Limit Theorem for Self-intersection Local Time Derivative of Multidimensional Fractional Brownian Motion

Abstract

The existence condition $H<1/d$ for first-order derivative of self-intersection local time for $d\ge 3$ dimensional fractional Brownian motion was obtained in Yu (J Theoret Probab 34(4):1749–1774, 2021). In this paper, we establish a limit theorem under the nonexistence critical condition $H=1/d$.

1 Introduction

Consider a d-dimensional fractional Brownian motion (fBm) with Hurst parameter $H\in (0,1)$, which is a d-dimensional centered Gaussian process $B^H=\{B_t^H, ~t\ge 0\}$ with component processes being independent copies of a 1-dimensional centered Gaussian process $B^{H,i}$, $i=1,2,\ldots ,d$ and the covariance function given by

$$\begin{aligned} \mathbb {E}[B_t^{H,i}B_s^{H,i}]=\frac{1}{2}\left[ t^{2H}+s^{2H}-|t-s|^{2H} \right] . \end{aligned}$$

Note that $B_t^{\frac{1}{2}}$ is a classical standard Brownian motion. Let $D=\{(r,s): 0<r<s<t\}$. The self-intersection local time (SLT) of fBm was first investigated in Rosen [13] and formally defined as

$$\begin{aligned} \alpha _t(y)=\int _{D}\delta (B^H_s-B^H_r-y)\text {d}r\text {d}s, \end{aligned}$$

where $B^H$ is a 2-dimensional fBm and $\delta$ is the Dirac delta function. It was further investigated in Hu [4], Hu and Nualart [6]. In particular, Hu and Nualart [6] showed its existence whenever $Hd<1$. Moreover, $\alpha _t(y)$ is Hölder continuous in time of any order strictly less than $1-H$ which can be derived from Xiao [15].

The derivative of self-intersection local time (DSLT) for fBm was first considered in the works by Yan et al. [16, 17], where the ideas were borrowed form Rosen [14]. The DSLT for fBm has two versions. One is extended by the Tanaka formula (see in Jung and Markowsky [9]):

$$\begin{aligned} \widetilde{\alpha }'_t(y)=-H\int _{D}\delta '(B^H_s-B^H_r-y)(s-r)^{2H-1}\text {d}r\text {d}s. \end{aligned}$$

The other is from the occupation-time formula (see Jung and Markowsky [10]):

$$\begin{aligned} \widehat{\alpha }'_t(y)=-\int _{D}\delta '(B^H_s-B^H_r-y)\text {d}r\text {d}s. \end{aligned}$$

Motivated by the first-order DSLT for fBm in Jung and Markowsky [10] and the k-th-order derivative of intersection local time (ILT) for fBm in Guo et al. [3], we will consider the following k-th-order DSLT for fBm in this paper,

$$\begin{aligned} \widehat{\alpha }^{(k)}_t(y)&=\frac{\partial ^k}{\partial _{y_1}^{k_1}\ldots \partial _{y_d}^{k_d}}\int _{D}\delta (B^H_s-B^H_r-y)\text {d}r\text {d}s\\&=(-1)^{|k|}\int _{D}\delta ^{(k)}(B^H_s-B^H_r-y)\text {d}r\text {d}s, \end{aligned}$$

where $k=(k_1,\ldots ,k_d)$ is a multi-index with all $k_i$ being nonnegative integers and $|k|=k_1+k_2+\cdots +k_d$, $\delta$ is the Dirac delta function of d variables and $\delta ^{(k)}(y)=\frac{\partial ^k}{\partial _{y_1}^{k_1}\ldots \partial _{y_d}^{k_d}}\delta (y)$ is the k-th-order partial derivative of $\delta$.

Set

$$\begin{aligned} f_\varepsilon (x)=\frac{1}{(2\pi \varepsilon )^{\frac{d}{2}}}e^{-\frac{|x|^2}{2\varepsilon }}=\frac{1}{(2\pi )^d}\int _{\mathbb {R}^d}e^{i\langle p,x\rangle }e^{-\varepsilon \frac{|p|^2}{2}}\text {d}p, \end{aligned}$$

where $\langle p,x\rangle =\sum _{j=1}^dp_jx_j$ and $|p|^2=\sum _{j=1}^dp_j^2$.

Since the Dirac delta function $\delta$ can be approximated by $f_\varepsilon (x)$, we approximate $\delta ^{(k)}$ and $\widehat{\alpha }_t^{(k)}(y)$ by

$$\begin{aligned} f^{(k)}_\varepsilon (x)=\frac{i^{|k|}}{(2\pi )^d}\int _{\mathbb {R}^d}p_1^{k_1}\ldots p_d^{k_d}e^{i\langle p,x\rangle }e^{-\varepsilon \frac{|p|^2}{2}}\text {d}p \end{aligned}$$

and

$$\begin{aligned} \widehat{\alpha }^{(k)}_{t,\varepsilon }(y)=(-1)^{|k|}\int _{D}f^{(k)}_\varepsilon (B^H_s-B^H_r-y)\text {d}r\text {d}s, \end{aligned}$$

(1.1)

respectively.

If $\widehat{\alpha }^{(k)}_{t,\varepsilon }(y)$ converges to a random variable in $L^p$ as $\varepsilon \rightarrow 0$, we denote the limit by $\widehat{\alpha }_t^{(k)}(y)$ and call it the k-th DSLT of $B^H$.

Recently, Yu [18] studied the existence and Hölder continuity conditions of $\widehat{\alpha }_t^{(k)}(y)$ and related limit theorem in critical case. We recall the existence condition for $\widehat{\alpha }_t^{(k)}(y)$ in $L^2$ as follows.

Theorem 1.1

[18] For $0<H<1$ and $\widehat{\alpha }^{(k)}_{t,\varepsilon }(y)$ defined in (1.1), let $\#:=\#\{k_i ~is ~odd, ~i=1, 2, \ldots d\}$ denotes the odd number of $k_i$, for $i=1, 2, \ldots , d$. If $H<\min \{\frac{2}{2|k|+d},\frac{1}{|k|+d-\#}, \frac{1}{d}\}$ for $|k|=\sum _{j=1}^dk_j$, then $\widehat{\alpha }^{(k)}_{t}(0)$ exists in $L^2$.

Note that, if $|k|=1$, the existence condition of $\widehat{\alpha }^{(k)}_{t}(0)$ is $H<1/d$, and $Hd=1$ is the critical condition of $\widehat{\alpha }_t^{(k)}(y)$ for any $d\ge 2$. When $Hd=1$ for $d=2$, Markowsky [11] proved the limit theorem for $|k|=1$.

Theorem 1.2

[11] $\widehat{\alpha }^{(k)}_{t,\varepsilon }(y)$ is defined in (1.1) with $y=0$. Suppose that $H=\frac{1}{2}$, $d=2$ and $|k|=1$, then as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \Big (\log 1/\varepsilon \Big )^{-1}\widehat{\alpha }^{(k)}_{t,\varepsilon }(0)\overset{law}{\rightarrow } N\left( 0,\sigma ^2_0\right) . \end{aligned}$$

In this paper, we will consider the case of $Hd=1$ for any $d\ge 3$ and $|k|=1$, and prove a limit theorem for $\widehat{\alpha }^{(k)}_{t,\varepsilon }(0)$. Without loss of generality, we assume that $k_1=1,k_2=0,\ldots ,k_d=0$ for the multi-index $k=(k_1,\ldots ,k_d)$, and for the convenience of writing, we will abbreviate $\widehat{\alpha }^{(1,0,\ldots ,0)}_{t,\varepsilon }(0)$ as $\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)$ in the subsequent content of this paper without causing confusion.

Theorem 1.3

$\widehat{\alpha }^{(k)}_{t,\varepsilon }(y)$ is defined in (1.1) with $y=0$. Suppose that $Hd=1$ for any $d\ge 3$ and $|k|=1$. Then, as $\varepsilon \rightarrow 0$, we have

$$\begin{aligned} \Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{H-\frac{1}{2}}\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)\overset{law}{\rightarrow }N(0,\sigma ^2), \end{aligned}$$

where $\sigma ^2=\frac{2Ht^{3-4H}}{(2\pi )^d(1-2H)^2}$.

When $|k|=1$, under the condition $H>1/d$, the behavior of $\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)$ as $\varepsilon \rightarrow 0$ is also of interest. One would expect a central limit theorem to exist, but this remains unproved. Nevertheless, we venture the following conjecture

(1) If $H=\frac{2}{d+2}>\frac{1}{d}$ and $d\ge 3$, $(\log \frac{1}{\varepsilon })^{\gamma _1(H)}\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)$ converges in distribution to a normal law for some $\gamma _1(H)<0$;

(2) If $H>\frac{1}{2}\ge \frac{2}{d+2}$ and $d\ge 2$, $\varepsilon ^{\gamma _2(H)}\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)$ converges in distribution to a normal law for some $\gamma _2(H)>0$;

(3) If $\frac{2}{d+2}<H<\frac{1}{2}$ and $d\ge 3$, $\varepsilon ^{\gamma _3(H)}(\log \frac{1}{\varepsilon })^{\gamma _4(H)}\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)$ converges in distribution to a normal law for some $\gamma _3(H)>0$ and $\gamma _4(H)<0$.

The paper has the following structure. We state some preliminary lemmas in Sect. 2. Section 3 is to prove the main result. Throughout this paper, if not mentioned otherwise, the letter C, with or without a subscript, denotes a generic positive finite constant and may change from line to line.

2 Preliminaries

In this section, we present two basic lemmas, which will be used in Sect. 3. The first lemma gives the bounds on the quantity of $\lambda \rho -\mu ^2$, which could be obtained from the Appendix B in [9] or Lemma 3.1 in [4]. In fact, $\lambda , \rho$ and $\mu$ represent the three quantities of the covariance matrix of the increment of fBm, and the bound estimation of $\lambda \rho -\mu ^2$ is beneficial for the subsequent calculation of the convergence of multiple integrals, which will bring a lot of convenience to the proof in Sect. 3.

Lemma 2.1

Let

$$\begin{aligned} \lambda =|s-r|^{2H}, ~~\rho =|s'-r'|^{2H}, \end{aligned}$$

and

$$\begin{aligned} \mu =\frac{1}{2}\Big (|s'-r|^{2H}+|s-r'|^{2H}-|s'-s|^{2H}-|r-r'|^{2H}\Big ). \end{aligned}$$

Case (i) Suppose that $D_1=\{(r,r',s,s')\in [0,t]^4 ~|~ r<r'<s<s'\}$, let $r'-r=a$, $s-r'=b$, $s'-s=c$. Then, there exists a constant $K_1$ such that

$$\begin{aligned} \lambda \rho -\mu ^2\ge K_1\,\left( (a+b)^{2H}c^{2H}+a^{2H}(b+c)^{2H}\right) \end{aligned}$$

and

$$\begin{aligned} 2\mu =(a+b+c)^{2H}+b^{2H}-a^{2H}-c^{2H}. \end{aligned}$$

Case (ii) Suppose that $D_2=\{(r,r',s,s')\in [0,t]^4 ~|~ r<r'<s'<s\}$, let $r'-r=a$, $s'-r'=b$, $s-s'=c$. Then, there exists a constant $K_2$ such that

$$\begin{aligned} \lambda \rho -\mu ^2\ge K_2\,b^{2H}\left( a^{2H}+c^{2H}\right) \end{aligned}$$

and

$$\begin{aligned} 2\mu =(a+b)^{2H}+(b+c)^{2H}-a^{2H}-c^{2H}. \end{aligned}$$

Case (iii) Suppose that $D_3=\{(r,r',s,s')\in [0,t]^4 ~|~ r<s<r'<s'\}$, let $s-r=a$, $r'-s=b$, $s'-r'=c$. Then, there exists a constant $K_3$ such that

$$\begin{aligned} \lambda \rho -\mu ^2\ge K_3(ac)^{2H} \end{aligned}$$

and

$$\begin{aligned} 2\mu =(a+b+c)^{2H}+b^{2H}-(a+b)^{2H}-(c+b)^{2H}. \end{aligned}$$

The second lemma shows the Wiener chaos expansion of $\widehat{\alpha }^{(k)}_{t,\varepsilon }(0)$ with $|k|=1$. Before that, we need to explain some notations. We will denote by $\mathcal {H}$ the Hilbert space obtained by taking the completion of the space of step functions endowed with the inner product

$$\begin{aligned} \langle \mathbbm {1}_{[a,b]}, \mathbbm {1}_{[c,d]}\rangle _{\mathcal {H}}=\mathbb {E}[(B_b^{H,1}-B_a^{H,1})(B_d^{H,1}-B_c^{H,1})], \end{aligned}$$

(2.1)

where $B^{H,1}$ is a 1-dimensional fBm. The mapping $\mathbbm {1}_{[0,t]}\rightarrow B_t^{H,1}$ can be extended to a linear isometry between $\mathcal {H}$ and a Gaussian subspace $L^2(\Omega , \mathcal {F}, \mathbb {P})$. For any integer $q\in \mathbb {N}$, we denote by $\mathcal {H}^{\otimes q}$ and $\mathcal {H}^{\odot q}$ the q-th tensor product of $\mathcal {H}$, and the q-th symmetric tensor product of $\mathcal {H}$, respectively.

Similarly, for d-dimensional fBm $B^H=(B^{H,1},\ldots ,B^{H,d})$, we can define corresponding Hilbert space $\mathcal {H}^d$ and tensor product spaces $(\mathcal {H}^d)^{\otimes q}$ and $(\mathcal {H}^d)^{\odot q}$. If $h=(h^1,\ldots , h^d)\in \mathcal {H}^d$, we set $B^H(h)=\sum _{j=1}^dB^{H,j}(h^j)$. Then $h\mapsto B^H(h)$ is a linear isometry between $\mathcal {H}^d$ and the Gaussian subspace of $L^2(\Omega _1\times \cdots \times \Omega _d, \mathcal {F}_1\times \cdots \times \mathcal {F}_d, \mathbb {P}_1\times \cdots \times \mathbb {P}_d)$ generated by $B^H$. The q-th Wiener chaos of $L^2(\Omega _1\times \cdots \times \Omega _d, \mathcal {F}_1\times \cdots \times \mathcal {F}_d, \mathbb {P}_1\times \cdots \times \mathbb {P}_d)$, denoted by $\mathfrak {H}_q$, is the closed subspace of $L^2(\Omega _1\times \cdots \times \Omega _d, \mathcal {F}_1\times \cdots \times \mathcal {F}_d, \mathbb {P}_1\times \cdots \times \mathbb {P}_d)$ generated by the variables

$$\begin{aligned} \Big \{\prod _{j=1}^dH_{q_j}(B^{H,j}(h^j))| \sum _{j=1}^dq_j=q, h^j\in \mathcal {H}, \Vert h^j\Vert _{\mathcal {H}}=1\Big \}, \end{aligned}$$

where $H_q$ is the q-th Hermite polynomial, defined by

$$\begin{aligned} H_q(x)=(-1)^qe^{x^2/2}\frac{d^q}{d x^q}e^{-x^2/2}. \end{aligned}$$

For $q\in \mathbb {N}$, $q\ge 1$ and $h\in \mathcal {H}^d$ of the form $h=(h^1,\ldots ,h^d)$ with $\Vert h^j\Vert _{\mathcal {H}}=1$, we can write

$$\begin{aligned} h^{\otimes q}=\sum _{i_1,\ldots ,i_q=1}^dh^{i_1}\otimes \cdots \otimes h^{i_q}. \end{aligned}$$

For such h, we define the mapping

$$\begin{aligned} I_q(h^{\otimes q})=\sum _{i_1,\ldots ,i_q=1}^d\prod _{j=1}^dH_{q_j(i_1,\ldots ,i_q)}(B^{H,j}(h^j)), j=1,\ldots ,d \end{aligned}$$

where $q_j(i_1,\ldots ,i_q)$ denotes the number of indices in $(i_1,\ldots ,i_q)$ equal to j. The range of $I_q$ is contained in $\mathfrak {H}_q$. This mapping provides a linear isometry between $(\mathcal {H}^d)^{\odot q}$ (equipped with the norm $\sqrt{q!}\Vert \cdot \Vert _{(\mathcal {H}^d)^{\otimes q}}$) and $\mathfrak {H}_q$ (equipped with the $L^2$-norm). Here multiple stochastic integral $I_n$ is the d-dimensional version see in Jaramillo and Nualart [8] (or in Flandoli and Tudor [2]).

It also holds that $I_n(f)=I_n(\widetilde{f})$, where $\widetilde{f}$ denotes the symmetrization of f. We recall that any square integrable random variable F which is measurable with respect to the $\sigma$-algebra generated by $B^H$ can be expanded into an orthogonal sum of multiple stochastic integrals

$$\begin{aligned} F=\sum _{n=0}^{\infty }I_n(f_n), \end{aligned}$$

where $f_n\in (\mathcal {H}^d)^{\odot n}$ are (uniquely determined) symmetric functions and $I_0(f)=\mathbb {E}(F)$.

The proof process of Wiener chaos expansion also requires the knowledge of Malliavin derivative $\mathbb {D}$ with respect to fBm $B^H$. Denote by $C_b^{\infty }(\mathbb {R}^n)$ the space of bounded smooth functions on $\mathbb {R}^n$. Consider the space of random variables

$$\begin{aligned} \mathcal {S}:=\{F=g(B^{H}(f_1),\ldots ,B^{H}(f_n)), g\in C_b^{\infty }(\mathbb {R}^n), f_j\in \mathcal {H}^d, j=1,\ldots ,d\}. \end{aligned}$$

The Malliavin derivative of $F\in \mathcal {S}$, denoted by $\mathbb {D}F$, is given by

$$\begin{aligned} \mathbb {D}F=\sum _{j=1}^n\partial _j g(B^{H}(f_1),\ldots ,B^{H}(f_n))f_j. \end{aligned}$$

By iteration, we can define the n-th derivatives $D^n$ for every $n\ge 2$, which is an element of $L^2(\Omega ,(\mathcal {H}^d)^{\otimes n})$. For example, we write for the smooth function f,

$$\begin{aligned} \mathbb {D}f(B_t^{H,1},\ldots ,B_t^{H,d})&=\mathbb {D}f(B^H(h_1),\ldots ,B^H(h_d))\\&=\sum _{j=1}^d\partial _j f(B^{H}(h_1),\ldots ,B^{H}(h_d))h_j, ~~h_j\in \mathcal {H}^d, j=1,2,\ldots ,d, \end{aligned}$$

where $h_j=\mathbbm {1}_{[0,t]}e_j, j=1,2,\ldots ,d$ and

$$\begin{aligned} e_1=(1,0,\ldots ,0), e_2=(0,1,0,\ldots ,0), \ldots , e_d=(0,\ldots ,0,1). \end{aligned}$$

(2.2)

Similarly,

$$\begin{aligned} \begin{aligned}&\mathbb {D}^{n}f(B_t^{H,1},\ldots ,B_t^{H,d})\\&=\sum _{i_1,\ldots ,i_n=1}^{d}\partial _{i_1}\cdots \partial _{i_n} f(B_t^{H,1},\ldots ,B_t^{H,d})\bigotimes _{j=1}^n(\mathbbm {1}_{[0,t]}e_{i_j}), \end{aligned} \end{aligned}$$

(2.3)

where $\mathbbm {1}_{[0,t]}e_{i_j}\in \mathcal {H}^d$, $\bigotimes _{j=1}^n(\mathbbm {1}_{[0,t]}e_{i_j})\in (\mathcal {H}^d)^{\otimes n}, i_j\in \{1,2,\ldots ,d\}, j=1,2,\ldots ,n$.

More detailed introductions to Malliavin derivative and multiple stochastic integral can be found in Nualart [12], Hu [5] and the references therein.

Lemma 2.2

Let $\widehat{\alpha }^{(k)}_{t,\varepsilon }(y)$ be defined in (1.1), then we have the Wiener chaos expansion for $k=(1,0\ldots ,0)$,

$$\begin{aligned} \widehat{\alpha }^{(k)}_{t,\varepsilon }(0)=\sum _{q=1}^{+\infty }I_{2q-1}(f_{2q-1,\varepsilon }). \end{aligned}$$

(i) If $d=2$, $f_{2q-1,\varepsilon }$ is the element of $(\mathcal {H}^2)^{\otimes (2q-1)}$ given by

$$\begin{aligned} f_{2q-1,\varepsilon }(x_1,\ldots ,x_{2q-1})=\beta _q\int _{0<r<s<t}(|s-r|^{2H}+\varepsilon )^{-q-1}\bigotimes _{j=1}^{2q-1}(\mathbbm {1}_{[r,s]}e_{i_j})(x_j)\text {d}r\text {d}s, \end{aligned}$$

where $\beta _q=\frac{(-1)^q}{2\pi (2q-1)!}\sum _{q_1+q_2=q, q_1\ge 1}\frac{(2q-1)!(2q_1)!}{(2q_1-1)!(q_1)!(q_2)!2^q}$ and $\mathbbm {1}_{[r,s]}e_{i_j}\in \mathcal {H}^2, i_j\in \{1,2\}, j=1,2,\ldots ,2q-1$, ($e_{i_j}$ defined in (2.2)).

(ii) If $d\ge 3$, $f_{2q-1,\varepsilon }\in (\mathcal {H}^d)^{\otimes (2q-1)}$

$$\begin{aligned} f_{2q-1,\varepsilon }(x_1,\ldots ,x_{2q-1})=\beta _{q,d}\int _{0<r<s<t}(|s-r|^{2H}+\varepsilon )^{-q-d/2}\bigotimes _{j=1}^{2q-1}(\mathbbm {1}_{[r,s]}e_{i_j})(x_j)\text {d}r\text {d}s, \end{aligned}$$

where $\beta _{q,d}=\frac{(-1)^q}{(2q-1)!(2\pi )^{d/2}}\sum _{q_1+\cdots +q_d=q, q_1\ge 1}\frac{(2q-1)!(2q_1)!}{(2q_1-1)!(q_1)!\cdots (q_d)!2^q}$ and $\mathbbm {1}_{[r,s]}e_{i_j}\in \mathcal {H}^d, i_j\in \{1,2,\ldots ,d\}, j=1,2,\ldots ,2q-1$.

Proof

The proof adopts a method similar to Lemma 7 in Hu and Nualart [6] (or the Appendix A in Das and Markowsky [1]).

(i) For the case $d=2$, by Stroock’s formula,

$$\begin{aligned} \widehat{\alpha }^{(k)}_{t,\varepsilon }(0)&=\frac{i}{(2\pi )^2}\int _0^t\int _0^s\int _{\mathbb {R}^2}e^{i\langle \xi ,B_s^H-B_r^H\rangle }\xi _1e^{-\varepsilon |\xi |^2/2}d\xi \text {d}r\text {d}s\\&=\sum _{q=1}^{+\infty }I_{2q-1}(f_{2q-1,\varepsilon }), \end{aligned}$$

where $f_{2q-1,\varepsilon }\in (\mathcal {H}^2)^{\otimes (2q-1)}$ and

$$\begin{aligned} f_{n,\varepsilon }\equiv f_{n,\varepsilon }(x_1,\ldots ,x_n)=\frac{1}{n!}\int _{0<r<s<t}\mathbb {E}[\mathbb {D}_{x_1,\ldots ,x_n}^{n}\partial _1f_{\varepsilon }(B_s^H-B_r^H)]\text {d}r\text {d}s \end{aligned}$$

with $x_j\in [0,t]$ for all $j=1,2,\ldots ,n$.

Let $i_j\in \{1,2\}$ for all $j=1,2,\ldots ,n$. Then by (2.3), we can compute the expectation

$$\begin{aligned} \mathbb {E}[\mathbb {D}_{x_1,\ldots ,x_n}^{n}\partial _1f_{\varepsilon }(B_s^H-B_r^H)]=\sum _{i_1,\ldots ,i_n=1}^{2}\mathbb {E}[\partial _{i_1}\cdots \partial _{i_n}\partial _1f_{\varepsilon }(B_s^H-B_r^H)]\bigotimes _{j=1}^n(\mathbbm {1}_{[r,s]}e_{i_j})(x_j), \end{aligned}$$

where $(\mathbbm {1}_{[r,s]}e_{i_j})(x_j)\in \mathcal {H}^2, i_j\in \{1,2\}, j=1,2,\ldots ,n$, ($e_{i_j}$ defined in (2.2)) and

$$\begin{aligned}&\mathbb {E}[\partial _{i_1}\cdots \partial _{i_n}\partial _1f_{\varepsilon }(B_s^H-B_r^H)]\\&=\frac{i^{n+1}}{(2\pi )^2}\int _{\mathbb {R}^2}\xi _1(\xi _{i_1}\xi _{i_2}\cdots \xi _{i_n})\mathbb {E}[e^{i\langle \xi ,B_s^H-B_r^H\rangle }]e^{-\varepsilon |\xi |^2/2}\text {d}\xi \\&=\frac{i^{n+1}}{(2\pi )^2}\int _{\mathbb {R}^2}\xi _1(\xi _{i_1}\xi _{i_2}\cdots \xi _{i_n})e^{-\frac{1}{2}(|s-r|^{2H}+\varepsilon )|\xi |^2}\text {d}\xi \\&=(i)^{n+1}(2\pi )^{-1}(|s-r|^{2H}+\varepsilon )^{-1-\frac{n+1}{2}}\mathbb {E}[X_1X_{i_1}X_{i_2}\cdots X_{i_n}], \end{aligned}$$

with the independent identical distribution standard Gaussian random variables $X_{i_n}$ and

$$\begin{aligned} \mathbb {E}[X_1X_{i_1}X_{i_2}\cdots X_{i_n}]= {\left\{ \begin{array}{ll} \frac{(2m_1)!(2m_2)!}{(m_1)!(m_2)!2^m}, &{}{\text {if} n=2(m_1+m_2)-1,}\\ &{} \text { the number of }i_k=1 \text { is }2m_1-1\\ &{} \text{ and } \text{ the } \text{ number } \text{ of } i_k=2 \text {is }2m_2,\\ 0, &{}{\text {otherwise}.} \end{array}\right. } \end{aligned}$$

Then, for $n=2q-1=2(q_1+q_2)-1$ with the number of $i_k=1$ is $2q_1-1$ and the number of $i_k=2$ is $2q_2$, the summation

$$\begin{aligned} \sum _{i_1,\ldots ,i_n=1}^{2}\mathbbm {1}_{\{n=2(q_1+q_2)-1\}}\mathbbm {1}_{\{\#\{i_k=1\}=2q_1-1\}}\mathbbm {1}_{\{\#\{i_k=2\}=2q_2\}}=\sum _{q_1+q_2=q, q_1\ge 1}\frac{(2q-1)!}{(2q_1-1)!(2q_2)!}, \end{aligned}$$

where $\#\{i_k=x\}$ denotes the number of $i_k=x$. This gives

$$\begin{aligned}&\sum _{i_1,\ldots ,i_n=1}^{2}\mathbb {E}[X_1X_{i_1}X_{i_2}\cdots X_{i_n}]\\&=\sum _{i_1,\ldots ,i_n=1}^{2}\mathbbm {1}_{\{n=2(q_1+q_2)-1\}}\mathbbm {1}_{\{\#\{i_k=1\}=2q_1-1\}}\mathbbm {1}_{\{\#\{i_k=2\}=2q_2\}}\frac{(2q_1)!(2q_2)!}{(q_1)!(q_2)!2^q}\\&=\sum _{q_1+q_2=q, q_1\ge 1}\frac{(2q-1)!}{(2q_1-1)!(2q_2)!}\frac{(2q_1)!(2q_2)!}{(q_1)!(q_2)!2^q}. \end{aligned}$$

Thus, we have

$$\begin{aligned} f_{2q-1,\varepsilon }(x_1,\ldots ,x_{2q-1})=\beta _q\int _{0<r<s<t}(|s-r|^{2H}+\varepsilon )^{-q-1}\bigotimes _{j=1}^{2q-1}(\mathbbm {1}_{[r,s]}e_{i_j})(x_j)\text {d}r\text {d}s, \end{aligned}$$

where $\beta _q=\frac{(-1)^q}{2\pi (2q-1)!}\sum _{q_1+q_2=q, q_1\ge 1}\frac{(2q-1)!(2q_1)!}{(2q_1-1)!(q_1)!(q_2)!2^q}$.

(ii) Similarly, we can prove the case of $d\ge 3$.

$$\begin{aligned} \widehat{\alpha }^{(k)}_{t,\varepsilon }(0)=\sum _{q=1}^{+\infty }I_{2q-1}(f_{2q-1,\varepsilon }), \end{aligned}$$

where $f_{2q-1,\varepsilon }\in (\mathcal {H}^d)^{\otimes (2q-1)}$ and

$$\begin{aligned} f_{n,\varepsilon }(x_1,\ldots ,x_n)=&\,\frac{(\iota )^{n+1}}{n!}\frac{1}{(2\pi )^{d/2}}\int _{0<r<s<t}(|s-r|^{2H}+\varepsilon )^{-(n+1/2-d/2)}\bigotimes _{j=1}^{n}(\mathbbm {1}_{[r,s]}e_{i_j})(x_j)\text {d}r\text {d}s\\&\times \sum _{i_1,\ldots ,i_n=1}^{d}\mathbb {E}[X_1X_{i_1}X_{i_2}\cdots X_{i_n}], \end{aligned}$$

with $(\mathbbm {1}_{[r,s]}e_{i_j})(x_j)\in \mathcal {H}^d, i_j\in \{1,2,\ldots ,d\}, j=1,2,\ldots ,n$.

Note that

$$\begin{aligned} \mathbb {E}[X_1X_{i_1}X_{i_2}\cdots X_{i_n}]= {\left\{ \begin{array}{ll} \frac{(2m_1)!\cdots (2m_d)!}{(m_1)!\cdots (m_d)!2^m}, &{}\text {if } n=2(m_1+\cdots +m_2)-1,\\ &{} \text { the number of }i_k=1 \text { is }2m_1-1\\ &{} \text{ and } \text{ the } \text{ number } \text{ of } i_k=\ell \text { is }2m_\ell \\ &{}\text {for }\ell =2,\ldots ,d,\\ 0, &{}{\text {otherwise}.} \end{array}\right. } \end{aligned}$$

Then, for $n=2q-1=2(q_1+\cdots +q_d)-1$ with the number of $i_k=1$ is $2q_1-1$ and the number of $i_k=\ell$ is $2q_{\ell }$, the summation

$$\begin{aligned}&\sum _{i_1,\ldots ,i_n=1}^{d}\mathbbm {1}_{\{n=2(q_1+\cdots +q_d)-1\}}\mathbbm {1}_{\{\#\{i_k=1\}=2q_1-1\}}\mathbbm {1}_{\{\#\{i_k=2\}=2q_2\}}\times \cdots \times \mathbbm {1}_{\{\#\{i_k=d\}=2q_d\}}\\&=\sum _{q_1+\cdots +q_d=q, q_1\ge 1}\frac{(2q-1)!}{(2q_1-1)!(2q_2)!\cdots (2q_d)!}. \end{aligned}$$

This gives

$$\begin{aligned} \sum _{i_1,\ldots ,i_n=1}^{d}\mathbb {E}[X_1X_{i_1}X_{i_2}\cdots X_{i_n}]&=\sum _{q_1+\cdots +q_d=q, q_1\ge 1}\frac{(2q-1)!}{(2q_1-1)!(2q_2)!\cdots (2q_d)!}\frac{(2q_1)!\cdots (2q_d)!}{(q_1)!\cdots (q_d)!2^q}\\&=\sum _{q_1+\cdots +q_d=q, q_1\ge 1}\frac{(2q-1)!(2q_1)!}{(2q_1-1)!(q_1)!\cdots (q_d)!2^q}. \end{aligned}$$

Thus,

$$\begin{aligned} f_{2q-1,\varepsilon }(x_1,\ldots ,x_{2q-1})=\beta _{q,d}\int _{0<r<s<t}(|s-r|^{2H}+\varepsilon )^{-q-d/2}\bigotimes _{j=1}^{2q-1}(\mathbbm {1}_{[r,s]}e_{i_j})(x_j)\text {d}r\text {d}s, \end{aligned}$$

where $\beta _{q,d}=\frac{(-1)^q}{(2q-1)!(2\pi )^{d/2}}\sum _{q_1+\cdots +q_d=q, q_1\ge 1}\frac{(2q-1)!(2q_1)!}{(2q_1-1)!(q_1)!\cdots (q_d)!2^q}$. $\square$

Lemma 2.3

If $Hd=1$, as $\varepsilon \rightarrow 0$, we have

(i)

$$\begin{aligned} \int _0^{\varepsilon ^{-\frac{1}{H}}}x^{H-\frac{1}{2}}(1+x^{H})^{-\frac{d}{2}-1}\text {d}x=O\left( \log \frac{1}{\varepsilon }\right) \end{aligned}$$

and

(ii)

$$\begin{aligned} \int _0^1x^{2H}(\varepsilon +x^{2H})^{-\frac{d}{2}-1}\text {d}x=O\left( \log \frac{1}{\varepsilon }\right) . \end{aligned}$$

Proof

For (i), by L’Hôspital’s rule, we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1}{\log \frac{1}{\varepsilon }}\int _0^{\varepsilon ^{-\frac{1}{H}}}x^{H-\frac{1}{2}}(1+x^{H})^{-\frac{d}{2}-1}\text {d}x&=\lim _{\varepsilon \rightarrow 0}\frac{1}{H}\varepsilon ^{-1-\frac{1}{2H}}(1+\varepsilon ^{-1})^{-\frac{d}{2}-1}\\&=\lim _{\varepsilon \rightarrow 0}\frac{1}{H}(\varepsilon +1)^{-\frac{d}{2}-1}\\&=\frac{1}{H}, \end{aligned}$$

where we use the condition $Hd=1$ in the second equality.

For (ii), take the variable transformation $x=y\varepsilon ^{\frac{1}{2H}}$,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1}{\log \frac{1}{\varepsilon }}\int _0^1x^{2H}(\varepsilon +x^{2H})^{-\frac{d}{2}-1}\text {d}x&=\lim _{\varepsilon \rightarrow 0}\frac{1}{\log \frac{1}{\varepsilon }}\int _0^{\varepsilon ^{-\frac{1}{2H}}}y^{2H}(1+y^{2H})^{-\frac{d}{2}-1}\text {d}y\\&=\lim _{\varepsilon \rightarrow 0}\frac{1}{2H}(\varepsilon +1)^{-\frac{d}{2}-1}\\&=\frac{1}{2H}, \end{aligned}$$

where we use L’Hôspital’s rule and the condition $Hd=1$ in the second equality.

$\square$

3 Proof of Theorem 1.3

In this section, the proof of Theorem 1.3 is taken into account, we will consider the case of $Hd=1$ for any $d\ge 3$ and $|k|=1$. By Lemma 2.2, $\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)$ has the following chaos decomposition

$$\begin{aligned} \widehat{\alpha }^{(1)}_{t,\varepsilon }(0)=\sum _{q=1}^\infty I_{2q-1}(f_{2q-1,\varepsilon }), \end{aligned}$$

where

$$\begin{aligned} f_{2q-1,\varepsilon }(x_1,\ldots ,x_{2q-1})=\int _{D}f_{2q-1,\varepsilon ,s,r}(x_1,\ldots ,x_{2q-1})\text {d}r\text {d}s \end{aligned}$$

with $D=\{(r,s): 0<r<s<t\}$, where

$$\begin{aligned} f_{2q-1,\varepsilon ,s,r}(x_1,\ldots ,x_{2q-1}):=\beta _{q,d}(|s-r|^{2H}+\varepsilon )^{-q-d/2}\bigotimes _{j=1}^{2q-1}(\mathbbm {1}_{[r,s]}e_{i_j})(x_j). \end{aligned}$$

For $q=1$,

$$\begin{aligned} \mathbb {E}\Big [\Big |I_1(f_{1,\varepsilon })\Big |^2\Big ]=\int _{D^2}\langle f_{1,\varepsilon ,s,r},f_{1,\varepsilon ,s',r'}\rangle _{\mathcal {H}^d}\text {d}r\text {d}s\text {d}r'\text {d}s', \end{aligned}$$

(3.1)

where $\mathcal {H}^d$ is the Hilbert space obtained by taking the completion of the step functions (see in Sect. 2).

For $q>1$, we have to describe the terms $\langle f_{2q-1,\varepsilon ,s_1,r_1},f_{2q-1,\varepsilon ,s_2,r_2}\rangle _{(\mathcal {H}^d)^{\otimes (2q-1)}}$, where $(\mathcal {H}^d)^{\otimes (2q-1)}$ is the $(2q-1)$-th tensor product of $\mathcal {H}^d$. For every $x, ~u_1, ~u_2>0$, we define

$$\begin{aligned} \mu (x,u_1,u_2)=|\mathbb {E}[B_{u_1}^{H,1}(B_{x+u_2}^{H,1}-B_x^{H,1})]|. \end{aligned}$$

(3.2)

For $j=1,2,\ldots ,2q-1$, $i_j\in \{1,2,\ldots ,d\}$,

$$\begin{aligned} \langle \mathbbm {1}_{[r,s]}e_{i_j},\mathbbm {1}_{[r,s]}e_{i_j}\rangle _{\mathcal {H}^{d}}=\langle \mathbbm {1}_{[r,s]}, \mathbbm {1}_{[r,s]}\rangle _{\mathcal {H}}. \end{aligned}$$

(3.3)

Then, we have

$$\begin{aligned}&\langle f_{2q-1,\varepsilon ,s_1,r_1},f_{2q-1,\varepsilon ,s_2,r_2}\rangle _{(\mathcal {H}^d)^{\otimes (2q-1)}}\nonumber \\&=\beta _{q,d}^2 (|s_1-r_1|^{2H}+\varepsilon )^{-q-d/2}(|s_2-r_2|^{2H}+\varepsilon )^{-q-d/2}\nonumber \\&\quad \times \langle \mathbbm {1}_{[r_1,s_1]}^{2q-1},\mathbbm {1}_{[r_2,s_2]}^{2q-1}\rangle _{\mathcal {H}^{\otimes (2q-1)}}\nonumber \\&=:\beta _{q,d}^2G^{(q,d)}_{\varepsilon ,r_2-r_1}(s_1-r_1,s_2-r_2), \end{aligned}$$

(3.4)

where

$$\begin{aligned} G^{(q,d)}_{\varepsilon ,x}(u_1,u_2)=\Big (\varepsilon +u_1^{2H}\Big )^{-\frac{d}{2}-q}\Big (\varepsilon +u_2^{2H}\Big )^{-\frac{d}{2}-q}\mu (x,u_1,u_2)^{2q-1}. \end{aligned}$$

Note that equations (3.2)–(3.4) here can degenerate into the case $d=1$ of the equations (2.18)–(2.19) in Jaramillo and Nualart [7].

Before completing the proof of the main result, we give some useful lemmas below.

Lemma 3.1

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbb {E}\Big [\Big (\frac{1}{\log \frac{1}{\varepsilon }}\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)\Big )^2\Big ]=\sigma ^2, \end{aligned}$$

where $\sigma ^2=\frac{2Ht^{3-4H}}{(2\pi )^d(1-2H)^2}$.

Proof

Let $(X,Y)\in \mathbb {R}\times \mathbb {R}$ be a jointly Gaussian vector with mean zero and covariance $A=(A_{i,j})_{i,j=1,2}$, let $f_A$ is the density of (X, Y) and $f_{1,\varepsilon }(x)=\frac{1}{\sqrt{2\pi \varepsilon }}e^{-\frac{x^2}{2\varepsilon }}, x\in \mathbb {R}$ be a 1-dimensional density function. Then,

$$\begin{aligned} \mathbb {E}[XYf_{1,\varepsilon }(X)f_{1,\varepsilon }(Y)]&=\int _{\mathbb {R}^2}xyf_{1,\varepsilon }(x)f_{1,\varepsilon }(y)f_{A}(x,y)\text {d}x\text {d}y\\&=(2\pi )^{-2}\varepsilon ^{-1}|A|^{-1/2}\int _{\mathbb {R}^2}xye^{-\frac{1}{2}(x,y)(\varepsilon ^{-1}I+A^{-1})(x,y)^T}\text {d}x\text {d}y\\&=(2\pi )^{-1}\varepsilon ^{-1}|A|^{-1/2}|\widetilde{A}|^{1/2}\int _{\mathbb {R}^2}xyf_{\widetilde{A}}(x,y)\text {d}x\text {d}y\\&=(2\pi )^{-1}|\varepsilon I+A|^{-1/2}\widetilde{A}_{1,2}\\&=(2\pi )^{-1}\varepsilon ^2|\varepsilon I+A|^{-\frac{3}{2}}A_{1,2}, \end{aligned}$$

where $\widetilde{A}:=(\varepsilon ^{-1}I+A^{-1})^{-1}$, $f_{\widetilde{A}}$ denotes the density of a Gaussian vector with mean zero and covariance $\widetilde{A}=(\widetilde{A}_{i,j})_{i,j=1,2}$.

Similarly, let $\Sigma =(\Sigma _{i,j})_{i,j=1,2}$ be the covariance matrix of $(B^{H,1}_s-B^{H,1}_r,B^{H,1}_{s'}-B^{H,1}_{r'})$, and $\Sigma ^{d-1}$ be the covariance matrix of $(\widetilde{B}^{H}_s-\widetilde{B}^{H}_r,\widetilde{B}^{H}_{s'}-\widetilde{B}^{H}_{r'})$ ($\widetilde{B}^H$ denotes the $(d-1)$-dimensional fBm). The notations $f_{\Sigma }$ and $f_{\Sigma ^{d-1}}$ represent their density functions, respectively. It is easy to find that $\Sigma ^{d-1}$ is a block diagonal matrix, and that the dimension of it is $2(d-1)\times 2(d-1)$. Then,

$$\begin{aligned}&(2\pi \varepsilon )^{-(d-1)}\int _{\mathbb {R}^{2(d-1)}}e^{-\frac{x_2^2+y_2^2+\cdots +x_d^2+y_d^2}{2\varepsilon }}f_{\Sigma ^{d-1}}(x_2,\ldots ,x_d,y_2,\ldots ,y_d)\text {d}x_2\cdots \text {d}x_d\text {d}y_2\cdots \text {d}y_d\\&=\left( (2\pi \varepsilon )^{-1}\int _{\mathbb {R}^{2}}e^{-\frac{x^2+y^2}{2\varepsilon }}f_{\Sigma }(x,y)\text {d}x\text {d}y\right) ^{d-1}\\&=\left( (2\pi )^{-1}|\varepsilon I+\Sigma |^{-1/2}\right) ^{d-1}. \end{aligned}$$

Thus, for any Gaussian vector $(X, Y)\in \mathbb {R}^d\times \mathbb {R}^d$ and k-th ($k=(1,0,\ldots ,0)$) order derivative, we have

$$\begin{aligned} \mathbb {E}[f^{(1)}_{\varepsilon }(X)f^{(1)}_{\varepsilon }(Y)]&=\frac{1}{\varepsilon ^2}(2\pi \varepsilon )^{-d}\mathbb {E}\left[ X_1Y_1e^{-\frac{X_1^2+\cdots +X_d^2+Y_1^2+\cdots +Y_d^2}{2\varepsilon }}\right] \\&=\frac{1}{\varepsilon ^2}(2\pi \varepsilon )^{-1}\int _{\mathbb {R}^2}x_1y_1e^{-\frac{x_1^2+y_1^2}{2\varepsilon }}f_{\Sigma }(x_1,y_1)\text {d}x_1\text {d}y_1\\&\qquad \times (2\pi \varepsilon )^{-(d-1)}\int _{\mathbb {R}^{2(d-1)}}e^{-\frac{x_2^2+y_2^2+\cdots +x_d^2+y_d^2}{2\varepsilon }}f_{\Sigma ^{d-1}}(\widetilde{x},\widetilde{y})d\widetilde{x}d\widetilde{y}\\&=\varepsilon ^{-2}(2\pi )^{-1}\varepsilon ^2|\varepsilon I+\Sigma |^{-\frac{3}{2}}\Sigma _{1,2}\times (2\pi )^{-(d-1)}|\varepsilon I+\Sigma |^{-\frac{d-1}{2}}\\&=(2\pi )^{-d}|\varepsilon I+\Sigma |^{-\frac{d}{2}-1}\Sigma _{1,2}, \end{aligned}$$

where $\widetilde{x}=(x_2,\ldots ,x_d), \widetilde{y}=(y_2,\ldots ,y_d)$.

Thus,

$$\begin{aligned} \mathbb {E}\Big [\Big |\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)\Big |^2\Big ]=V_1(\varepsilon )+V_2(\varepsilon )+V_3(\varepsilon ) \end{aligned}$$

with

$$\begin{aligned} V_i(\varepsilon )=\frac{2}{(2\pi )^d}\int _{D_i}|\varepsilon I+\Sigma |^{-\frac{d}{2}-1}|\mu | \text {d}r\text {d}s\text {d}r'\text {d}s', \end{aligned}$$

(3.5)

where $D_i$ (i=1, 2, 3) defined in Lemma 2.1 and $\Sigma$ is a covariance matrix with $\Sigma _{1,1}=\lambda$, $\Sigma _{2,2}=\rho$, $\Sigma _{1,2}=\mu$ given in Lemma 2.1.

Next, we will split the proof into three parts to consider $V_1(\varepsilon )$, $V_2(\varepsilon )$ and $V_3(\varepsilon )$, respectively.

For the $V_1(\varepsilon )$ term, changing the coordinates $(r, r', s, s')$ by $(r, a=r'-r, b=s-r', c=s'-s)$ and integrating the r variable, we get

$$\begin{aligned} V_1(\varepsilon )&\le \,C\,\int _{[0,t]^4}|\varepsilon I+\Sigma |^{-\frac{d}{2}-1}|\mu | \text {d}r\text {d}a\text {d}b\text {d}c\\&=\,C\,\int _{[0,t]^3}|\varepsilon I+\Sigma |^{-\frac{d}{2}-1}|\mu | \text {d}a\text {d}b\text {d}c\\&=:\widetilde{V_1}(\varepsilon ). \end{aligned}$$

Applying Lemma 2.1 Case (i), for some $C>0$, we get

$$\begin{aligned} |\varepsilon I+\Sigma |&=(\varepsilon +\Sigma _{1,1})(\varepsilon +\Sigma _{2,2})-\Sigma ^2_{1,2}=\varepsilon ^2+\varepsilon (\Sigma _{1,1}+\Sigma _{2,2})-|\Sigma |\nonumber \\&\ge C\Big [\varepsilon ^2+\varepsilon ((a+b)^{2H}+(b+c)^{2H})+a^{2H}(c+b)^{2H}+c^{2H}(a+b)^{2H}\Big ]\nonumber \\&\ge C\Big [\varepsilon ^2+(a+b)^{H}(b+c)^{H}(\varepsilon +(ac)^{H})\Big ]\nonumber \\&\ge C(a+b)^{H}(b+c)^{H}(\varepsilon +(ac)^{H}), \end{aligned}$$

(3.6)

where we use the Young’s inequality in the second to last inequality.

Substituting (3.6) and

$$\begin{aligned} |\mu |=\frac{1}{2}\left| (a+b+c)^{2H}+b^{2H}-a^{2H}-c^{2H}\right| \le \sqrt{\lambda \rho }=(a+b)^{H}(b+c)^{H} \end{aligned}$$

into the integrand of $\widetilde{V_1}(\varepsilon )$,

$$\begin{aligned} \widetilde{V_1}(\varepsilon )&\le C\int _{[0,t]^3}(a+b)^{-\frac{Hd}{2}}(b+c)^{-\frac{Hd}{2}}\Big (\varepsilon +(ac)^{H}\Big )^{-\frac{d}{2}-1}\text {d}a\text {d}b\text {d}c\\&\le C\int _{[0,t]^3}(a+b)^{H-\frac{Hd}{2}}(a+b)^{-H}(b+c)^{-\frac{Hd}{2}}\Big (\varepsilon +(ac)^{H}\Big )^{-\frac{d}{2}-1}\text {d}a\text {d}b\text {d}c\\&\le C\int _{[0,t]^3}b^{-H-\frac{Hd}{2}}a^{H-\frac{Hd}{2}}\Big (\varepsilon +(ac)^{H}\Big )^{-\frac{d}{2}-1}\text {d}a\text {d}b\text {d}c\\&\le C \varepsilon ^{\frac{1}{H}-\frac{d}{2}-1}\int _0^{t\varepsilon ^{-\frac{1}{H}}}\int _0^ta^{H-\frac{Hd}{2}}\Big (1+(ac)^{H}\Big )^{-\frac{d}{2}-1}\text {d}a\text {d}c, \end{aligned}$$

where we make the change of variable $c=c\,\varepsilon ^{-\frac{1}{H}}$ in the last inequality.

By L’Hôspital’s rule, we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\widetilde{V_{1}}(\varepsilon )&\le \lim _{\varepsilon \rightarrow 0}\frac{-\frac{Ct}{H}\varepsilon ^{-1-\frac{1}{H}}\int _0^ta^{H-\frac{1}{2}}(1+t^Ha^H\varepsilon ^{-1})^{-\frac{d}{2}-1}da}{(1-\frac{1}{2H})\varepsilon ^{-\frac{1}{2H}}}\\&=\lim _{\varepsilon \rightarrow 0}\frac{\frac{Ct}{H}}{\frac{1}{2H}-1}\int _0^{t\varepsilon ^{-\frac{1}{H}}}a^{H-\frac{1}{2}}(1+t^Ha^H)^{-\frac{d}{2}-1}\text {d}a\\&=O\left( \log \frac{1}{\varepsilon }\right) , \end{aligned}$$

where we have used Lemma 2.3 in the last equality.

So, we can obtain

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}V_{1}(\varepsilon )=0. \end{aligned}$$

(3.7)

For the $V_2(\varepsilon )$ term, changing the coordinates $(r, r', s, s')$ by $(r, a=r'-r, b=s'-r', c=s-s')$ and integrating the r variable, then by (3.5), we get

$$\begin{aligned} V_2(\varepsilon ) \le \,C \int _{[0,t]^3}|\varepsilon I+\Sigma |^{-\frac{d}{2}-1}|\mu | \text {d}a\text {d}b\text {d}c=:\widetilde{V_2}(\varepsilon ). \end{aligned}$$

By Lemma 2.1 Case (ii),

$$\begin{aligned} |\varepsilon I+\Sigma |= & {} (\varepsilon +\Sigma _{1,1})(\varepsilon +\Sigma _{2,2})-\Sigma ^2_{1,2}\ge \varepsilon ^2+\varepsilon ((a+b+c)^{2H}+b^{2H})\\{} & {} +K_2 b^{2H}(a^{2H}+c^{2H}). \end{aligned}$$

Then, we have

$$\begin{aligned} \widetilde{V_2}(\varepsilon )\le C\int _{[0,t]^3}|\mu |\Big (\varepsilon ((a+b+c)^{2H}+b^{2H}) + b^{2H}(a^{2H}+c^{2H})\Big )^{-\frac{d}{2}-1}\text {d}a\text {d}b\text {d}c. \end{aligned}$$

Next, we need to estimate this integral over the regions $\{b\le (a\vee c)\}$ and $\{b > (a\vee c)\}$ separately, and denote these two integrals by $\widetilde{V_{2,1}}(\varepsilon )$ and $\widetilde{V_{2,2}}(\varepsilon )$, respectively. Note that

$$\begin{aligned} |\mu |&=\frac{1}{2}\Big ((a+b)^{2H}+(b+c)^{2H}-a^{2H}-c^{2H}\Big )\\&=Hb\int _0^1\left( (a+bv)^{2H-1}+(c+bv)^{2H-1}\right) \text {d}v\\&\le b^{2H}\wedge \left( 2Hb(a\wedge c)^{2H-1}\right) . \end{aligned}$$

If $b\le (a\vee c)$, we choose $|\mu |\le b^{2H}$. Thus,

$$\begin{aligned} \widetilde{V_{2,1}}(\varepsilon )&\le C\int _{[0,t]^3}b^{2H}\Big (\varepsilon (a\vee c)^{2H}+b^{2H}(a\vee c)^{2H}\Big )^{-\frac{d}{2}-1}\text {d}a\text {d}b\text {d}c\nonumber \\&\le C\int _{[0,t]^3}(a\vee c)^{-1-2H}b^{2H}\Big (\varepsilon +b^{2H}\Big )^{-\frac{d}{2}-1}\text {d}a\text {d}b\text {d}c\nonumber \\&\le C \int _0^tb^{2H}\Big (\varepsilon +b^{2H}\Big )^{-\frac{d}{2}-1}\text {d}b\nonumber \\&=O(\log \frac{1}{\varepsilon }), ~as~ \varepsilon \rightarrow 0, \end{aligned}$$

where we have used Lemma 2.3 in the last equality and the following fact

$$\begin{aligned} \varepsilon ^2+\varepsilon ((a+b+c)^{2H}+b^{2H})+ b^{2H}(a^{2H}+c^{2H})\ge \varepsilon (a\vee c)^{2H}+b^{2H}(a\vee c)^{2H}. \end{aligned}$$

If $b>(a\vee c)$, we choose $|\mu |\le 2Hb(a\wedge c)^{2H-1}$. Similarly, we have

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{\widetilde{V_{2,2}}(\varepsilon )}{\log \frac{1}{\varepsilon }}&\le \limsup _{\varepsilon \rightarrow 0}\frac{C}{\log \frac{1}{\varepsilon }}\int _{[0,t]^3}b(a\wedge c)^{2H-1}[b^{2H}(\varepsilon +(a\vee c)^{2H})]^{-\frac{d}{2}-1}\text {d}a\text {d}b\text {d}c\\&\le \limsup _{\varepsilon \rightarrow 0}\frac{C}{\log \frac{1}{\varepsilon }}\int _0^t\!\!b^{-2H}db\int _{[0,t]^2}\!\!(a\wedge c)^{2H-1}(\varepsilon +(a\vee c)^{2H})]^{-\frac{d}{2}-1}\text {d}c\text {d}a\\&\le \limsup _{\varepsilon \rightarrow 0}\frac{C}{\log \frac{1}{\varepsilon }}\int _0^t\int _0^a c^{2H-1}[\varepsilon +a^{2H}]^{-\frac{d}{2}-1}\text {d}c\text {d}a\\&=\limsup _{\varepsilon \rightarrow 0}\frac{C}{\log \frac{1}{\varepsilon }}\int _0^ta^{2H}(\varepsilon +a^{2H})^{-\frac{d}{2}-1}\text {d}a<\infty , \end{aligned}$$

where we have used the following fact

$$\begin{aligned} \varepsilon ^2+\varepsilon ((a+b+c)^{2H}+b^{2H})+ b^{2H}(a^{2H}+c^{2H})\ge \varepsilon b^{2H}+b^{2H}(a\vee c)^{2H}. \end{aligned}$$

So, by the above result, we can obtain

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}V_{2}(\varepsilon )=0. \end{aligned}$$

(3.8)

For the $V_3(\varepsilon )$ term.

$$\begin{aligned} V_3(\varepsilon ) =\frac{2}{(2\pi )^d}\int _{D_3}|\varepsilon I+\Sigma |^{-d/2-1}|\mu | \text {d}s\text {d}r\text {d}s'\text {d}r'. \end{aligned}$$

By changing the coordinates $(r, r', s, s')$ by $(r, a=s-r, b=r'-s, c=s'-r')$, then from (3.2) and Lemma 2.1 Case (iii), we can write

$$\begin{aligned} \mu (a+b,a,c)=|\mu |&=\frac{1}{2}\Big |(a+b+c)^{2H}+b^{2H}-(b+c)^{2H}-(a+b)^{2H}\Big |\\&=H(1-2H)ac\int _0^1\int _0^1(b+ax+cy)^{2H-2}\text {d}x\text {d}y. \end{aligned}$$

and $|\varepsilon I+\Sigma |=\varepsilon ^2+\varepsilon (a^{2\,H}+c^{2\,H})+(ac)^{2\,H}-\mu (a+b,a,c)^2$. It is not hard to see that

$$\begin{aligned} V_3(\varepsilon )&=\frac{2}{(2\pi )^d}\int _{[0,t]^3}\mathbbm {1}_{(0,t)}(a+b+c)(t-a-b-c)|\varepsilon I+\Sigma |^{-d/2-1}|\mu | \text {d}a\text {d}b\text {d}c\\&=\frac{2}{(2\pi )^d} \int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2\times [0,t]}\mathbbm {1}_{(0,t)}(b+\varepsilon ^{\frac{1}{2H}}(a+c))\\&\quad \times \frac{(t-b-\varepsilon ^{\frac{1}{2H}}(a+c))\mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)}{\Big [(1+a^{2H})(1+c^{2H})-\varepsilon ^{-2}\mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)^2\Big ]^{\frac{d}{2}+1}} \varepsilon ^{\frac{1}{H}-2(d/2+1)}\text {d}a\text {d}c\text {d}b, \end{aligned}$$

where we change the coordinates (a, b, c) by $(\varepsilon ^{-\frac{1}{2H}}a, b, \varepsilon ^{-\frac{1}{2H}}c)$ in the last equality. Denote

$$\begin{aligned} \widetilde{V_3}(\varepsilon )&=\frac{2}{(2\pi )^d}\int _{O_{\varepsilon ,3}}\mathbbm {1}_{(0,t)}(b+\varepsilon ^{\frac{1}{2H}}(a+c))\\&\qquad \times \frac{(t-b-\varepsilon ^{\frac{1}{2H}}(a+c))\mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)}{\Big [(1+a^{2H})(1+c^{2H})-\varepsilon ^{-2}\mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)^2\Big ]^{\frac{d}{2}+1}} \varepsilon ^{\frac{1}{H}-2(d/2+1)}\text {d}a\text {d}c\text {d}b, \end{aligned}$$

where $O_{\varepsilon ,3}=\{[0,t\varepsilon ^{-\frac{1}{2H}}]^2\times [(\log \frac{1}{\varepsilon })^{-1},t]\}$.

We conclude that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}V_3(\varepsilon )=\lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}\widetilde{V_3}(\varepsilon ). \end{aligned}$$

(3.9)

Indeed,

$$\begin{aligned}&\limsup _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}|V_3(\varepsilon )-\widetilde{V_3}(\varepsilon )|\\&\le \limsup _{\varepsilon \rightarrow 0}C_{H,d,t}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2\times [0,(\log \frac{1}{\varepsilon })^{-1}]}\varepsilon \mu (a,a,c)\\&\quad \times \Big [(1+a^{2H})(1+c^{2H})-\mu ^2(a,a,c)\Big ]^{-\frac{d}{2}-1} \varepsilon ^{\frac{1}{H}-2(d/2+1)}\text {d}a\text {d}c\text {d}b\\&\le \limsup _{\varepsilon \rightarrow 0}C_{H,d,t}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}(\varepsilon \log \frac{1}{\varepsilon })^{-1}\\&\quad \times \int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}(a\wedge c)^{2H}\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-\frac{d}{2}-1}\text {d}a\text {d}c\\&\le \limsup _{\varepsilon \rightarrow 0}C_{H,d,t}\varepsilon ^{\frac{1}{H}-3}\Big (\log 1/\varepsilon \Big )^{2H-2}\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}(ac)^{H}\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-\frac{d}{2}-1}\text {d}a\text {d}c\\&\le \limsup _{\varepsilon \rightarrow 0}C_{H,d,t}\varepsilon ^{\frac{1}{H}-3}\Big (\log 1/\varepsilon \Big )^{2H-2}\\&=0, \end{aligned}$$

where we use

$$\begin{aligned} \mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)\le \mu (\varepsilon ^{\frac{1}{2H}}a+0,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)=\varepsilon \mu (a,a,c) \end{aligned}$$

in the first inequality and use $\mu (a,a,c)\le (a\wedge c)^{2H}$,

$$\begin{aligned} (1+a^{2H})(1+c^{2H})-\mu ^2(a,a,c)\ge \frac{3}{4}(1+a^{2H})(1+c^{2H}) \end{aligned}$$

in the second inequality.

By the definition of $\mu (a+b,a,c)$, it is easy to find

$$\begin{aligned} \mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)&= H(1-2H) \varepsilon ^{\frac{1}{H}}ac\int _{[0,1]^2}(b+\varepsilon ^{\frac{1}{2H}}av_1+\varepsilon ^{\frac{1}{2H}}cv_2)^{2H-2}\text {d}v_1\text {d}v_2 \end{aligned}$$

and use the Taylor’s theorem for integrand,

$$\begin{aligned} \varepsilon ^{-\frac{1}{H}}\mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)=H(1-2H)acb^{2H-2}+O(\varepsilon ^{\frac{1}{2H}}ac(a+c)). \end{aligned}$$

Similarly, the denominator of the integrand in $V_3(\varepsilon )$ can be rewritten as

$$\begin{aligned}&\Big [(1+a^{2H})(1+c^{2H})-\varepsilon ^{-2}\mu (\varepsilon ^{\frac{1}{2H}}a+b,\varepsilon ^{\frac{1}{2H}}a,\varepsilon ^{\frac{1}{2H}}c)^2\Big ]^{-\frac{d}{2}-1}\\&\qquad =\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-\frac{d}{2}-1}+O\Big (\varepsilon ^{\frac{2}{H}-2} a^2c^2[(1+a^{2H})(1+c^{2H})]^{-\frac{d}{2}-3}\Big ). \end{aligned}$$

It is easy to see that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Big (\log 1/\varepsilon \Big )^{2H-1}\int _{(\log \frac{1}{\varepsilon })^{-1}}^tb^{2H-2}\text {d}b=\frac{1}{1-2H}, \end{aligned}$$

and

$$\begin{aligned}{} & {} \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\Big )^{2H-1}\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}\varepsilon ^{\frac{1}{2H}+\frac{2}{H}-d-2}ac(a+c)\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-\frac{d}{2}-1}\text {d}a\text {d}c\nonumber \\{} & {} \quad +\lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\Big )^{2H-1}\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}\varepsilon ^{\frac{2}{H}-2+\frac{2}{H}-d-2} a^3c^3\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-\frac{d}{2}-3}\text {d}a\text {d}c\nonumber \\{} & {} =0. \end{aligned}$$

(3.10)

Then, by L’Hôspital’s rule, we have

$$\begin{aligned} \begin{aligned}&\lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}\widetilde{V_3}(\varepsilon )\\&=H(1-2H)\frac{2}{(2\pi )^d}\lim _{\varepsilon \rightarrow 0}\Big (\log 1/\varepsilon \Big )^{2H-1}\int _{(\log \frac{1}{\varepsilon })^{-1}}^t(t-b)b^{2H-2}\text {d}b\\&\quad \times \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\Big )^{2H-1}\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}ac\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-\frac{d}{2}-1}\text {d}a\text {d}c\\&=H(1-2H)\frac{2}{(2\pi )^d}\times \frac{t}{1-2H}\times \frac{t^{2-4H}}{(1-2H)^2}. \end{aligned} \end{aligned}$$

(3.11)

Together (3.7), (3.8), (3.9) and (3.11), we can see

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbb {E}\Big [\Big |\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{H-1/2}\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)\Big |^2\Big ] =\frac{2Ht^{3-4H}}{(2\pi )^d(1-2H)^2}=:\sigma ^2. \end{aligned}$$

$\square$

Lemma 3.2

For $I_1(f_{1,\varepsilon })$ given in (3.1), then

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbb {E}\Big [\Big |\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{H-1/2}I_1(f_{1,\varepsilon })\Big |^2\Big ]=\sigma ^2. \end{aligned}$$

Proof

From (3.1), we can find

$$\begin{aligned} \mathbb {E}\Big [\Big |I_1(f_{1,\varepsilon })\Big |^2\Big ]=\Big (V_1^{(1)}(\varepsilon )+V_2^{(1)}(\varepsilon )+V_3^{(1)}(\varepsilon )\Big ), \end{aligned}$$

(3.12)

where $V_i^{(1)}(\varepsilon )=2\int _{D_i}\langle f_{1,\varepsilon ,s_1,r_1},f_{1,\varepsilon ,s_2,r_2}\rangle _{\mathcal {H}^d}dr_1dr_2ds_1ds_2$ for $i=1, 2, 3$, and $\langle f_{1,\varepsilon ,s_1,r_1},f_{1,\varepsilon ,s_2,r_2}\rangle _{\mathcal {H}^d}$ was defined in (3.4). Then we have

$$\begin{aligned} 0\le V_i^{(1)}(\varepsilon )\le V_i(\varepsilon ). \end{aligned}$$

(3.13)

Combining (3.13) with (3.7) and (3.8), we can see

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}\Big (V_1^{(1)}(\varepsilon )+V_2^{(1)}(\varepsilon )\Big )=0. \end{aligned}$$

Thus, we only need to consider $\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}V_3^{(1)}(\varepsilon )$ as $\varepsilon \rightarrow 0$.

By (3.1), (3.4) and (3.12) we have

$$\begin{aligned} V_3^{(1)}(\varepsilon )&=2\beta _{1,d}^2\int _{D_3}G^{(1)}_{\varepsilon ,r'-r}(s-r,s'-r')\text {d}r\text {d}s\text {d}r'\text {d}s'\\&=2\beta _{1,d}^2\int _{[0,t]^3}\int _0^{t-(a+b+c)}\mathbbm {1}_{(0,t)}(a+b+c)(\varepsilon +a^{2H})^{-d/2-1}\\&\quad \times (\varepsilon +c^{2H})^{-d/2-1}\mu (a+b,a,c)ds_1\text {d}a\text {d}b\text {d}c\\&=2H(1-2H)\beta _{1,d}^2\int _0^t\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}\int _{[0,1]^2}\mathbbm {1}_{(0,t)}\Big ((b+\varepsilon ^{\frac{1}{2H}}(a+c)\Big )\\&\quad \times \Big (t-b-\varepsilon ^{\frac{1}{2H}}(a+c)\Big )\\&\quad \times \Big [(1+a^{2H})(1+c^{2H})\Big ]^{-d/2-1}ac\Big (b+\varepsilon ^{\frac{1}{2H}}(av_1+cv_2)\Big )^{2H-2}dv_1dv_2\text {d}a\text {d}c\text {d}b. \end{aligned}$$

Note that

$$\begin{aligned} \int _{[0,1]^2}\Big (b+\varepsilon ^{\frac{1}{2H}}(av_1+cv_2)\Big )^{2H-2}\text {d}v_1\text {d}v_2=b^{2H-2}+O(\varepsilon ^{\frac{1}{2H}}(a+c)) \end{aligned}$$

and

$$\begin{aligned}&\int _{[0,1]^2}\Big (t-b-\varepsilon ^{\frac{1}{2H}}(a+c)\Big ) \Big [(1+a^{2H})(1+c^{2H})\Big ]^{-d/2-1}\\&\quad ac \Big (b+\varepsilon ^{\frac{1}{2H}}(av_1+cv_2)\Big )^{2H-2}\text {d}v_1\text {d}v_2\\&\quad =(t-b)b^{2H-2}ac\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-d/2-1}\\&\quad +O\left( \varepsilon ^{\frac{1}{2H}}(a+c)ac \Big [(1+a^{2H})(1+c^{2H})\Big ]^{-d/2-1}\right) . \end{aligned}$$

Similar to (3.9), we get

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}V^{(1)}_3(\varepsilon )=\lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}\widetilde{V}^{(1)}_3(\varepsilon ), \end{aligned}$$

where

$$\begin{aligned} \widetilde{V}_3^{(1)}(\varepsilon )&=2H(1-2H)\beta _{1,d}^2\int _{(\log \frac{1}{\varepsilon })^{-1}}^t\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}\int _{[0,1]^2} \mathbbm {1}_{(0,t)}\Big ((b+\varepsilon ^{\frac{1}{2H}}(a+c)\Big )\\&\quad \Big (t-b-\varepsilon ^{\frac{1}{2H}}(a+c)\Big )\\&\quad \times \Big [(1+a^{2H})(1+c^{2H})\Big ]^{-d/2-1}ac\Big (b+\varepsilon ^{\frac{1}{2H}}(av_1+cv_2)\Big )^{2H-2}\text {d}v_1\text {d}v_2\text {d}a\text {d}c\text {d}b. \end{aligned}$$

According to (3.10) and (3.11), we can find that

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}\widetilde{V}^{(1)}_3(\varepsilon )\\&=2H(1-2H)\beta ^2_{1,d}\lim _{\varepsilon \rightarrow 0}\Big (\log 1/\varepsilon \Big )^{2H-1}\int _{(\log \frac{1}{\varepsilon })^{-1}}^t(t-b)b^{2H-2}\text {d}b\\&\quad \times \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\Big )^{2H-1}\int _{[0,t\varepsilon ^{-\frac{1}{2H}}]^2}ac\Big [(1+a^{2H})(1+c^{2H})\Big ]^{-\frac{d}{2}-1}\text {d}a\text {d}c\\&=H(1-2H)\frac{2}{(2\pi )^d}\times \frac{t}{1-2H}\times \frac{t^{2-4H}}{(1-2H)^2}=\sigma ^2, \end{aligned}$$

where we use $\beta _{1,d}^2=\frac{1}{(2\pi )^d}$ in the second equality.

Thus,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{2H-1}V^{(1)}_3(\varepsilon )=\sigma ^2. \end{aligned}$$

$\square$

Proof of Theorem 1.3

By Lemmas 3.1–3.2 and

$$\begin{aligned} \widehat{\alpha }^{(1)}_{t,\varepsilon }(0)=I_1(f_{1,\varepsilon })+\sum _{q=2}^\infty I_{2q-1}(f_{2q-1,\varepsilon }), \end{aligned}$$

we can see

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbb {E}\Big [\Big |\Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{H-1/2}\sum _{q=2}^\infty I_{2q-1}(f_{2q-1,\varepsilon })\Big |^2\Big ]=0. \end{aligned}$$

Since $I_1(f_{1,\varepsilon })$ is Gaussian, we have, as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{H-1/2}I_1(f_{1,\varepsilon })\overset{law}{\rightarrow }N(0,\sigma ^2). \end{aligned}$$

Thus,

$$\begin{aligned} \Big (\varepsilon ^{-\frac{1}{H}}\log 1/\varepsilon \Big )^{H-1/2}\widehat{\alpha }^{(1)}_{t,\varepsilon }(0)\overset{law}{\rightarrow }N(0,\sigma ^2), \end{aligned}$$

as $\varepsilon \rightarrow 0$. This completes the proof. $\square$