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Weak convergence of the complex fractional Brownian motion

Abstract

In this paper, we obtain two approximations in law of the complex fractional Brownian motion by processes constructed from a Poisson process and a Lévy process, respectively.

Introduction

Statistics and econometrics have made great progress in the last two decades. It became necessary to study the distributions or the asymptotic distributions of some complex statistics, so the weak convergence of a stochastic process has been widely studying as an important subject of modern probability theory.

Kac [13] described the solution of the telegrapher’s equation in terms of a Poisson process. Later, Stroock [18] gave that the law of the continuous processes \(\{X_{\varepsilon}(t), t\geq0\}\) given by

$$ X_{\epsilon}(t)=\varepsilon \int_{0}^{\frac{t}{{\varepsilon }^{2}}}(-1)^{N(r)}\,dr= \frac{1}{\varepsilon} \int_{0}^{t}(-1)^{N(\frac {r}{{\varepsilon}^{2}})}\,dr,\quad t\geq0, $$
(1)

where \(\{N(t),t \geq0\}\) is a standard Poisson process, weakly converges when ε tends to zero, in the Banach space \(\mathcal{C}([0,T])\) of continuous functions on \([0, T]\), to the law of a standard Brownian motion.

This result of Stroock [18] has been extended to obtain approximations of other processes such as, among others: Brownian sheet (cf. Bardina and Jolis [5]), m-dimensional Brownian motion (cf. Bardina and Rovira [7]), fractional Brownian motion (cf. Delgado and Jolis [11], Li and Dai [14]), fractional Brownian sheet (cf. Tudor [19], Bardina et al. [6], Wang et al. [20]) and so on. We can refer to Dai [10], Mishura and Banna [16], Nieminen [12], Ouahra [17], Wang et al. [21] and the references therein for more information about weak convergence.

On the other hand, by the fact \((-1)^{N(r)}=\cos(\pi N(r))=e^{i\pi N(r)}\), equality (1) can be rewritten by

$$ X_{\epsilon}(t)=\frac{1}{\varepsilon} \int_{0}^{t}e^{i\pi N(\frac {r}{{\varepsilon}^{2}})}\,dr,\quad t \geq0. $$
(2)

In this case, \(X_{\epsilon}(t)\) can also be written by Euler’s formula as \(X_{\epsilon}(t)=\operatorname{Re}X_{\epsilon}(t)+i\operatorname{Im}X_{\epsilon}(t)\), where

$$ \operatorname{Re}X_{\epsilon}(t)=\frac{1}{\varepsilon} \int_{0}^{t}\cos\biggl(\pi N\biggl( \frac {r}{{\varepsilon}^{2}}\biggr)\biggr)\,dr \quad \mbox{and}\quad \operatorname{Im} X_{\epsilon}(t)=\frac {1}{\varepsilon} \int_{0}^{t}\sin\biggl(\pi N\biggl( \frac{r}{{\varepsilon}^{2}}\biggr)\biggr)\,dr $$

are the real part and the imaginary part, respectively.

Then, some authors considered the weak convergence to the complex Brownian motion by the angles θ replacing the π, where \(\theta\in(0, 2\pi)\) in equation (2). For example, Bardina [2] and Bardina et al. [4] constructed the process from a standard Poisson process which respectively weakly or strongly converges in law to a complex Brownian motion, and got that the real part and the imaginary part of this process are two independent Brownian motions. Bardina and Bascompte [3] obtained the weak convergence towards two independent Gaussian processes from a Poisson process (see also Bardina and Rovira [7], a d-dimensional Brownian motion of this result). In addition, it is well known that some properties of the Poisson process can be found from a Lévy process (cf. Applebaum [1]), so there are some literature works which research an approximation of a complex Brownian motion from the Lévy process (cf. Bardina and Rovira [8]).

Inspired by all the above works, the purpose of this paper is to research a weak approximation of a complex fractional Brownian motion from a standard Poisson process and from a Lévy process, respectively, by the method in Delgado and Jolis [11].

Let \(\{M_{t}, t\geq0\}\) be a Poisson process of parameter 2. We define \(\{ N_{t}, t\geq0\}\) and \(\{N'_{t}, t\geq0\}\) two other counter processes where, at each jump of M, each of them jumps or does not jump with probability \(\frac{1}{2}\), independently of the jumps of the other process and of its past. In Bardina et al. [4], they proved that N and \(N'\) are Poisson processes of parameter 1 with independent increments on disjoint intervals.

Then, for \(\theta\in(0, \pi)\cup(\pi, 2\pi)\), we consider the first processes \(Z_{\varepsilon}^{\theta}=\{Z^{\theta}_{\varepsilon}(t), t\in [0,1]\}\) with

$$ Z_{\varepsilon}^{\theta}(t)=\frac{2}{\varepsilon}(-1)^{G} \int _{0}^{1}K_{H}(t,r) (-1)^{N_{\frac{2r}{{\varepsilon}^{2}}}'} e^{i\theta N_{\frac{2r}{{\varepsilon}^{2}}}}\,dr,\quad t\in[0,1], $$
(3)

where N and \(N'\) are the processes defined above, G is a random variable independent of N and \(N'\), with Bernoulli distribution of parameter \(\frac{1}{2}\), and

$$ K_{H}(t,r)=c_{H}r^{\frac{1}{2}-H} \int_{r}^{t}(u-r)^{H-\frac{3}{2}}u^{H-\frac {1}{2}} \,du $$
(4)

which is the kernel of fractional Brownian motion with \(H \in(\frac {1}{2},1)\) and \(c_{H}\) is the following normalizing constant:

$$c_{H}= \biggl[\frac{H(2H-1)}{\beta(2-2H,H-\frac{1}{2})} \biggr]^{\frac{1}{2}} $$

(cf. Mandelbrot and Van Ness [15]).

And consider the other processes \(Y_{\varepsilon}^{\theta}=\{Y^{\theta }_{\varepsilon}(t), t\in[0,1]\}\) from a Lévy process with

$$ Y_{\varepsilon}^{\theta}(t) =\frac{2}{\varepsilon}c(\theta) \int_{0}^{1} K_{H}(t,r)e^{i\theta X_{\frac {2r}{{\varepsilon}^{2}}}} \,dr,\quad t\geq0, $$
(5)

where \(\{X_{s}, s\geq0\}\) is a Lévy process with Lévy exponent \(\psi_{X}(u)\), \(c(\theta)=\sqrt{\frac{\|\psi_{X}(\theta)\|^{2}}{2\operatorname{Re}[\psi_{X}(\theta)]}}\) is a constant depending on θ, \(\theta\in(0, 2\pi)\), and \(K_{H}(t,r)\) is defined in (4).

It is well known that Lévy exponent \(\psi_{X}(u)\) can be expressed by the Lévy–Khinchine formula as follows:

$$\psi_{X}(u)=-aiu+\frac{1}{2}{\sigma}^{2}u^{2}- \int_{R\setminus\{0\} }\bigl(e^{iux}-1-iuxI_{|x|< 1}\bigr) \eta(dx), $$

where \(a\in R\), \(\sigma\geq0\), and η is a Lévy measure, that is, \(\int_{R\setminus\{0\}} \min\{x^{2},1\}\eta(dx)<\infty\). For the sake of simplicity, let

$$ a(u):=\operatorname{Re}\psi_{X}(u)=\frac{1}{2}{ \sigma}^{2}u^{2}- \int_{R\setminus\{0\} }\bigl(\cos(ux)-1\bigr)\eta(dx) $$
(6)

and

$$ b(u):=\operatorname{Im}\psi_{X}(u)=-au- \frac{1}{2}{\sigma}^{2}u^{2}- \int_{R\setminus\{0\} }\bigl(\sin(ux)-uxI_{|x|< 1}\bigr)\eta(dx). $$
(7)

In addition, denote \(\phi_{X_{t}}(u)=\mathrm{E}(e^{iuX_{t}})=e^{-t\psi _{X}(u)}\) as the characteristic function of a Lévy process. According to (6) and (7), it is easy to get

$$ \bigl\Vert \phi_{X_{t}}(u) \bigr\Vert =e^{-ta(u)}. $$
(8)

The one aim of this paper is to extend the result in Bardina et al. [4] to the case of the complex fractional Brownian motion from the unique standard Poisson process and a sequence of independent random variables with common distribution Bernoulli \(\frac{1}{2}\), that is:

Theorem 1.1

Let \(\{P_{\varepsilon}^{1}, \varepsilon>0\}\) be the family of laws of the processes \(Z_{\varepsilon}^{\theta}\) given by (3) in the Banach space \(\mathcal{C}([0,1], \mathbb{C})\). Then \(P_{\varepsilon }^{1}\) converges weakly as ε tends to zero to the law \(P^{\theta}\) in the Banach space \(\mathcal{C}([0,1], \mathbb{C})\) of a complex fractional Brownian motion \(Z=\{Z(t), t\in[0,1]\}\):

$$ Z(t)= \int_{0}^{1}K_{H}(t,r)\,dW(r)= \int_{0}^{1}K_{H}(t,r) \,dW^{1}(r)+i \int_{0}^{1}K_{H}(t,r) \,dW^{2}(r), $$
(9)

where \(W(r)=W^{1}(r)+iW^{2}(r)\) is a complex Brownian motion, \(W^{1}(r)\) and \(W^{2}(r)\) are two independent standard Brownian motions.

The other aim of this paper is to extend the result of Bardina and Rovira [8] to a slightly more general setting applicable to the complex fractional Brownian motion. So, for our processes \(Y_{\varepsilon}^{\theta}\), we get the following weak convergence of realizations of these processes, which is stated as follows.

Theorem 1.2

The family \(\{P_{\varepsilon}^{2}, \varepsilon>0\}\) of laws of the processes \(Y_{\varepsilon}^{\theta}\) in \(\mathcal{C}([0,1], \mathbb {C})\) converges weakly when ε tends to zero to the family \(P^{\theta}\) of laws of a complex fractional Brownian motion Z.

The rest of the paper is organized as follows. Section 2 is devoted to proving the tightness of the family \(\{P_{\varepsilon}^{1}, \varepsilon >0\}\) and \(\{P_{\varepsilon}^{2}, \varepsilon>0\}\). In Sect. 3, we give the proof of our main result.

In addition, throughout the paper C denotes positive constants, not depending on ε, which may change from one expression to another.

Main lemmas

In order to prove that the family \(P^{1}_{\varepsilon}\) is tight, we need to prove that the laws corresponding to the real part and the imaginary part of processes \(Z_{\varepsilon}^{\theta}\) are tight. Using the Billingsley criterium (see Billingsley [9]) and that our processes are null on the origin, it suffices to prove the following.

Lemma 2.1

For any \(t>s\), \(\varepsilon>0 \), there exists a constant C such that

$$ \sup_{\varepsilon} \bigl(\mathrm{E}\bigl( \operatorname{Re}Z_{\varepsilon}^{\theta }(t)-\operatorname{Re}Z_{\varepsilon}^{\theta}(s) \bigr)^{4}+ \mathrm{E}\bigl(\operatorname{Im}Z_{\varepsilon}^{\theta}(t)- \operatorname{Im}Z_{\varepsilon}^{\theta }(s)\bigr)^{4} \bigr) \leq C(t-s)^{4H}. $$
(10)

Proof

From the definition and the independence of N and \(N'\), it is easy to calculate (see Bardina et al. [4]) that, for any \(0 \leq x_{1} \leq x_{2}\),

$$ \mathrm{E}\bigl[(-1)^{N_{x_{2}}'-N_{x_{1}}'}e^{i{\theta }(N_{x_{2}}-N_{x_{1}})} \bigr]=e^{-2(x_{2}-x_{1})}. $$
(11)

Following the representation of complex fractional Brownian motions \(Z_{\varepsilon}^{\theta}\), the real part and the imaginary part can be written as follows:

$$ \operatorname{Re}Z_{\varepsilon}^{\theta}(t)= \frac{2}{\varepsilon}(-1)^{G} \int _{0}^{1}K_{H}(t,r) (-1)^{N_{\frac{2r}{{\varepsilon}^{2}}}'} \cos(\theta N_{\frac{2r}{{\varepsilon}^{2}}})\,dr $$
(12)

and

$$ \operatorname{Im}Z_{\varepsilon}^{\theta}(t)= \frac{2}{\varepsilon}(-1)^{G} \int _{0}^{1}K_{H}(t,r) (-1)^{N_{\frac{2r}{{\varepsilon}^{2}}}'} \sin(\theta N_{\frac{2r}{{\varepsilon}^{2}}})\,dr, $$
(13)

respectively. Then \(Z_{\varepsilon}^{\theta}(t)=\operatorname{Re}Z_{\varepsilon }^{\theta}(t)+i\operatorname{Im}Z_{\varepsilon}^{\theta}(t)\). Furthermore, the increments of the real part and the imaginary part of the processes \(Z_{\varepsilon}^{\theta}\) can be expressed as follows:

$$\begin{aligned} \operatorname{Re}Z_{\varepsilon}^{\theta}(t)-\operatorname{Re}Z_{\varepsilon}^{\theta}(s)&= \frac {2}{\varepsilon}(-1)^{G} \int_{0}^{1}\bigl(K_{H}(t,r)-K_{H}(s,r) \bigr) (-1)^{N_{\frac{2r}{{\varepsilon }^{2}}}'}\cos(\theta N_{\frac{2r}{{\varepsilon}^{2}}})\,dr \\ &=\frac{2}{\varepsilon}(-1)^{G} \int_{0}^{1}\Delta K_{H}(t,s,r) (-1)^{N_{\frac{2r}{{\varepsilon}^{2}}}'}\cos(\theta N_{\frac {2r}{{\varepsilon}^{2}}})\,dr, \end{aligned}$$
(14)

and

$$\begin{aligned} \operatorname{Im}Z_{\varepsilon}^{\theta}(t)-\operatorname{Im}Z_{\varepsilon}^{\theta}(s)&= \frac {2}{\varepsilon}(-1)^{G} \int_{0}^{1}\bigl(K_{H}(t,r)-K_{H}(s,r) \bigr) (-1)^{N_{\frac{2r}{{\varepsilon }^{2}}}'}\sin(\theta N_{\frac{2r}{{\varepsilon}^{2}}})\,dr \\ &=\frac{2}{\varepsilon}(-1)^{G} \int_{0}^{1}\Delta K_{H}(t,s,r) (-1)^{N_{\frac{2r}{{\varepsilon}^{2}}}'}\sin(\theta N_{\frac {2r}{{\varepsilon}^{2}}})\,dr, \end{aligned}$$
(15)

where \(\Delta K(t,s,r)=K_{H}(t,r)-K_{H}(s,r)\).

Considering

$$\begin{aligned} &2(\cos x_{1}\cos x_{2}\cos x_{3}\cos x_{4}+\sin x_{1}\sin x_{2}\sin x_{3}\sin x_{4}) \\ &\quad =\cos(x_{4}-x_{3})\cos(x_{2}- x_{1})+\cos(x_{4}+x_{3})\cos(x_{2}+ x_{1}) \end{aligned}$$
(16)

and the independent increments of \(N'\), we can get equality (17):

$$\begin{aligned} (-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{2}}{{\varepsilon}^{2}}}' +N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'} &= (-1)^{(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac{2r_{1}}{{\varepsilon }^{2}}}')+(N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'-N_{\frac{2r_{3}}{{\varepsilon}^{2}}}') +2(N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon }^{2}}}')} \\ &=(-1)^{(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac{2r_{1}}{{\varepsilon }^{2}}}')+(N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'-N_{\frac{2r_{3}}{{\varepsilon}^{2}}}')}. \end{aligned}$$
(17)

Therefore, the left-hand side of inequality (10) can be calculated as follows:

$$\begin{aligned} &\mathrm{E}\bigl(\operatorname{Re}Z_{\varepsilon}^{\theta}(t)- \operatorname{Re}Z_{\varepsilon}^{\theta}(s)\bigr)^{4}+ \mathrm{E}\bigl(\operatorname{Im}Z_{\varepsilon}^{\theta}(t)- \operatorname{Im}Z_{\varepsilon}^{\theta }(s)\bigr)^{4} \\ &\quad =\frac{16}{{\varepsilon}^{4}}\mathrm{E} \Biggl[(-1)^{4G} \int_{[0,1]^{4}}\prod_{i=1}^{4} \Delta K_{H}(t,s,r_{i}) (-1)^{N_{\frac {2r_{1}}{{\varepsilon}^{2}}}' +N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon }^{2}}}'+N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'} \\ &\qquad {} \times\cos(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\cos(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}) \cos(\theta N_{\frac{2r_{3}}{{\varepsilon}^{2}}})\cos(\theta N_{\frac {2r_{4}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &\qquad {} +\frac{16}{{\varepsilon}^{4}}\mathrm{E} \Biggl[(-1)^{4G} \int_{[0,1]^{4}}\prod_{i=1}^{4} \Delta K_{H}(t,s,r_{i}) (-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}' +N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon }^{2}}}'+N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'} \\ &\qquad {} \times\sin(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\sin(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}) \sin(\theta N_{\frac{2r_{3}}{{\varepsilon}^{2}}})\sin(\theta N_{\frac {2r_{4}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &\quad =\frac{16}{{\varepsilon}^{4}}\times4! \mathrm{E} \Biggl[(-1)^{4G} \int_{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\qquad {} \times(-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}' +N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon }^{2}}}'+N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'} \\ &\qquad {} \times\bigl[\cos(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\cos(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}) \cos(\theta N_{\frac{2r_{3}}{{\varepsilon}^{2}}})\cos(\theta N_{\frac {2r_{4}}{{\varepsilon}^{2}}}) \\ &\qquad {} +\sin(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\sin(\theta N_{\frac {2r_{2}}{{\varepsilon}^{2}}}) \sin( \theta N_{\frac{2r_{3}}{{\varepsilon}^{2}}})\sin(\theta N_{\frac {2r_{4}}{{\varepsilon}^{2}}})\bigr]\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &\quad =\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \mathrm{E} \Biggl[ \int _{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\qquad {} \times(-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{2}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{4}}{{\varepsilon}^{2}}}'} \\ &\qquad {} \times\cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}}) \cos( \theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{1}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &\qquad {} +\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \mathrm{E} \Biggl[ \int _{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\qquad {} \times(-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{2}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{4}}{{\varepsilon}^{2}}}'} \\ &\qquad {} \times\cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}+\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}}) \cos( \theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}+\theta N_{\frac {2r_{1}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &\quad :=I_{1}+I_{2}, \end{aligned}$$
(18)

where

$$\begin{aligned} I_{1}&=\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \mathrm{E} \Biggl[ \int _{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times(-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{2}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{4}}{{\varepsilon}^{2}}}'} \\ &\quad {} \times\cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}})\cos( \theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \end{aligned}$$
(19)

and

$$\begin{aligned} I_{2}&=\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \mathrm{E} \Biggl[ \int _{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times(-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{2}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac {2r_{4}}{{\varepsilon}^{2}}}'} \\ &\quad {} \times\cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}+\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}})\cos( \theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}+\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr]. \end{aligned}$$
(20)

According to equality (11) and \(|\cos{\theta x}|\leq |e^{i\theta x}|\), it is easy to get

$$ \mathrm{E}(-1)^{N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}'}e^{i\theta(N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}})} =e^{-2(\frac{2r_{2}}{{\varepsilon}^{2}}-\frac{2r_{1}}{{\varepsilon }^{2}})}=e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}}. $$
(21)

Then

$$\begin{aligned} I_{1}&=\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \mathrm{E} \Biggl[ \int _{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times (-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{2}}{{\varepsilon }^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'} \\ &\quad {} \times\cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}})\cos( \theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &=\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \int_{[0,1]^{4}}\mathbf {1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times\mathrm{E} \bigl[(-1)^{N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}'}\cos(\theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \bigr] \\ &\quad {} \times\mathrm{E} \bigl[(-1)^{N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{3}}{{\varepsilon}^{2}}}'} \cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}}) \bigr]\bigotimes _{i=1}^{4}\,dr_{i} \\ &\leq\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \Biggl\{ \int _{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2}\}}\prod _{i=1}^{2}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times\mathrm{E}\bigl[(-1)^{N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}'}\cos(\theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigr]\,dr_{1} \,dr_{2} \Biggr\} ^{2} \\ &\leq\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \Biggl\{ \int _{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2}\}}\prod _{i=1}^{2}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times\mathrm{E}\bigl[(-1)^{N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}'}e^{i\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}- N_{\frac{2r_{1}}{{\varepsilon}^{2}}})}\bigr] \,dr_{1}\,dr_{2} \Biggr\} ^{2} \\ &= \frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \Biggl\{ \int _{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2}\}}\prod _{i=1}^{2}\Delta K_{H}(t,s,r_{i})e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1}\,dr_{2} \Biggr\} ^{2}. \end{aligned}$$
(22)

Using the inequality \(|ab|\leq\frac{1}{2}(a^{2}+b^{2})\), the last expression of (22) is easily bounded by

$$\begin{aligned} &\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2}\times\frac{1}{2^{2}} \biggl\{ \int_{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2}\}}\Delta K_{H}^{2}(t,s,r_{1})e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1}\,dr_{2} \\ &\qquad {} + \int_{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2}\}}\Delta K_{H}^{2}(t,s,r_{2})e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1}\,dr_{2} \biggr\} ^{2} \\ &\quad = \frac{4!}{2}\times\frac{1}{2^{2}} \biggl\{ \int_{0}^{1}\Delta K_{H}^{2}(t,s,r_{1}) \,dr_{1} \int_{r_{1}}^{1}\frac{4}{{\varepsilon}^{2}}e^{-\frac {4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{2} \\ &\qquad {} + \int_{0}^{1}\Delta K_{H}^{2}(t,s,r_{2}) \,dr_{2} \int_{0}^{r_{2}}\frac {4}{{\varepsilon}^{2}}e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1} \biggr\} ^{2} \\ &\quad \leq\frac{4!}{2}\times\frac{1}{2^{2}} \biggl\{ \int_{0}^{1}\Delta K_{H}^{2}(t,s,r) \,dr \biggr\} ^{2} \end{aligned}$$
(23)

as ε tends to zero. It is well known that

$$ \int_{0}^{1}\bigl(K_{H}(t,r)-K_{H}(s,r) \bigr)^{2}\,dr=E\bigl[B_{H}(t)-B_{H}(s) \bigr]^{2}=(t-s)^{2H}. $$
(24)

Then there exists a constant \(C_{1}\) such that

$$ \mathrm{E}\bigl(\operatorname{Re}Z_{\varepsilon}^{\theta}(t)- \operatorname{Re}Z_{\varepsilon}^{\theta }(s)\bigr)^{4} \leq C_{1}(t-s)^{4H}. $$
(25)

Moreover, considering the fact that

$$\begin{aligned} &\cos(x_{4}+x_{3})\cos(x_{2}+x_{1}) \\ &\quad =\bigl[\cos\bigl((x_{4}-x_{3})+(x_{2}-x_{1})+(2x_{3}+x_{1}-x_{2}) \bigr)\cos(x_{2}+x_{1})\bigr] \\ &\quad =\bigl[\cos\bigl((x_{4}-x_{3})+(x_{2}-x_{1}) \bigr)\bigr] \bigl[\cos(2x_{3}+x_{1}-x_{2}) \cos(x_{2}+x_{1})\bigr] \\ &\qquad {} -\bigl[\sin\bigl((x_{4}-x_{3})+(x_{2}-x_{1}) \bigr)\bigr] \bigl[\sin(2x_{3}+x_{1}-x_{2}) \cos(x_{2}+x_{1})\bigr] \\ &\quad \leq \bigl\vert \bigl[\cos\bigl((x_{4}-x_{3})+(x_{2}-x_{1}) \bigr)\bigr] \bigr\vert + \bigl\vert \bigl[\sin\bigl((x_{4}-x_{3})+(x_{2}-x_{1}) \bigr)\bigr] \bigr\vert \\ &\quad \leq\bigl[ \bigl\vert \cos(x_{4}-x_{3}) \bigr\vert + \bigl\vert \sin(x_{4}-x_{3}) \bigr\vert \bigr] \times\bigl[ \bigl\vert \cos(x_{2}-x_{1}) \bigr\vert + \bigl\vert \sin(x_{2}-x_{1}) \bigr\vert \bigr] \end{aligned}$$
(26)

and using the inequality \(|x+y|\leq\sqrt{2(x^{2}+y^{2})}\), we get

$$\begin{aligned} \bigl\vert \cos(x_{4}-x_{3})+\sin(x_{4}-x_{3}) \bigr\vert &\leq\sqrt{2}\bigl(\cos^{2}(x_{4}-x_{3})+ \sin ^{2}(x_{4}-x_{3})\bigr)^{\frac{1}{2}} \\ &=\sqrt{2} \bigl\vert e^{i(x_{4}-x_{3})} \bigr\vert . \end{aligned}$$
(27)

Then, for the term \(I_{2}\), we have that

$$\begin{aligned} I_{2}&=\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \mathrm{E} \Biggl[ \int _{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times (-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{2}}{{\varepsilon }^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'} \\ &\quad {} \times\cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}+\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}})\cos( \theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}+\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &\leq\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \mathrm{E} \Biggl[ \int _{[0,1]^{4}}\mathbf{1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times (-1)^{N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{2}}{{\varepsilon }^{2}}}'+N_{\frac{2r_{3}}{{\varepsilon}^{2}}}'+N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'} \\ &\quad {} \times\bigl[\cos(\theta N_{\frac{2r_{4}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{3}}{{\varepsilon}^{2}}})+ \sin(\theta N_{\frac{2r_{4}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{3}}{{\varepsilon}^{2}}})\bigr] \\ &\quad {} \times\bigl[\cos(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{1}}{{\varepsilon}^{2}}})+ \sin(\theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigr]\bigotimes _{i=1}^{4}\,dr_{i} \Biggr] \\ &\leq\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \int_{[0,1]^{4}}\mathbf {1}_{\{r_{1} \leq r_{2} \leq r_{3} \leq r_{4}\}}\prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \bigotimes_{i=1}^{4}\,dr_{i} \\ &\quad {} \times\mathrm{E} \bigl[(-1)^{N_{\frac{2r_{4}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{3}}{{\varepsilon}^{2}}}'}\bigl(\cos(\theta N_{\frac{2r_{4}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{3}}{{\varepsilon}^{2}}})+\sin(\theta N_{\frac {2r_{4}}{{\varepsilon}^{2}}}-\theta N_{\frac{2r_{3}}{{\varepsilon}^{2}}})\bigr) \bigr] \\ &\quad {} \times\mathrm{E} \bigl[(-1)^{N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}'}\bigl(\cos(\theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})+\sin(\theta N_{\frac {2r_{2}}{{\varepsilon}^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigr) \bigr] \\ &= \frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \Biggl\{ \int _{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2} \}}\prod _{i=1}^{2}\Delta K_{H}(t,s,r_{i}) \mathrm{E} \bigl[(-1)^{N_{\frac{2r_{2}}{{\varepsilon }^{2}}}'-N_{\frac{2r_{1}}{{\varepsilon}^{2}}}'} \\ &\quad {} \times\bigl(\cos(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-\theta N_{\frac {2r_{1}}{{\varepsilon}^{2}}})+ \sin(\theta N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\bigr) \bigr] \,dr_{1}\,dr_{2} \Biggr\} ^{2} \\ &\leq\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2} \Biggl\{ \int _{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2} \}}\prod _{i=1}^{2}\Delta K_{H}(t,s,r_{i}) \\ &\quad {} \times \mathrm{E}\bigl[(-1)^{N_{\frac{2r_{2}}{{\varepsilon}^{2}}}'-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}'}e^{i\theta( N_{\frac{2r_{2}}{{\varepsilon}^{2}}}- N_{\frac{2r_{1}}{{\varepsilon}^{2}}})}\bigr] \,dr_{1}\,dr_{2} \Biggr\} ^{2} \\ &\leq\frac{16}{{\varepsilon}^{4}}\times\frac{4!}{2}\times\frac {1}{2^{2}} \biggl\{ \int_{[0,1]^{2}}\mathbf{1}_{\{r_{1} \leq r_{2}\}}\Delta K_{H}^{2}(t,s,r_{1})e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1}\,dr_{2} \biggr\} ^{2}. \end{aligned}$$
(28)

Similar to the proof of the term \(I_{1}\), using equality (23), we easily get the bound of the last integral of (28) as follows:

$$ \frac{4!}{2}\times\frac{1}{2^{2}} \biggl\{ \int_{0}^{1}\Delta K_{H}^{2}(t,s,r) \,dr \biggr\} ^{2}\leq C_{2} (t-s)^{4H}, $$
(29)

where \(C_{2}\) is a constant. Combining (25) and (29), we obtain that there exists a constant C such that

$$ \sup_{\varepsilon}\bigl(\mathrm{E}\bigl( \operatorname{Re}Z_{\varepsilon}^{\theta }(t)-\operatorname{Re}Z_{\varepsilon}^{\theta}(s) \bigr)^{4}+ \mathrm{E}\bigl(\operatorname{Im}Z_{\varepsilon}^{\theta}(t)- \operatorname{Im}Z_{\varepsilon}^{\theta }(s)\bigr)^{4}\bigr) \leq C(t-s)^{4H}. $$
(30)

This completes the proof. □

Next, we consider the tightness of the processes \(Y_{\varepsilon }^{\theta}\).

Lemma 2.2

For any \(t>s\), \(\varepsilon>0\), there exists a constant C such that

$$ \sup_{\varepsilon} \bigl(\mathrm{E}\bigl[ \operatorname{Re}Y_{\varepsilon}^{\theta }(t)-\operatorname{Re}Y_{\varepsilon}^{\theta}(s) \bigr]^{4}+\mathrm{E}\bigl[\operatorname{Im}Y_{\varepsilon }^{\theta}(t)- \operatorname{Im}Y_{\varepsilon}^{\theta}(s)\bigr]^{4} \bigr) \leq C(t-s)^{4H}. $$
(31)

Proof

For the complex function \(a(t)=e^{i\theta t}\), we can obtain by using fundamental operations:

$$ (\cos\theta t-\cos\theta s)^{4}+(\sin\theta t-\sin\theta s)^{4}\leq \bigl\vert a(t)-a(s) \bigr\vert ^{4}. $$
(32)

By (32), we have

$$\begin{aligned} &\mathrm{E}\bigl[\operatorname{Re}Y_{\varepsilon}^{\theta}(t)- \operatorname{Re}Y_{\varepsilon}^{\theta }(s)\bigr]^{4}+ \mathrm{E}\bigl[\operatorname{Im}Y_{\varepsilon}^{\theta}(t)- \operatorname{Im}Y_{\varepsilon }^{\theta}(s)\bigr]^{4} \\ &\quad \leq\mathrm{E} \bigl\vert Y_{\varepsilon}^{\theta}(t)-Y_{\varepsilon}^{\theta }(s) \bigr\vert ^{4} \\ &\quad =c^{4}(\theta)\frac{2^{4}}{\varepsilon^{4}}\mathrm{E} \biggl( \int_{0}^{1} \Delta K_{H}(t,s,r_{1})e^{i\theta X_{\frac{2r_{1}}{{\varepsilon}^{2}}}} \,dr_{1} \int _{0}^{1}\Delta K_{H}(t,s,r_{2})e^{-i\theta X_{\frac{2r_{2}}{{\varepsilon }^{2}}}} \,dr_{2} \biggr)^{2} \\ &\quad =c^{4}(\theta)\frac{2^{4}}{\varepsilon^{4}} \Biggl( \int_{[0,1]^{4}} \prod_{i=1}^{4} \Delta K_{H}(t,s,r_{i})\mathrm{E}\bigl[e^{i\theta(X_{\frac {2r_{1}}{{\varepsilon}^{2}}}- X_{\frac{2r_{2}}{{\varepsilon}^{2}}})}e^{i\theta (X_{\frac{2r_{3}}{{\varepsilon}^{2}}}- X_{\frac{2r_{4}}{{\varepsilon }^{2}}})} \bigr]\bigotimes_{i=1}^{4} \,dr_{i} \Biggr) \\ &\quad =c^{4}(\theta)\frac{2^{4}}{\varepsilon^{4}}\times4! \int_{[0,1]^{4}}\mathbf {1}_{\{r_{1}\leq r_{2}\leq r_{3} \leq r_{4}\}} \prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \\ &\qquad {} \times\mathrm{E} \bigl[e^{i\theta(X_{\frac{2r_{2}}{{\varepsilon}^{2}}}- X_{\frac{2r_{1}}{{\varepsilon}^{2}}})}e^{i\theta(X_{\frac {2r_{4}}{{\varepsilon}^{2}}}- X_{\frac{2r_{3}}{{\varepsilon}^{2}}})} \bigr] \bigotimes_{i=1}^{4}\,dr_{i}. \end{aligned}$$
(33)

Because the process X has independent increments, we have

$$ \mathrm{E} \bigl[e^{i\theta[(X_{\frac{2r_{4}}{{\varepsilon}^{2}}}-X_{\frac {2r_{3}}{{\varepsilon}^{2}}})+(X_{\frac{2r_{2}}{{\varepsilon}^{2}}}- X_{\frac {2r_{1}}{{\varepsilon}^{2}}})]} \bigr] = \bigl\Vert \phi_{X_{\frac{2r_{4}}{{\varepsilon}^{2}}}-X_{\frac{2r_{3}}{{\varepsilon }^{2}}}}( \theta) \bigr\Vert \bigl\Vert \phi_{X_{\frac{2r_{2}}{{\varepsilon}^{2}}}-X_{\frac {2r_{1}}{{\varepsilon}^{2}}}}(\theta) \bigr\Vert . $$
(34)

So, the last expression of (33) is easily bounded by

$$\begin{aligned} &c^{4}(\theta)\frac{2^{4}}{\varepsilon^{4}}\times4! \int_{[0,1]^{4}}\mathbf{1}_{\{ r_{1}\leq r_{2}\leq r_{3} \leq r_{4}\}} \prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) \bigl\Vert \phi _{X_{\frac{2r_{2}}{{\varepsilon}^{2}}}-X_{\frac{2r_{1}}{{\varepsilon }^{2}}}}(\theta) \bigr\Vert \\ &\quad {} \times \bigl\Vert \phi_{X_{\frac{2r_{4}}{{\varepsilon}^{2}}}-X_{\frac {2r_{3}}{{\varepsilon}^{2}}}}(\theta) \bigr\Vert \bigotimes_{i=1}^{4}\,dr_{i}. \end{aligned}$$
(35)

According to equality (8) of the Lévy process, we get \(\|\phi_{X_{u}-X_{v}}(\theta)\|=e^{-(u-v)a(\theta)}\). Thus, expression (35) is equal to

$$\begin{aligned} &c^{4}(\theta)\frac{2^{4}}{\varepsilon^{4}}\times4! \int_{[0,1]^{4}}\mathbf {1}_{\{r_{1}\leq r_{2}\leq r_{3} \leq r_{4}\}} \prod _{i=1}^{4}\Delta K_{H}(t,s,r_{i}) e^{-\frac{2(r_{2}-r_{1})}{{\varepsilon}^{2}}a(\theta)} \\ &\qquad {} \times e^{-\frac{2(r_{4}-r_{3})}{{\varepsilon}^{2}}a(\theta)}\bigotimes_{i=1}^{4} \,dr_{i} \\ &\quad =c^{4}(\theta)\frac{2^{3}}{\varepsilon^{4}}\times4! \Biggl( \int_{[0,1]^{2}}\mathbf {1}_{\{r_{1}\leq r_{2}\}} \prod _{i=1}^{2}\Delta K_{H}(t,s,r_{i}) e^{-\frac {2(r_{2}-r_{1})}{{\varepsilon}^{2}}a(\theta)}\,dr_{1}\,dr_{2} \Biggr)^{2} \\ &\quad \leq c^{4}(\theta)\frac{2^{4}}{\varepsilon^{4}}\times4! \biggl( \int _{[0,1]^{2}}\mathbf{1}_{\{r_{1}\leq r_{2}\}} \Delta K_{H}^{2}(t,s,r_{1}) e^{-\frac {2(r_{2}-r_{1})}{{\varepsilon}^{2}}a(\theta)} \,dr_{1}\,dr_{2} \\ &\qquad {} + \int_{[0,1]^{2}}\mathbf{1}_{\{r_{1}\leq r_{2}\}} \Delta K_{H}^{2}(t,s,r_{2}) e^{-\frac{2(r_{2}-r_{1})}{{\varepsilon}^{2}}a(\theta)} \,dr_{1}\,dr_{2} \biggr)^{2} \\ &\quad = c^{4}(\theta)2\times4! \biggl( \int_{0}^{1} \Delta K_{H}^{2}(t,s,r_{1}) \biggl( \int _{r_{1}}^{1}\frac{2}{\varepsilon^{2}}e^{-\frac{2(r_{2}-r_{1})}{{\varepsilon }^{2}}a(\theta)} \,dr_{2} \biggr)\,dr_{1} \\ &\qquad {} + \int_{0}^{1} \Delta K_{H}^{2}(t,s,r_{2}) \biggl( \int_{0}^{r_{2}}\frac {2}{\varepsilon^{2}}e^{-\frac{2(r_{2}-r_{1})}{{\varepsilon}^{2}}a(\theta )} \,dr_{1} \biggr)\,dr_{2} \biggr)^{2} \\ &\quad \leq4c^{4}(\theta)4! \biggl( \int_{0}^{1}\Delta K_{H}^{2}(t,s,r) \,dr \biggr)^{2}. \end{aligned}$$
(36)

By equality (24), there exists a constant C such that the last integral of (36) can be bounded by \(C(t-s)^{4H}\). The proof has been completed. □

For the proof of Theorem 1.1, we need the following lemma.

Lemma 2.3

For any \(f(r)\in L^{2}([0,1])\) and \(\varepsilon>0\), let

$$F(t)=\frac{2}{\varepsilon}(-1)^{G} \int_{0}^{t}f(r) (-1)^{N'_{\frac {2r}{{\varepsilon}^{2}}}}e^{i\theta N_{\frac{2r}{{\varepsilon}^{2}}}} \,dr,\quad t\in[0,1]. $$

Then there exists a constant C such that \(\mathrm{E}[\operatorname{Re}F(t)]^{2}\leq C\int_{0}^{t}f^{2}(r)\,dr\) and \(\mathrm{E}[\operatorname{Im}F(t)]^{2}\leq C\int_{0}^{t}f^{2}(r)\,dr\).

Proof

Following the definition of \(F(t)\), we get

$$\begin{aligned} \mathrm{E}\bigl[\operatorname{Re}F(t)\bigr]^{2}&=\mathrm{E} \biggl[ \frac{2}{\varepsilon}(-1)^{G} \int _{0}^{t}f(r) (-1)^{N'_{\frac{2r}{{\varepsilon}^{2}}}}\cos(\theta N_{\frac {2r}{{\varepsilon}^{2}}})\,dr \biggr]^{2} \\ &=\mathrm{E} \biggl[\frac{4}{{\varepsilon}^{2}}(-1)^{2G} \int _{0}^{t}f(r_{1}) (-1)^{N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}}\cos(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\,dr_{1} \\ &\quad {} \times \int_{0}^{t}f(r_{2}) (-1)^{N'_{\frac{2r_{2}}{{\varepsilon}^{2}}}} \cos(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}})\,dr_{2} \biggr] \\ &=\mathrm{E} \biggl[\frac{4}{{\varepsilon}^{2}}\times2! \int_{[0,t]^{2}}\mathbf {1}_{\{r_{1}\leq r_{2}\}}f(r_{1})f(r_{2}) (-1)^{N'_{\frac{2r_{1}}{{\varepsilon }^{2}}}+N'_{\frac{2r_{2}}{{\varepsilon}^{2}}}} \\ &\quad {} \times\cos(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\cos(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}}) \,dr_{1} \,dr_{2} \biggr]. \end{aligned}$$
(37)

Because

$$ (-1)^{N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}+N'_{\frac{2r_{2}}{{\varepsilon}^{2}}}} =(-1)^{(N'_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N'_{\frac{2r_{1}}{{\varepsilon }^{2}}})+2N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}} =(-1)^{(N'_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N'_{\frac{2r_{1}}{{\varepsilon}^{2}}})} $$

and

$$ \cos(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\cos(\theta N_{\frac {2r_{2}}{{\varepsilon}^{2}}}) =\frac{1}{2} \bigl[\cos\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N_{\frac {2r_{1}}{{\varepsilon}^{2}}})+\cos\theta(N_{\frac{2r_{2}}{{\varepsilon }^{2}}}+N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \bigr]. $$

Then equation (37) is equal to

$$\begin{aligned} &\mathrm{E} \biggl[\frac{4}{{\varepsilon}^{2}} \int_{[0,t]^{2}}\mathbf{1}_{\{ r_{1}\leq r_{2}\}}f(r_{1})f(r_{2}) (-1)^{N'_{\frac{2r_{2}}{{\varepsilon }^{2}}}-N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}}\bigl[\cos\theta(N_{\frac {2r_{2}}{{\varepsilon}^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \\ &\qquad {} +\cos\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}+N_{\frac {2r_{1}}{{\varepsilon}^{2}}})\bigr] \,dr_{1} \,dr_{2} \biggr] \\ &\quad =\frac{4}{{\varepsilon}^{2}} \int_{[0,t]^{2}}\mathbf{1}_{\{r_{1}\leq r_{2}\} }f(r_{1})f(r_{2}) \mathrm{E} \bigl[(-1)^{N'_{\frac{2r_{2}}{{\varepsilon }^{2}}}-N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}}\bigl[\cos\theta(N_{\frac {2r_{2}}{{\varepsilon}^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \bigr] \bigr]\,dr_{1} \,dr_{2} \\ &\qquad {} +\frac{4}{{\varepsilon}^{2}} \int_{[0,t]^{2}}\mathbf{1}_{\{r_{1}\leq r_{2}\} }f(r_{1})f(r_{2}) \mathrm{E} \bigl[(-1)^{N'_{\frac{2r_{2}}{{\varepsilon }^{2}}}-N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}}\bigl[\cos\theta(N_{\frac {2r_{2}}{{\varepsilon}^{2}}}+N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \bigr] \bigr]\,dr_{1} \,dr_{2} \\ &\quad :=I_{3}+I_{4}. \end{aligned}$$

Using (21), we get

$$\begin{aligned} I_{3}&=\frac{4}{{\varepsilon}^{2}} \int_{[0,t]^{2}}\mathbf{1}_{\{r_{1}\leq r_{2}\} }f(r_{1})f(r_{2})e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1} \,dr_{2} \\ &=\frac{4}{{\varepsilon}^{2}}\times\frac{1}{2} \biggl[ \int_{[0,t]^{2}}\mathbf {1}_{\{r_{1}\leq r_{2}\}}f^{2}(r_{1})e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon }^{2}}} \,dr_{1} \,dr_{2} \\ &\quad {} + \int_{[0,t]^{2}}\mathbf{1}_{\{r_{1}\leq r_{2}\}}f^{2}(r_{2})e^{-\frac {4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1} \,dr_{2} \biggr] \\ &=\frac{1}{2} \biggl[ \int_{0}^{t}f^{2}(r_{1}) \,dr_{1} \int_{r_{1}}^{t}\frac {4}{{\varepsilon}^{2}}e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{2} + \int_{0}^{t}f^{2}(r_{2}) \,dr_{2} \int_{0}^{r_{2}}\frac{4}{{\varepsilon }^{2}}e^{-\frac{4(r_{2}-r_{1})}{{\varepsilon}^{2}}} \,dr_{1} \biggr] \\ &=\frac{1}{2} \biggl[ \int_{0}^{t}f^{2}(r_{1}) \bigl(1-e^{-\frac {4(t-r_{1})}{{\varepsilon}^{2}}}\bigr)\,dr_{1}+ \int_{0}^{t}f^{2}(r_{2}) \bigl(1-e^{-\frac {4r_{2}}{{\varepsilon}^{2}}}\bigr)\,dr_{2} \biggr] \\ &\leq\frac{1}{2} \biggl[ \int_{0}^{t}f^{2}(r_{1}) \,dr_{1}+ \int _{0}^{t}f^{2}(r_{2}) \,dr_{2} \biggr]= \int_{0}^{t}f^{2}(r)\,dr \end{aligned}$$

and

$$\begin{aligned} I_{4}&=\frac{4}{{\varepsilon}^{2}} \int_{[0,t]^{2}}\mathbf{1}_{\{r_{1}\leq r_{2}\} }f(r_{1})f(r_{2}) \mathrm{E} \bigl[(-1)^{N'_{\frac{2r_{2}}{{\varepsilon }^{2}}}-N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}}\bigl[\cos\theta(N_{\frac {2r_{2}}{{\varepsilon}^{2}}}+N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \bigr] \bigr]\,dr_{1} \,dr_{2} \\ &\leq\frac{4}{{\varepsilon}^{2}}\sqrt{2} \int_{[0,t]^{2}}\mathbf{1}_{\{ r_{1}\leq r_{2}\}}f(r_{1})f(r_{2})\bigl| \mathrm{E} \bigl[(-1)^{N'_{\frac {2r_{2}}{{\varepsilon}^{2}}}-N'_{\frac{2r_{1}}{{\varepsilon}^{2}}}}e^{i\theta (N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}})} \bigr]\bigr|\,dr_{1} \,dr_{2} \\ &\leq\sqrt{2} \int_{0}^{t}f^{2}(r)\,dr \end{aligned}$$

since

$$\begin{aligned} &\cos\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}+N_{\frac {2r_{2}}{{\varepsilon}^{2}}}) \\ &\quad =\cos\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}+2N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \\ &\quad =\cos\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N_{\frac {2r_{1}}{{\varepsilon}^{2}}})\cos(2N_{\frac{2r_{1}}{{\varepsilon}^{2}}})- \sin \theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N_{\frac{2r_{1}}{{\varepsilon }^{2}}})\sin(2N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \\ &\quad \leq \bigl\vert \cos\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N_{\frac {2r_{1}}{{\varepsilon}^{2}}}) \bigr\vert + \bigl\vert \sin\theta(N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \bigr\vert \\ &\quad \leq\sqrt{2} \bigl(\cos^{2}\theta(N_{\frac{2r_{2}}{{\varepsilon }^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}})+ \sin^{2}\theta(N_{\frac {2r_{2}}{{\varepsilon}^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}}) \bigr) \\ &\quad =\sqrt{2} \bigl\vert e^{i\theta(N_{\frac{2r_{2}}{{\varepsilon}^{2}}}-N_{\frac {2r_{1}}{{\varepsilon}^{2}}})} \bigr\vert . \end{aligned}$$

For the term \(\mathrm{E}[\operatorname{Im}F(t)]^{2}\), we have

$$\begin{aligned} \mathrm{E}\bigl[\operatorname{Im}F(t)\bigr]^{2}&=\mathrm{E} \biggl[ \frac{2}{\varepsilon}(-1)^{G} \int _{0}^{t}f(r) (-1)^{N'_{\frac{2r}{{\varepsilon}^{2}}}}\sin(\theta N_{\frac {2r}{{\varepsilon}^{2}}})\,dr \biggr]^{2} \\ &=\mathrm{E} \biggl[\frac{4}{{\varepsilon}^{2}}\times2! \int_{[0,t]^{2}}\mathbf {1}_{\{r_{1}\leq r_{2}\}}f(r_{1})f(r_{2}) (-1)^{N'_{\frac{2r_{1}}{{\varepsilon }^{2}}}+N'_{\frac{2r_{2}}{{\varepsilon}^{2}}}}\sin(\theta N_{\frac {2r_{1}}{{\varepsilon}^{2}}}) \\ &\quad {} \times\sin(\theta N_{\frac{2r_{2}}{{\varepsilon}^{2}}})\,dr_{1} \,dr_{2} \biggr]. \end{aligned}$$

Using the equation

$$ \sin(\theta N_{\frac{2r_{1}}{{\varepsilon}^{2}}})\sin(\theta N_{\frac {2r_{2}}{{\varepsilon}^{2}}})=\frac{1}{2} \bigl(\cos\theta(N_{\frac {2r_{2}}{{\varepsilon}^{2}}}-N_{\frac{2r_{1}}{{\varepsilon}^{2}}})-\cos\theta (N_{\frac{2r_{2}}{{\varepsilon}^{2}}}+N_{\frac{2r_{2}}{{\varepsilon}^{2}}})\bigr), $$

similar to the calculation of \(\mathrm{E}[\operatorname{Re}F(t)]^{2}\), we can get \(\mathrm{E}[\operatorname{Im}F(t)]^{2}\leq C \int_{0}^{t}f^{2}(r)\,dr\). The proof of this lemma is accomplished. □

Weak convergence to the complex fractional Brownian motion

In this section, we give the proof of Theorem 1.1 and Theorem 1.2 by checking that the families of laws of the processes \(Z^{\theta}_{\varepsilon}\) and \(Y^{\theta}_{\varepsilon}\) are tight respectively and that any weakly convergent subsequence converges to the law of the complex fractional Brownian motion.

Proof of Theorem 1.1

Firstly, following Lemma 2.1, we have proved the tightness of the family \(P_{\varepsilon}^{1}\) of laws of the processes \(Z^{\theta}_{\varepsilon}\) by applying Theorem 12.3 of Billingsley [9].

Next we will prove that the family of stochastic processes \(Z^{\theta }_{\varepsilon}\) converges in the sense of finite dimensional distribution function to the process Z. That is, for any integer number \(N\geq1\), considering arbitrary real numbers \(a_{1},\ldots,a_{N} \in\mathbb{R}\) and \(t_{1}, \ldots, t_{N} \in[0,1]\), we have

$$S^{\theta}_{\varepsilon}:=\sum_{k=1}^{N}a_{k}Z_{\varepsilon}^{\theta }(t_{k}) \rightarrow S:=\sum_{k=1}^{N}a_{k}Z(t_{k}) $$

as ε tends to zero. To prove this, the convergence of the corresponding characteristic functions must be checked.

For the sake of simplicity, we denote

$$\begin{aligned} \theta(r)&=\frac{2}{\varepsilon}(-1)^{G+N'_{\frac{2r}{{\epsilon }^{2}}}}e^{i\theta N_{\frac{2r}{{\epsilon}^{2}}}} \\ &=\frac{2}{\varepsilon}(-1)^{G+N_{\frac{2r}{{\epsilon}^{2}}}'} \cos (\theta N_{\frac{2r}{{\epsilon}^{2}}}) +i \frac{2}{\varepsilon}(-1)^{G+N_{\frac{2r}{{\epsilon}^{2}}}'} \sin (\theta N_{\frac{2r}{{\epsilon}^{2}}}) \\ &:=\operatorname{Re}\theta(r)+i\operatorname{Im}\theta(r). \end{aligned}$$

Note that

$$ S^{\theta}_{\varepsilon}= \int_{0}^{1}K^{*}(r)\theta(r)\,dr= \int _{0}^{1}K^{*}(r)\operatorname{Re} \theta(r)\,dr+i \int_{0}^{1}K^{*}(r)\operatorname{Im} \theta(r)\,dr $$
(38)

and

$$ S= \int_{0}^{1}K^{*}(r)\,dW(r)= \int_{0}^{1}K^{*}(r) \,dW^{1}(r)+i \int_{0}^{1}K^{*}(r) \,dW^{2}(r), $$
(39)

where \(K^{*}(r)=\sum_{k=1}^{N}a_{k}K(t_{k},r)\), \(W(r)=W^{1}(r)+iW^{2}(r)\) is a complex Brownian motion, \(W^{1}(r)\) and \(W^{2}(r)\) are two independent standard Brownian motions.

The function \(K^{*}(r)\in L^{2}([0,1])\) can be approximated by a sequence of step functions of the form

$$ K^{n}(r)=\sum_{i=0}^{m_{n}-1}K^{n}_{i}1_{(r^{n}_{i},r^{n}_{i+1}]}(r), $$
(40)

with \(0=r^{n}_{0}< r^{n}_{1}<\cdots<r^{n}_{m_{n}-1}<r^{n}_{m_{n}}=1\) and \(K^{n}_{i}\), \(i=0,\ldots, m_{n}-1\) being constants that are chosen such that

$$ \int_{0}^{1}\bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\,dr\leq\frac{1}{n} \quad \text{for any }n\in \mathbb{N}. $$
(41)

Define now

$$ S^{n}_{\varepsilon}= \int_{0}^{1}K^{n}(r) \theta_{\varepsilon}(r)\,dr= \int _{0}^{1}K^{n}(r)\operatorname{Re} \theta_{\varepsilon}(r)\,dr+i \int_{0}^{1}K^{n}(r)\operatorname{Im} \theta _{\varepsilon}(r)\,dr $$
(42)

and

$$ S^{n}= \int_{0}^{1}K^{n}(r)\,dW(r)= \int_{0}^{1}K^{n}(r) \,dW^{1}(r)+i \int_{0}^{1}K^{n}(r) \,dW^{2}(r). $$
(43)

By taking \(f(r)=K^{*}(r)-K^{n}(r)\) in Lemma 2.3, we have that there exists a positive constant C, which does not depend on n, such that

$$ \mathrm{E}\bigl[\bigl(\operatorname{Re}S^{\theta}_{\varepsilon}- \operatorname{Re}S^{n}_{\varepsilon}\bigr)^{2}\bigr] \leq C \int_{0}^{1}\bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\,dr\leq C \frac{1}{n} $$
(44)

and

$$ \mathrm{E}\bigl[\bigl(\operatorname{Im}S^{\theta}_{\varepsilon}- \operatorname{Im}S^{n}_{\varepsilon}\bigr)^{2}\bigr] \leq C \int_{0}^{1}\bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\,dr\leq C \frac{1}{n} $$
(45)

for any \(\varepsilon> 0\).

On the other hand, for fixed \(n\in\mathbb{N}\),

$$\begin{aligned} S^{n}_{\varepsilon}&=\sum_{i=0}^{m_{n}-1} \int_{0}^{1}K^{n}_{i} \theta_{\varepsilon }(r)\,dr \\ &=\sum_{i=0}^{m_{n}-1} \int_{0}^{1}K^{n}_{i} \operatorname{Re}\theta_{\varepsilon}(r)\,dr+i\sum_{i=0}^{m_{n}-1} \int_{0}^{1}K^{n}_{i} \operatorname{Im}\theta_{\varepsilon}(r)\,dr \end{aligned}$$

converges in law as ε tends to zero to

$$\begin{aligned} S^{n}&= \int_{0}^{1}K^{n}(r) \,dW^{1}(r)+i \int_{0}^{1}K^{n}(r) \,dW^{2}(r) \\ &=\sum_{i=0}^{m_{n}-1} \int_{0}^{1}K^{n}_{i} \,dW^{1}(r)+i\sum_{i=0}^{m_{n}-1} \int _{0}^{1}K^{n}_{i} \,dW^{2}(r) \end{aligned}$$

due to the result established by Bardina et al. [4]. Then we have the convergence of the corresponding characteristic function: for any \(x\in\mathbb{R}\) and \(n \in\mathbb{N}\),

$$ \mathrm{E}\bigl[e^{ixS^{n}_{\varepsilon}}\bigr] \rightarrow\mathrm{E} \bigl[e^{ixS^{n}}\bigr]\quad \text{as } \varepsilon\rightarrow0. $$
(46)

From Bardina et al. [4], it is easy to get that \(\operatorname{Re}Z^{\theta}_{\varepsilon}(t)\) and \(\operatorname{Im}Z^{\theta }_{\varepsilon}(t)\) are two independent centered Gaussian processes. So, we get that

$$\begin{aligned} & \bigl\vert \mathrm{E}\bigl[e^{i\lambda S_{\varepsilon}^{\theta}}\bigr]- \mathrm{E} \bigl[e^{i\lambda S}\bigr] \bigr\vert \\ &\quad = \bigl\vert \mathrm{E}\bigl[e^{i\lambda(\operatorname{Re}S_{\varepsilon}^{\theta}+i\operatorname{Im}S_{\varepsilon }^{\theta})}\bigr] - \mathrm{E} \bigl[e^{i\lambda(\operatorname{Re}S+i\operatorname{Im}S)}\bigr] \bigr\vert \\ &\quad = \bigl\vert \mathrm{E}\bigl[e^{i\lambda \operatorname{Re}S_{\varepsilon}^{\theta}}e^{-\lambda \operatorname{Im}S_{\varepsilon}^{\theta}}\bigr] - \mathrm{E}\bigl[e^{i\lambda \operatorname{Re}S}e^{-\lambda \operatorname{Im}S}\bigr] \bigr\vert \\ &\quad = \bigl\vert \mathrm{E}\bigl[\bigl(e^{i\lambda \operatorname{Re}S_{\varepsilon}^{\theta}}-e^{i\lambda \operatorname{Re}S} \bigr) e^{-\lambda \operatorname{Im}S_{\varepsilon}^{\theta}}\bigr] \\ &\qquad {} + \mathrm{E}\bigl[e^{i\lambda \operatorname{Re}S} \bigl(e^{-\lambda \operatorname{Im}S_{\varepsilon}^{\theta}}-e^{-\lambda \operatorname{Im}S} \bigr)\bigr] \bigr\vert \\ &\quad \leq \bigl\vert \mathrm{E}\bigl[e^{-\lambda \operatorname{Im}S_{\varepsilon}^{\theta}}\bigr] \bigr\vert I_{5}+ \bigl\vert \mathrm{E} \bigl[e^{i\lambda \operatorname{Re}S}\bigr] \bigr\vert I_{6}, \end{aligned}$$
(47)

where \(I_{5}:=|\mathrm{E}(e^{i\lambda \operatorname{Re}S_{\varepsilon}^{\theta}}-e^{i\lambda \operatorname{Re}S})| \) and \(I_{6}:=|\mathrm{E}(e^{-\lambda \operatorname{Im}S_{\varepsilon}^{\theta}}-e^{-\lambda \operatorname{Im}S})| \).

Using the mean value theorem, there exists \(\xi\in(a,b)\) for \(a\leq b\) such that

$$e^{ib}-e^{ia}= \int_{a}^{b}ie^{ix}dx=ie^{i\xi}(b-a)=e^{i(\frac{\pi }{2}+\xi)}(b-a). $$

Then it is easy to get

$$\begin{aligned} \bigl\vert \mathrm{E}e^{iX_{2}}-\mathrm{E}e^{iX_{1}} \bigr\vert &= \bigl\vert \mathrm {E}\bigl(e^{iX_{2}}-e^{iX_{1}}\bigr) \bigr\vert = \bigl\vert \mathrm{E}e^{i(\frac{\pi}{2}+\xi)}(X_{2}-X_{1}) \bigr\vert \\ &\leq\mathrm{E}\bigl[ \bigl\vert e^{i(\frac{\pi}{2}+\xi)} \bigr\vert \vert X_{2}-X_{1} \vert \bigr]\leq\mathrm{E} \vert X_{2}-X_{1} \vert , \end{aligned}$$

where \(X_{1}\), \(X_{2}\) are two random variables. So, there exists a constant \(C>0\) for the term \(I_{5}\) such that

$$I_{5} \leq \vert \lambda \vert \bigl\{ \mathrm{E} \bigl\vert \operatorname{Re}S_{\varepsilon}^{\theta}-\operatorname{Re}S \bigr\vert \bigr\} \leq C\bigl\{ \mathrm{E} \bigl\vert \operatorname{Re}S_{\varepsilon}^{\theta}- \operatorname{Re}S \bigr\vert \bigr\} . $$

Meanwhile, by (38) and (39), we have

$$\mathrm{E} \bigl\vert \operatorname{Re}S_{\varepsilon}^{\theta}- \operatorname{Re}S \bigr\vert \leq I_{51}+I_{52}+I_{53}, $$

where \(I_{51}=\mathrm{E}|\operatorname{Re}S_{\varepsilon}^{\theta}-\operatorname{Re}S^{n}_{\epsilon}|\), \(I_{52}=\mathrm{E}|\operatorname{Re}S^{n}_{\epsilon}-\operatorname{Re}S^{n}|\), and \(I_{53}=\mathrm{E}|\operatorname{Re}S^{n}-\operatorname{Re}S|\).

Using the Schwarz inequality \((\mathrm{E}|\xi\eta|)^{2} \leq\mathrm {E}|\xi|^{2}\mathrm{E}|\eta|^{2}\) and (44), we can get, for any \(\varepsilon>0\) and \(n \in\mathbb{N}\),

$$\begin{aligned} I_{51} &=\mathrm{E} \bigl\vert \operatorname{Re}S_{\varepsilon}^{\theta}- \operatorname{Re}S^{n}_{\epsilon} \bigr\vert \\ &=\mathrm{E} \biggl\vert \int_{0}^{1} \bigl(K^{*}(r)-K^{n}(r)\bigr) \operatorname{Re}\theta(r)\,dr \biggr\vert \\ &\leq \biggl( \int_{0}^{1} \bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\operatorname{Re}\theta(r)\,dr \biggr)^{\frac {1}{2}} \biggl( \mathrm{E} \int_{0}^{1} \bigl(\operatorname{Re}\theta(r) \bigr)^{2}\,dr \biggr)^{\frac{1}{2}} \\ &\leq C \biggl( \int_{0}^{1} \bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\,dr\biggr)^{\frac{1}{2}} \leq C \frac{1}{\sqrt{n}}. \end{aligned}$$

By Bardina et al. [4], we easily get that the real part and the imaginary part of the processes \(Z^{\theta }_{\varepsilon}\) are two independent Brownian motions. So, for the term \(I_{52}\), we have

$$\begin{aligned} I_{52}&=\mathrm{E} \bigl\vert \operatorname{Re}S^{n}_{\epsilon}- \operatorname{Re}S^{n} \bigr\vert \\ &=\mathrm{E} \biggl\vert \int_{0}^{1} K^{n}(r)\operatorname{Re} \theta(r) \,dr- \int_{0}^{1} K^{n}(r)\,dW(r) \biggr\vert \\ &\to0,\quad \epsilon\rightarrow0. \end{aligned}$$

According to the Schwarz inequality and the isometric property of the Wiener integral with respect to the term \(I_{53}\), we obtain that, for any \(n \in\mathbb{N}\),

$$ I_{53} \leq\bigl(\mathrm{E} \bigl\vert \operatorname{Re}S^{n}- \operatorname{Re}S \bigr\vert ^{2}\bigr)^{\frac{1}{2}} \leq C \biggl( \int _{0}^{1}\bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\,dr \biggr)^{1/2}\leq C \frac{1}{\sqrt{n}}. $$

Then both \(I_{51}\) and \(I_{53}\) become arbitrarily small by taking \(n\geq n_{0}\) for some \(n_{0} \in\mathbb{N}\). Similarly, we can prove \(I_{6}\) converging to 0 as n tends to infinity. This completes the proof. □

Proof of Theorem 1.2

The tightness of the processes \(Y^{\theta}_{\varepsilon}\) comes from Lemma 2.3. Next, we identify the limit law by proving that the family of stochastic processes \(Y^{\theta}_{\varepsilon}\) converges in the sense of finite dimensional distribution function to the process Z as ε tends to zero, that is, we prove

$$T_{\varepsilon}:=\sum_{k=1}^{N}a_{k}Y_{\varepsilon}^{\theta }(t_{k}) \rightarrow T:=\sum_{k=1}^{N}a_{k}Z(t_{k}) $$

in distribution when ε tends to zero, where \(a_{1},\ldots,a_{N} \in\mathbb{R}\) and \(t_{1}, \ldots, t_{N} \in[0,1]\).

Similar to the proof of Theorem 1.1, note that

$$T_{\varepsilon}= \int_{0}^{1}K^{*}(r)\frac{2}{\varepsilon} c( \theta)e^{i\theta X_{\frac{2r}{{\varepsilon}^{2}}}}\,dr\quad \mbox{and}\quad T= \int_{0}^{1}K^{*}(r)\,dW(r), $$

where \(K^{*}(r)=\sum_{k=1}^{N}a_{k}K(t_{k},r)\).

Because \(K^{*}(r)\in L_{2}([0,1])\), there exists a simple function

$$ K^{n}(r)=\sum_{k=0}^{m_{n}-1}K^{n}_{k} \mathbf{1}_{(r_{k}^{n},r_{k+1}^{n}]}(r) $$
(48)

with \(0=r_{0}^{n}< r_{1}^{n}< \cdots<r_{m_{n}-1}^{n}<r_{m_{n}}^{n}=1 \) such that

$$\int_{0}^{1}\bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\,dr\leq\frac{1}{n},\quad n\geq1. $$

Now, define two variables \(T_{\varepsilon}^{n}=\int_{0}^{1}K^{n}(r)\frac{2}{\varepsilon} c(\theta )e^{i\theta X_{\frac{2r}{{\varepsilon}^{2}}}}\,dr\) and \(T^{n}=\int _{0}^{1}K^{n}(r)\,dW(r)\), then

$$\begin{aligned} \mathrm{E} \bigl\vert T_{\varepsilon}^{n}-T_{\varepsilon} \bigr\vert &\leq\mathrm{E} \int _{0}^{1} \biggl\vert \bigl(K^{n}(r)-K^{*}(r) \bigr)\frac{2}{\varepsilon} c(\theta)e^{i\theta X_{\frac {2r}{{\varepsilon}^{2}}}} \biggr\vert \,dr \\ &\leq c(\theta) \biggl( \int_{0}^{1} \bigl\vert \bigl(K^{n}(r)-K^{*}(r) \bigr) \bigr\vert ^{2}\,dr \biggr)^{\frac{1}{2}} \\ &\leq c(\theta) \frac{1}{\sqrt{n}}. \end{aligned}$$
(49)

By (48), we have

$$\begin{aligned} T_{\varepsilon}^{n}&= \int_{0}^{1}K^{n}(r)\frac{2}{\varepsilon} c( \theta )e^{i\theta X_{\frac{2r}{{\varepsilon}^{2}}}}\,dr \\ &=\sum_{k=0}^{m_{n}-1}K^{n}_{k} \int_{r_{k}^{n}}^{r_{k+1}^{n}}\frac {2}{\varepsilon} c( \theta)e^{i\theta X_{\frac{2r}{{\varepsilon}^{2}}}}\,dr. \end{aligned}$$
(50)

According to the result obtained by Bardina and Rovira [8], it is easy to get the weak convergence of \(T_{\varepsilon }^{n}\) to \(T^{n}\), where \(T^{n}=\int_{0}^{1}K^{n}(r)\,dW(r)=\sum_{k=0}^{m_{n}-1}K^{n}_{k}\int _{r_{k}^{n}}^{r_{k+1}^{n}}\,dW(r)\).

So,

$$ \mathrm{E}\bigl(e^{iuT_{\varepsilon}^{n}}\bigr)\rightarrow\mathrm {E} \bigl(e^{iuT^{n}}\bigr),\quad {\varepsilon\rightarrow0}. $$
(51)

By the triangle inequality, we have

$$ \bigl\vert \mathrm{E}\bigl(e^{iuT_{\varepsilon}}\bigr)-\mathrm{E} \bigl(e^{iuT}\bigr) \bigr\vert \leq\alpha _{\varepsilon}^{n}+ \beta_{\varepsilon}^{n}+\gamma^{n},\quad u\in R, \varepsilon>0, n\in N, $$
(52)

where \(\alpha_{\varepsilon}^{n}=|\mathrm{E}(e^{iuT_{\varepsilon }})-\mathrm{E}(e^{iuT_{\varepsilon}^{n}})|\), \(\beta_{\varepsilon}^{n}=|\mathrm{E}(e^{iuT_{\varepsilon}^{n}})-\mathrm {E}(e^{iuT^{n}})|\), \(\gamma^{n}=|\mathrm{E}(e^{iuT^{n}})-\mathrm{E}(e^{iuT})|\).

Using the mean value theorem and inequality (49), we obtain \(\alpha_{\varepsilon}^{n}\leq u\mathrm{E}|T_{\varepsilon }-T_{\varepsilon}^{n}|\) converging to 0 as n tends to infinity. With respect to the term \(\beta_{\varepsilon}^{n}\), it is easy to get the convergence of \(\beta_{\varepsilon}^{n}\) to 0 as ε tends to 0 from (51).

Next we consider the term \(\gamma^{n}\). By the Schwarz inequality and the isometric property of the Wiener integer, we get

$$\begin{aligned} \gamma^{n}&\leq u\mathrm{E} \bigl\vert T^{n}-T \bigr\vert \\ &\leq u\bigl[\mathrm{E} \bigl\vert \operatorname{Re}T^{n}- \operatorname{Re}T \bigr\vert +\mathrm{E} \bigl\vert \operatorname{Im}T^{n}- \operatorname{Im}T \bigr\vert \bigr] \\ &\leq \vert u \vert \bigl[\bigl(\mathrm{E} \bigl\vert \operatorname{Re}T^{n}-\operatorname{Re}T \bigr\vert ^{2} \bigr)^{\frac{1}{2}}+\bigl(\mathrm {E} \bigl\vert \operatorname{Im}T^{n}- \operatorname{Im}T \bigr\vert ^{2}\bigr)^{\frac{1}{2}}\bigr] \\ &= \vert u \vert \biggl( \int_{0}^{1}\bigl(K^{*}(r)-K^{n}(r) \bigr)^{2}\,dr \biggr)^{\frac{1}{2}} \\ &\leq \vert u \vert \frac{1}{\sqrt{n}} \rightarrow0 \end{aligned}$$
(53)

as n tends to infinity. This completes the proof. □

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Acknowledgements

The authors would like to thank the anonymous referees and the associate editor for their careful reading of the manuscript and some valuable comments.

Funding

This research is supported by the Distinguished Young Scholars Foundation of Anhui Province (1608085J06), the National Natural Science Foundation of China (11271020), the Top Talent Project of University Discipline (Speciality) (gxbjZD03), the Natural Science Foundation of Chuzhou University (2016QD13), and the Natural Science Foundation of Universities of Anhui Province (KJ2016A527, KJ2017A426, KJ2018A0429), the First Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics).

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Sang, L., Shen, G. & Wang, Q. Weak convergence of the complex fractional Brownian motion. Adv Differ Equ 2018, 444 (2018). https://doi.org/10.1186/s13662-018-1892-4

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MSC

  • 60F05
  • 60G15
  • 60G22

Keywords

  • Complex fractional Brownian motion
  • Poisson process
  • Lévy process
  • Weak convergence