&\text {and}&\bigg |\frac{V^{(2)}_{2n,T}}{E[V^{(2)}_{2n,T}]}-1\bigg |<\frac{1}{2}\eta _0, \end{aligned}$$

(3.28)

where $\eta _0$ is a positive constant. The properties of the truncation functional $\psi _n$ in (i) of Proposition 2.1 are easy to verified. We choose a sufficiently small $\eta _0$ and a positive integer $n_0$ both depending on H so that $\widehat{H}^{(2)}_n\in (0,1)$ whenever $\psi _n>0$ and $n\ge n_0$. We will only consider $n\ge n_0$ in what follows. On the event $\{\psi _n>0\}$, we have

$$\begin{aligned} \widehat{H}^{(2)}_n= & {} H -\frac{1}{2\log 2}\log \frac{V^{(2)}_{2n,T}/E[V^{(2)}_{2n,T}]}{V^{(2)}_{n,T}/E[V^{(2)}_{n,T} ]}-\frac{1}{4n\log 2}+O(n^{-2}) \\ {}= & {} H {-} \frac{1}{2\log 2}\bigg \{\bigg [\frac{V^{(2)}_{2n,T}}{E[V^{(2)}_{2n,T}]}-1\bigg ] -\bigg [\frac{V^{(2)}_{n,T}}{E[V^{(2)}_{n,T}]}-1\bigg ] \\ {}{} & {} \quad {-}\frac{1}{2}\bigg [\frac{V^{(2)}_{2n,T}}{E[V^{(2)}_{2n,T}]}-1\bigg ]^2 {+}\frac{1}{2}\bigg [\frac{V^{(2)}_{n,T}}{E[V^{(2)}_{n,T}]}-1\bigg ]^2 +R_n \bigg \} -\frac{1}{4n\log 2}+O(n^{-2}) \end{aligned}$$

where

$$\begin{aligned} R_n= & {} \bigg [\frac{V^{(2)}_{2n,T}}{E[V^{(2)}_{2n,T}]}-1\bigg ]^3 \int _0^1(1-s)^2\bigg (1+ s\bigg [\frac{V^{(2)}_{2n,T}}{E[V^{(2)}_{2n,T}]}-1\bigg ]\bigg )^{-3}ds \\ {}{} & {} -\bigg [\frac{V^{(2)}_{n,T}}{E[V^{(2)}_{n,T}]}-1\bigg ]^3 \int _0^1(1-s)^2\bigg (1+s\bigg [\frac{V^{(2)}_{n,T}}{E[V^{(2)}_{n,T}]}-1\bigg ]\bigg )^{-3}ds. \end{aligned}$$

Since the family

$$\begin{aligned} {{\mathfrak {F}}}=\big \{n^{1/2}\big (V_{n,T}^{(2)}/E[V_{n,T}^{(2)}]-1\big )\big \}_{n\in {{\mathbb {N}}}} \end{aligned}$$

(3.29)

of Wiener functionals is bounded in ${{\mathbb {D}}}_\infty$, as is known by the hypercontractivity and stability of the ${{\mathfrak {F}}}$ under the Malliavin operator (i.e., $-2^{-1}L$ is the identity on the second chaos), we see $R_n\psi _n = O_M(n^{-1.5})$ with the help of (3.28). $\square$

3.4 Asymptotic expansion of $M_n$

We will apply Theorem 3 of Tudor and Yoshida (2019a) to the sequence $(M_n)_{n\in {{\mathbb {N}}}}$ by checking Conditions [C1]-[C3] therein. Condition [C1] is satisfied by Corollary 3.13. Conditions [C2] and [C3] for boundedness of $\{M_n\}_{n\in {{\mathbb {N}}}}$ and $\{n^{1/2}(G_n-G_\infty )\}_{n\in {{\mathbb {N}}}}$ are verified with the hypercontractivity and stability of a fixed chaos under the Malliavin operator, as mentioned in the last part of the proof of Proposition 2.1. Remark that $n^{1/2}(G_n-G_\infty )=n^{1/2}\widetilde{G}_n+O(n^{-1/2})$; $n^{1/2}\widetilde{G}_n$ is in the second chaos and the error term is deterministic.

According to Corollary 3.13, $T_n\rightarrow ^dT_\infty$ as $n\rightarrow \infty$, where $T_\infty =(T_{\infty ,1},T_{\infty ,2})$ is a centered two-dimensional Gaussian random variable defined on some probability space with variance matrix ${{\mathfrak {T}}}$. Then

$$\begin{aligned} {{\mathbb {E}}}[T_{\infty ,2}|T_{\infty ,1}]= & {} \theta T_{\infty ,1}, \end{aligned}$$

where $\theta ={{\mathfrak {T}}}_{12}/{{\mathfrak {T}}}_{11}$. Suggested by Formula (23) of Tudor and Yoshida (2019a), we define a symbol ${{\mathfrak {S}}}_n(\texttt{i}z)$ by

$$\begin{aligned} {{\mathfrak {S}}}_n(\texttt{i}z)= & {} 1+ \frac{1}{3}\theta {{\mathfrak {T}}}_{11}n^{-1/2}(\texttt{i}z)^3. \end{aligned}$$

We note that introducing the parameter $\theta$ makes the resulting formula look slightly simple, that involves many ${{\mathfrak {T}}}_{11}^{-1}$ or $G_\infty ^{-1}$ by derivatives. An approximate density $p_n^M(z)$ is then defined by

$$\begin{aligned} p_n^M(z) \>=\>{{\mathfrak {S}}}_n(-\partial _z) \phi \big (z;0,{{\mathfrak {T}}}_{11}\big ) \>=\>{{\mathfrak {S}}}_n(\partial _z)^* \phi \big (z;0,{{\mathfrak {T}}}_{11}\big ). \end{aligned}$$

In other words,

$$\begin{aligned} p_n^M(z)= & {} \phi (z;0,G_\infty )+ \frac{1}{3}n^{-1/2}\theta G_\infty \bigg (\frac{z^3}{G_\infty ^3}-\frac{3z}{G_\infty ^2}\bigg ) \phi (z;0,G_\infty ). \end{aligned}$$

(3.30)

The function $p_n^M(z)$ is the Fourier inversion of the function

$$\begin{aligned} \varphi _n(u)= & {} e^{-\frac{1}{2}{{\mathfrak {T}}}_{11}u^2}+n^{-1/2}\frac{1}{3}\theta {{\mathfrak {T}}}_{11}(\texttt{i}u)^3e^{-\frac{1}{2}{{\mathfrak {T}}}_{11}u^2}. \end{aligned}$$

Recall that $\mathcal{E}(K,\gamma )$ is the set of measurable functions on ${{\mathbb {R}}}$ such that $|f(z)|\le K(1+|z|^\gamma )$ for all $z\in {{\mathbb {R}}}$. Theorem 3 of Tudor and Yoshida (2019a) gives the following result.

Theorem 3.14

For any positive numbers K and $\gamma$,

$$\begin{aligned} \sup _{f\in \mathcal{E}(K,\gamma )} \bigg |E[f(M_n)]-\int _{{\mathbb {R}}}f(z)p_n^M(z)dz\bigg |= & {} o(n^{-1/2}) \end{aligned}$$

as $n\rightarrow \infty$.

3.5 Proof of Theorem 2.2

In this section, we will combine the asymptotic expansion of $M_n$ with the perturbation method to give asymptotic expansion for $Z_n$. The perturbation method was used in Yoshida (1997), Yoshida (2001), Yoshida (2013). For this method, we refer the reader to Sakamoto and Yoshida (2003) and Yoshida (2020). Following the formulation in Sakamoto and Yoshida (2003), it is possible to derive asymptotic expansion of ${{\mathbb {S}}}_n={{\mathbb {X}}}_n+r_n{{\mathbb {Y}}}_n$ from the joint convergence of $({{\mathbb {X}}}_n,{{\mathbb {Y}}}_n)$ once that of ${{\mathbb {X}}}_n$ was obtained. We set ${{\mathbb {X}}}_n=M_n$, ${{\mathbb {Y}}}_n=N_n$, ${{\mathbb {S}}}_n=Z_n$ and $r_n=n^{-1/2}$. Let

$$\begin{aligned} \xi _n= & {} \frac{4|G_n-G_\infty |^2}{G_\infty ^2}. \end{aligned}$$

Then,

$$\begin{aligned} |DM_n|_{{\mathfrak {H}}}^2 \>=\>2G_n > G_\infty \end{aligned}$$

whenever $|\xi _n|<1$. Therefore $\sup _nE\big [1_{\{|\xi _n|<1\}}|DM_n|_{{\mathfrak {H}}}^{-p}\big ]<\infty$ for any $p>1$. Moreover, $P\big [|\xi _n|>1/2\big ]=O(n^{-L})$ for every $L>0$ since $\widetilde{G}_n$ is bounded in $L^{\infty \text {--}}=\cap _{p>1}L_p$. It is easy to see $\xi _n$, as well as $M_n$ and $N_n$, is bounded in ${{\mathbb {D}}}_\infty$. On the other hand, since

$$\begin{aligned} \big (M_n^{[1]},M_n^{[2]}\big )&\rightarrow ^d&\big (M_\infty ^{[1]},M_\infty ^{[2]}\big ) \quad (n\rightarrow \infty ) \end{aligned}$$

from Proposition 3.12, we obtain

$$\begin{aligned}{} & {} (M_n,N_n)= \bigg (-M_n^{[2]}+M_n^{[1]},\>\frac{1}{2}\big \{ (M_n^{[2]})^2-(M_n^{[1]})^2\big \} -\frac{1}{2}\bigg )+o_p(1) \nonumber \\{} & {} \quad \rightarrow ^d \bigg (-M_\infty ^{[2]}+M_\infty ^{[1]},\>\frac{1}{2}\big \{ (M_\infty ^{[2]})^2-(M_\infty ^{[1]})^2\big \} -\frac{1}{2}\bigg ). \end{aligned}$$

(3.31)

Therefore

$$\begin{aligned} g_\infty (z)= & {} -\partial _z\bigg \{{{\mathbb {E}}}\bigg [\frac{1}{2}\big \{ (M_\infty ^{[2]})^2-(M_\infty ^{[1]})^2\big \} -\frac{1}{2}\bigg |-M_\infty ^{[2]}+M_\infty ^{[1]}=z\bigg ]\phi (z;0,G_\infty )\bigg \} \nonumber \\= & {} -\frac{1}{2}\partial _z\bigg \{{{\mathbb {E}}}\bigg [-z\big \{ M_\infty ^{[2]}+M_\infty ^{[1]}\big \} -1\bigg |-M_\infty ^{[2]}+M_\infty ^{[1]}=z\bigg ]\phi (z;0,G_\infty )\bigg \} \nonumber \\= & {} -\frac{1}{2}\partial _z\big \{\big [\tau z^2-1\big ]\phi (z;0,G_\infty )\big \} \nonumber \\= & {} \bigg (\frac{\tau }{2G_\infty }z^3-(\tau +\frac{1}{2G_\infty })z\bigg ) \phi (z;0,G_\infty ) \end{aligned}$$

(3.32)

with the coefficient

$$\begin{aligned} \tau= & {} \frac{\Sigma _{22}-\Sigma _{11}}{G_\infty } \>=\>\frac{{{\mathfrak {U}}}_{22}-{{\mathfrak {U}}}_{11}}{G_\infty }. \end{aligned}$$

From (3.30) and (3.32), we obtain the expansion formula for $Z_n$ as

$$\begin{aligned} p_n^Z(z)= & {} \big (1+n^{-1/2}q^Z(z)\big )\phi (z;0,G_\infty ), \end{aligned}$$

(3.33)

where

$$\begin{aligned} q^Z(z)= & {} \bigg (\frac{\theta }{3G_\infty ^2}+\frac{\tau }{2G_\infty }\bigg )z^3 -\bigg (\frac{2\theta +1}{2G_\infty }+\tau \bigg )z; \end{aligned}$$

(3.34)

recall $\theta ={{\mathfrak {T}}}_{12}/{{\mathfrak {T}}}_{11}$. That is,

$$\begin{aligned} \sup _{f\in \mathcal{E}(K,\gamma )}\bigg | E[f(Z_n)] -\int _{{\mathbb {R}}}f(z)p_n^Z(z)dz\bigg |= & {} o(n^{-1/2}) \end{aligned}$$

(3.35)

as $n\rightarrow \infty$. However, since $\widehat{H}_n^{(2)}\in [0,1]$ by definition, and the family ${{\mathfrak {F}}}$ of (3.29) is bounded in $L^{\infty \text {--}}$, we know

$$\begin{aligned} \sup _{f\in \mathcal{E}(K,\gamma )}\big | E\big [f\big (c\sqrt{n}(\widehat{H}_n^{(2)}-H)\big )\big ]-E[f(Z_n)]\big |\le & {} 2K(1+(c\sqrt{n})^\gamma ) P{\left[ {\psi _n<1} \right] } \nonumber \\ {}= & {} O(n^{-L}) \end{aligned}$$

(3.36)

as $n\rightarrow \infty$, for every $L>0$, where $c=2\log 2$. Consequently

$$\begin{aligned} \sup _{f\in \mathcal{E}(K,\gamma )}\bigg | E\big [f\big (c\sqrt{n}(\widehat{H}_n^{(2)}-H)\big )\big ]-\int _{{\mathbb {R}}}f(z)p_n^Z(z)dz\bigg |= & {} o(n^{-1/2}) \end{aligned}$$

(3.37)

as $n\rightarrow \infty$.

Finally we rescale the argument of $p_n^Z(z)$ by c to obtain the approximate density $p_n(z)$ for $\widehat{H}_n^{(2)}$ appearing in (2.6) with

$$\begin{aligned} q(z)\>=\>q^Z({ c}z)&\text {and}&v_H=c^{-2}G_\infty . \end{aligned}$$

(3.38)

This completes the proof of Theorem 2.2. $\square$

3.6 Proof of Theorem 2.3

The effect of the modification in (2.8) appears as the shift of $\widehat{H}_n^{(2)}$ by the constant $-b(H)/n$ in the asymptotic expansion. So, the estimate (2.10) is almost obvious if we replace the function f(z) by $f\big (z-n^{-1/2}b(H)\big )$ in (2.7) with some modified $(M,\gamma )$ and next by expansion after change of variables. Rigorous justification does not matter. We may cut off the event $\{\widehat{H}_n^{(2)}\not \in (H/2,(H+1)/2)\}$ since $\big \Vert \sqrt{n}\big (\widehat{H}_n^{(b)}-H\big )\big \Vert _\infty =O(n^{1/2})$. Just start with $N_n$ including $-cb\big (\widehat{H}_n^{(2)})$. Then a modification is adding $-cb(H)$ in (3.31) to the limit of $N_n$. This causes addition of $-cb(H)G_\infty ^{-1}z\phi (z;0,G_\infty )$ to (3.32) to give the same result as the one by the above intuition. $\square$

Remark 3.15

It is not difficult to show that $b^*$ and $b^{**}\in C^\infty (0,1)$ by using the representation of $\Sigma _{ij}$ ($i,j=1,2$) in H, and the representation (3.38) of q(z) depending on H, as well as the uniform positivity (3.27) of $G_\infty$ depending on H.

4 Numerical study

We give some results of simulation. The following two figures compare the histogram of $\widehat{H}_n^{(2)}$ with normal approximation and asymptotic expansion, in the case of $H=0.5$ and $n=32$. The left figure plots the histogram of $\widehat{H}_n^{(2)}$ and the right one the histogram but without cutoff as (1.1), i.e., $\widehat{H}_n^{(2)'}:= \frac{1}{2}-\frac{1}{2\log 2}\log \big ({V_{2n,T}^{(2)}}/{V_{n,T}^{(2)}}\big )$. The solid lines are obtained by the asymptotic expansion, and the dashed ones by the normal approximation. Each of the following numerical results is based on $10^5$ paths.

When n increases, the atoms become smaller as well as the difference between the asymptotic expansion and the normal approximation decreases.

Since the precision of approximation of the atoms is determined by that of density, we will leave the curves and omit to draw atoms by the approximations, in the following several plots for different values of H. In practical use, the density should be integrated on the tails to approximate the distribution of $\widehat{H}_n^{(2)}$ of (1.1), that has atoms on the endpoints 0 and 1.