Open Access

Fractional Brownian sheet and martingale difference random fields

Journal of Inequalities and Applications20162016:202

https://doi.org/10.1186/s13660-016-1144-7

Received: 27 March 2016

Accepted: 11 August 2016

Published: 22 August 2016

Abstract

In this paper, we prove a functional central limit theorem for the multidimensional parameter fractional Brownian sheet using martingale difference random fields. The proof is based on the invariance principle for the Brownian sheet due Poghosyan and Roelly (Stat. Probab. Lett. 38:235-245, 1998).

Keywords

martingale difference random fields multidimensional parameter fractional Brownian sheet weak convergence

MSC

60B10 60G15

1 Introduction

Self-similar stochastic processes with long range dependence (or long memory) are important aspect of stochastic models in various scientific areas, such as econometrics, network traffic analysis, hydrology, telecommunications, and so on. These are processes \(X=\{X_{t}, t\geq0\}\) whose dependence on the time parameter t is self-similar, in the sense that there exists a (self-similarity) parameter \(0< H<1\) such that for any constant \(c\geq0\), \(\{X_{ct}, t\geq0\}\) and \(\{ c^{H} X_{t}, t\geq0\}\) have the same distribution. These processes are often endowed with other distinctive properties.

Fractional Brownian motion (fBm) is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data. It is a suitable generalization of the standard Brownian motion B, which exhibits a long range dependence (when \(H>1/2\)), self-similarity, and Hölder’s continuity, and which has stationary increments. Some surveys and comprehensive literature concerning fBm could be found in Biagini et al. [2], Gradinaru et al. [3], Hu [4], Mishura [5] and Nualart [6].

The so-called fBm of Hurst parameter H is a continuous centered Gaussian process \(B^{H}=\{B_{t}^{H}, t\geq0\}\) with the covariance function
$$ R(t,s)=E\bigl[B^{H}_{t}B^{H}_{s} \bigr]=\frac{1}{2}\bigl[s^{2H}+t^{2H}-|t-s|^{2H} \bigr]. $$
Recall that \(B^{H}\) has the following integral representation with respect to the standard Brownian motion B (when \(H>\frac{1}{2}\)):
$$ B^{H}_{t}= \int_{0}^{t}K_{H}(t,s) \,dB_{s}, \quad t\geq0, $$
(1.1)
where \(K_{H}\) is the kernel defined by (see, e.g., Decreusefond and Üstünel [7])
$$ K_{H}(t,s)= \biggl(H-\frac{1}{2} \biggr)c_{H} s^{\frac{1}{2}-H} \int_{s}^{t}u^{H-\frac {1}{2}}(u-s)^{H-\frac{3}{2}} \,du $$
(1.2)
with \(c_{H}>0\) the following normalizing constant:
$$ c_{H}=\sqrt{\frac{2H\Gamma(\frac{3}{2}-H)}{ \Gamma(H+\frac{1}{2})\Gamma(2-2H)}}. $$
There are two possible multidimensional parameter extensions of the fBm. The first one is the Lévy fractional Brownian random field (see Ciesielski and Kamont [8]), and the second one is the anisotropic fractional Brownian random field introduced by Kamont [9] as a centered Gaussian process \(B^{\alpha}=\{B_{t}^{\alpha}, t\in\mathbb{R}_{+}^{d}\}\) with covariance function given by
$$ E\bigl[B^{\alpha}_{t}B^{\alpha}_{s} \bigr]=\prod_{k=1}^{d}\frac{1}{2} \bigl[s_{k}^{2\alpha _{k}}+t_{k}^{2\alpha_{k}}-|t_{k}-s_{k}|^{2\alpha_{k}} \bigr], $$
where \(\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{d})\in(0,1)^{d}\). We will call it a d-parameter fractional Brownian sheet. For \(\alpha_{1}=\alpha_{2}=\cdots=\alpha_{d}=\frac{1}{2}\), it coincides with the standard d-parameter Brownian sheet \(W=\{W_{t}, t\in\mathbb{R}_{+}^{d}\}\). This process is null on the axes and has a continuous version.

It is well known that a martingale difference random field is extremely useful because it imposes much milder restrictions on the memory of the sequence than under independence, yet most limit theorems that hold for an independent sequence will also hold for a martingale difference random field. Limit theorems for martingale differences were studied for example by Dai et al. [10], Nahapetian [11], Nieminen [12], Poghosyan [13], Shen and Yan [14], Shen et al. [15], Wang et al. [16] and so on. In this work, we will present a multidimensional parameter invariance principle for the fractional Brownian sheet, which is proved by a convergence criterion for random fields to multi-parameter Brownian sheet proved in Poghosyan and Roelly [1].

The rest of this paper is organized as follows. Section 2 contains some preliminaries on the multidimensional parameter stochastic processes and a precise statement of the main result of this paper. Finally, Section 3 is devoted to a proof of the main weak convergence theorem, Theorem 2.1.

2 Preliminaries and main results

We will use the definitions and notations introduced in the basic work of Bickel and Wichura [17]. Consider \([0,1]^{d}\) with the usual partial order. Let \((\Omega, \mathscr{F}, P)\) be a complete probability space and let \(\{\mathscr{F}_{t}; t\in[0,1]^{d}\}\) be a family of sub-σ-fields of \(\mathscr{F}\) such that \(\mathscr{F}_{s}\subseteq\mathscr{F}_{t}\) for any \(s\leq t\). Given \(s\leq t\), we denote by \(\Delta_{s}X_{t}\) the increment of the process X over the rectangle \((s, t]=\prod_{i=1}^{d}(s_{i}, t_{i}]\subset\mathbb{R}^{d}\).

Let Λ be the group of all mappings \(\lambda: [0,1]^{d}\rightarrow[0,1]^{d}\) of the form \(\lambda(t)=(\lambda_{1}(t_{1}),\ldots, \lambda_{d}(t_{d}))\), where each \(\lambda_{i}: [0,1]\rightarrow[0,1]\) is continuous, is strictly increasing, and fixes zero and one. Denote by \(D=D([0,1]^{d})\) the Skorohod space of functions on \([0,1]^{d}\) which are continuous from above with limits from below and equip D, as usual, with the metric
$$ d(x,y):=\inf\bigl\{ \min\bigl(\Vert x-y\lambda \Vert ,\Vert \lambda \Vert \bigr): \lambda\in\Lambda\bigr\} , $$
where \(\|x-y\lambda\|=\sup\{|x(t)-y(\lambda(t))| : t\in[0,1]^{d}\}\) and \(\|\lambda\|=\sup\{|\lambda(t)-t| : t\in[0,1]^{d}\}\). Under this metric, D is a separable and complete metric space. For more details we refer to Bickel and Wichura [17]. Let X, \(X^{n}\) be processes in D, we say \(X^{n}\) converges in law to X if \(Ef(X^{n})\rightarrow Ef(X)\) for all bounded and continuous functions \(f: D\rightarrow\mathbb{R}\) as n tends to infinity.
On the d-dimensional integer lattice \(\mathbb{Z}^{d}\), let I be the σ-algebra of invariant subsets of Ω:
$$ I=\bigl\{ A\in\mathscr{F}:\tau_{u}(A)=A \mbox{ for each } u\in \mathbb{Z}^{d}\bigr\} , $$
where \(\{\tau_{u}, u\in\mathbb{Z}^{d}\}\) is the group of translations, acting on Ω by \(\tau_{u}(X)=X(t-u)\), \(t\in\mathbb{Z}^{d}\).

Definition 2.1

A random field \(\{\xi(t), t\in\mathbb{Z}^{d}\}\) is called translation invariant (homogeneous) if \(P(\tau_{u}(A))=P(A)\) for each \(A\in\mathscr{F}\) and \(u\in\mathbb{Z}^{d}\).

Definition 2.2

A translation invariant random field \(\{\xi(t), t\in\mathbb{Z}^{d}\}\) is called ergodic if P is trivial on the σ-algebra of invariant subsets, i.e. \(P(A)=0\) or \(P(A)=1\) for each \(A\in I\).

For \(u=(u_{1},u_{2},\ldots,u_{d}) \in\mathbb{Z}^{d}\), let
$$ \mathbb{Z}^{d}_{-}(u)=\bigl\{ t\in\mathbb{Z}^{d}: \exists j, 1\leq j\leq d \mbox{ such that } t_{j}< u_{j}\bigr\} , $$
and let \(\mathbb{Z}^{d}_{+}(u)=\mathbb{Z}^{d}\backslash\mathbb{Z}^{d}_{-}(u)\). For a random field \(\{\xi(t), t\in\mathbb{Z}^{d}\}\), set \(\mathcal{P}(u)=\sigma\{\xi(t), t\in\mathbb{Z}^{d}_{-}(u)\}\).

Definition 2.3

A random field \(\{\xi(t), t\in\mathbb{Z}^{d}\}\) is called a martingale difference if, for each \(t\in\mathbb{Z}^{d}\),
$$ E\bigl(\xi(t)|\mathcal{P}(t-1)\bigr)=0\quad \mbox{a.s.}, $$
where \(t-1=(t_{1}-1,t_{2}-1,\ldots,t_{d}-1)\).
Set
$$ K^{n}_{H}(t,s):=n \int_{s-\frac{1}{n}}^{s} K_{H}\biggl( \frac{\lfloor nt\rfloor }{n},u\biggr)\,du,\quad n=1,2,\ldots, $$
(2.1)
where \(\lfloor x\rfloor\) denotes the greatest integer not exceeding x. It is an approximation of \(K_{H}(t,s)\).
Let \(\alpha_{k}>\frac{1}{2}\), \(k=1,2,\ldots,d\). Taking into account the integral representation (1.1) for the fBm, the d-parameter fractional Brownian sheet \(B^{\alpha}\) has also the following integral representation:
$$ B^{\alpha}_{t}= \int_{0}^{t_{d}}\cdots \int_{0}^{t_{1}}K_{\alpha _{1}}(t_{1},u_{1})K_{\alpha_{2}}(t_{2},u_{2}) \cdots K_{\alpha_{d}}(t_{d},u_{d})\, dW_{u}, $$
(2.2)
where \(\{K_{\alpha_{k}},k=1,\ldots,d\}\) is given by (1.2).

The following theorem is the main result of the present paper, it is a multidimensional extension of Nieminen [12].

Theorem 2.1

Let \(\alpha_{k}>\frac{1}{2}\), \(k=1,2,\ldots,d\). \(\{\xi_{i_{1},i_{2},\ldots,i_{d}}^{n},i_{k}=1,2,\ldots \}\) is a translation invariant, ergodic, martingale difference random field with finite second moment \(E(\xi_{i_{1},i_{2},\ldots ,i_{d}}^{n})^{2}<+\infty\) such that
$$ \lim_{n\rightarrow\infty}\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}=1 \quad \textit{a.s.} $$
(2.3)
for all \(1\leq i_{k} \leq n\) and
$$ \max_{1\leq i_{k}\leq n}\bigl|\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr| \leq C \quad \textit{a.s.} $$
(2.4)
for some \(C\geq1\). Define, for all \(n\geq1\), \(t=(t_{1},t_{2},\ldots,t_{d})\in[0,1]^{d}\),
$$ B^{n}_{t}:=\frac{1}{n^{\frac{d}{2}}}\sum _{k=1}^{d}\sum_{i_{k}=1}^{\lfloor nt_{k}\rfloor} \xi_{i_{1},i_{2},\ldots,i_{d}}^{n} $$
(2.5)
and
$$\begin{aligned}& Z^{n}_{t}: = \int_{0}^{t_{d}}\cdots \int_{0}^{t_{1}} K^{n}_{\alpha_{1}}(t_{1},u_{1}) \cdots K^{n}_{\alpha_{d}}(t_{d},u_{d}) \,dB^{n}_{u} \\& \quad =n^{\frac{d}{2}}\sum_{k=1}^{d}\sum _{i_{k}=1}^{\lfloor nt_{k}\rfloor} \xi_{i_{1},i_{2},\ldots,i_{d}}^{n} \\& \qquad {}\times \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}}\cdots \int_{\frac {i_{1}-1}{n}}^{\frac{i_{1}}{n}} K_{\alpha_{1}}\biggl( \frac{\lfloor nt_{1}\rfloor}{n},u_{1}\biggr)\cdots K_{\alpha_{d}}\biggl( \frac {\lfloor nt_{d}\rfloor}{n},u_{d}\biggr) \,du_{1}\cdots \,du_{d}, \end{aligned}$$
(2.6)
where the kernel \(K_{\alpha_{k}}\) is given by (1.2) and the sequence \(\{K^{n}_{\alpha_{k}}, n=1,2,\ldots\}\) defined by (2.1) is an approximation of \(K_{\alpha_{k}}\).

Then, \(\{Z^{n}\}\) converges weakly in the Skorohod space \(D([0,1]^{d})\) to the d-parameter fractional Brownian sheet \(B^{\alpha}\).

In the rest of this paper, most of the estimates contain unspecified constants. An unspecified positive and finite constant will be denoted by C, which may not be the same in each occurrence. Sometimes we shall emphasize the dependence of these constants upon parameters.

3 Proof of Theorem 2.1

In this section, we will prove Theorem 2.1. We verify weak convergence via the convergence of finite dimensional distributions and tightness. We first check the tightness. Since the \(Z^{n}\) are null on the axes, using the criterion established in Bickel and Wichura [17], it suffices to prove the following lemma.

Lemma 3.1

Let \(\{Z^{n}_{t}\}\) be the family of processes defined by (2.6). Then for any \(s, t\in[0,1]^{d}\) with \(s< t\) and any even number \(m\geq2\), there exists a constant \(C_{m}\) such that
$$ \sup_{n} E\bigl(\Delta_{s}Z^{n}_{t} \bigr)^{m}\leq C_{m} \prod_{k=1}^{d}(t_{k}-s_{k})^{m\alpha_{k}}. $$

Proof

Notice that
$$\begin{aligned} \Delta_{s}Z^{n}_{t} =& \int_{s_{d}}^{t_{d}}\cdots \int_{s_{1}}^{t_{1}} \bigl(K^{n}_{\alpha_{1}}(t_{1},u_{1})-K^{n}_{\alpha_{1}}(s_{1},u_{1}) \bigr)\times \cdots\times \bigl(K^{n}_{\alpha_{d}}(t_{d},u_{d})-K^{n}_{\alpha_{1}}(s_{d},u_{d}) \bigr) \,dB^{n}_{u} \\ =&n^{\frac{d}{2}}\sum_{k=1}^{d}\sum _{i_{k}=1}^{\lfloor nt_{k}\rfloor} \xi_{i_{1},i_{2},\ldots,i_{d}}^{n} \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}}\cdots \int_{\frac {i_{1}-1}{n}}^{\frac{i_{1}}{n}} \biggl(K_{\alpha_{1}}\biggl( \frac{\lfloor nt_{1}\rfloor}{n},u_{1}\biggr)-K_{\alpha_{1}}\biggl( \frac {\lfloor ns_{1}\rfloor}{n},u_{1}\biggr)\biggr) \\ &{}\times \cdots\times \biggl(K_{\alpha_{d}}\biggl(\frac{\lfloor nt_{d}\rfloor}{n},u_{d} \biggr)-K_{\alpha_{d}}\biggl(\frac {\lfloor ns_{d}\rfloor}{n},u_{d}\biggr)\biggr) \,du_{1}\cdots \,du_{d} \\ =&n^{\frac{d}{2}}\sum_{k=1}^{d}\sum _{i_{k}=1}^{\lfloor nt_{k}\rfloor} \xi_{i_{1},i_{2},\ldots,i_{d}}^{n} \int_{\frac{i_{1}-1}{n}}^{\frac{i_{1}}{n}} \biggl(K_{\alpha_{1}}\biggl( \frac{\lfloor nt_{1}\rfloor}{n},u_{1}\biggr)-K_{\alpha_{1}}\biggl( \frac {\lfloor ns_{1}\rfloor}{n},u_{1}\biggr)\biggr) \,du_{1} \\ &{} \times \cdots\times \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}} \biggl(K_{\alpha_{d}}\biggl( \frac{\lfloor nt_{d}\rfloor}{n},u_{d}\biggr)-K_{\alpha_{d}}\biggl( \frac {\lfloor ns_{d}\rfloor}{n},u_{d}\biggr)\biggr) \,du_{d}. \end{aligned}$$
Thus,
$$\begin{aligned} E\bigl(\Delta_{s}Z^{n}_{t}\bigr)^{m} =&n^{\frac{dm}{2}} E \Biggl[ \sum_{k=1}^{d} \sum_{i_{k}=1}^{\lfloor nt_{k}\rfloor} \xi_{i_{1},i_{2},\ldots,i_{d}}^{n} \int_{\frac{i_{1}-1}{n}}^{\frac{i_{1}}{n}} \biggl(K_{\alpha_{1}}\biggl( \frac{\lfloor nt_{1}\rfloor}{n},u_{1}\biggr)-K_{\alpha_{1}}\biggl( \frac {\lfloor ns_{1}\rfloor}{n},u_{1}\biggr)\biggr) \,du_{1} \\ &{}\times \cdots\times \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}} \biggl(K_{\alpha_{d}}\biggl( \frac{\lfloor nt_{d}\rfloor}{n},u_{d}\biggr)-K_{\alpha_{d}}\biggl( \frac {\lfloor ns_{d}\rfloor}{n},u_{d}\biggr)\biggr) \,du_{d} \Biggr]^{m} \\ \leq& n^{\frac{dm}{2}}C_{m} \Biggl[ \sum _{k=1}^{d}\sum_{i_{k}=1}^{\lfloor nt_{k}\rfloor} \int_{\frac{i_{1}-1}{n}}^{\frac{i_{1}}{n}} \biggl(K_{\alpha_{1}}\biggl( \frac{\lfloor nt_{1}\rfloor}{n},u_{1}\biggr)-K_{\alpha_{1}}\biggl( \frac {\lfloor ns_{1}\rfloor}{n},u_{1}\biggr)\biggr) \,du_{1} \\ &{}\times \cdots\times \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}}\cdots \biggl(K_{\alpha_{d}}\biggl( \frac{\lfloor nt_{d}\rfloor}{n},u_{d}\biggr)-K_{\alpha_{d}}\biggl( \frac {\lfloor ns_{d}\rfloor}{n},u_{d}\biggr)\biggr) \,du_{d} \Biggr]^{m} \\ =&C_{m} \Biggl[ \prod_{k=1}^{d} \Biggl( \sqrt{n}\sum_{i_{k}=1}^{\lfloor nt_{k}\rfloor} \int_{\frac{i_{k-1}}{n}}^{\frac{i_{k}}{n}} \biggl(K_{\alpha_{k}}\biggl( \frac{\lfloor nt_{k}\rfloor}{n},u_{k}\biggr)-K_{\alpha_{k}}\biggl( \frac {\lfloor ns_{k}\rfloor}{n},u_{k}\biggr)\biggr) \,du_{k} \Biggr)^{2} \Biggr]^{\frac{m}{2}}. \end{aligned}$$
By the Cauchy-Schwarz inequality, the last expression can be bounded by
$$\begin{aligned}& C_{m} \prod_{k=1}^{d} \Biggl[ \sum_{i_{k}=1}^{\lfloor nt_{k}\rfloor} \int_{\frac{i_{k-1}}{n}}^{\frac{i_{k}}{n}} \biggl(K_{\alpha_{k}}\biggl( \frac{\lfloor nt_{k}\rfloor}{n},u_{k}\biggr)-K_{\alpha_{k}}\biggl( \frac {\lfloor ns_{k}\rfloor}{n},u_{k}\biggr)\biggr)^{2} \,du_{k} \Biggr]^{\frac{m}{2}} \\& \quad \leq C_{m} \prod_{k=1}^{d} \biggl[ \int_{0}^{t_{k}} \biggl(K_{\alpha_{k}}\biggl( \frac{\lfloor nt_{k}\rfloor}{n},u_{k}\biggr)-K_{\alpha_{k}}\biggl( \frac {\lfloor ns_{k}\rfloor}{n},u_{k}\biggr)\biggr)^{2} \,du_{k} \biggr]^{\frac{m}{2}} \\& \quad = C_{m} \prod_{k=1}^{d} \biggl| \frac{\lfloor nt_{k}\rfloor-\lfloor ns_{k}\rfloor}{n} \biggr|^{m\alpha_{k}}. \end{aligned}$$
Let us have now arbitrary \(0< s_{k}< t_{k}\) and \(\frac{1}{2}<\alpha_{k}<1\). If \(nt_{k}-ns_{k}\geq1\), then we have \(\vert \frac{\lfloor nt_{k}\rfloor -\lfloor ns_{k}\rfloor}{n} \vert ^{2\alpha_{k}}\leq|2(t_{k}-s_{k})|^{2\alpha _{k}}\). On the other hand, if \(nt_{k}-ns_{k}<1\) then either \(t_{k}\) and \(s_{k}\) belong to a same subinterval \([\frac{m}{n},\frac{m+1}{n})\) for some integer m, which implies \(\vert \frac{\lfloor nt_{k}\rfloor -\lfloor ns_{k}\rfloor}{n} \vert ^{2\alpha_{k}}=0\). Therefore, we get
$$\biggl\vert \frac{\lfloor nt_{k}\rfloor-\lfloor ns_{k}\rfloor}{n}\biggr\vert ^{2\alpha _{k}} \leq\bigl|2(t_{k}-s_{k})\bigr|^{2\alpha_{k}} $$
for all \(n\geq1\). This completes the proof of this lemma. □

We now proceed with the identification of the limit law by proving the convergence of the finite-dimensional distributions of the processes \(\{Z^{n}_{t}\}\) to those of \(B^{\alpha}\).

Theorem 3.1

The family of processes \(\{Z^{n}_{t}\}\) defined by (2.6) converges, in the sense of a finite-dimensional distribution, to the d-parameter fractional Brownian sheet \(B^{\alpha}\).

Proof

For all \(N\in\mathbb{N}\), consider \(a_{1},\ldots,a_{N} \in\mathbb{R}\) and \(t^{1},\ldots,t^{N} \in[0,1]^{d}\). It suffices to prove that the linear combination
$$ Y^{n}:=\sum_{j=1}^{N}a_{j}Z^{n}_{t^{j}} $$
converges in distribution, as n tends to infinity, to a normally distributed random variable with zero mean and variance
$$ E \Biggl(\sum_{j=1}^{N}a_{j}B^{\alpha}_{t^{j}} \Biggr)^{2}. $$
Fact is that the zero mean is trivial. Next, we observe that
$$\begin{aligned} \bigl(\sigma^{n}\bigr)^{2} :=&E\bigl(Y^{n} \bigr)^{2} = \sum_{j,l=1}^{N}a_{j}a_{l} EZ^{n}_{t^{j}}Z^{n}_{t^{l}} \\ =& \sum_{j,l=1}^{N}a_{j}a_{l} n^{d} \sum_{k=1}^{d}\sum _{i_{k}=1}^{n} \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}}\cdots \int_{\frac {i_{1}-1}{n}}^{\frac{i_{1}}{n}} K_{\alpha_{1}}\biggl( \frac{\lfloor nt^{j}_{1}\rfloor}{n},u_{1}\biggr) \\ &{}\times\cdots\times K_{\alpha _{d}}\biggl( \frac{\lfloor nt^{j}_{d}\rfloor}{n},u_{d}\biggr) \,du_{1}\cdots \,du_{d} \\ &{}\times \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}}\cdots \int_{\frac {i_{1}-1}{n}}^{\frac{i_{1}}{n}} K_{\alpha_{1}}\biggl( \frac{\lfloor nt^{l}_{1}\rfloor}{n},u_{1}\biggr)\cdots K_{\alpha _{d}}\biggl( \frac{\lfloor nt^{l}_{d}\rfloor}{n},u_{d}\biggr) \,du_{1}\cdots \,du_{d}\bigl(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr)^{2} \\ =& \sum_{j,l=1}^{N}a_{j}a_{l} n^{d} \prod_{k=1}^{d}\sum _{i_{k}=1}^{n} \int_{\frac{i_{k}-1}{n}}^{\frac{i_{k}}{n}} K_{\alpha_{k}}\biggl( \frac{\lfloor nt^{j}_{k}\rfloor}{n},u\biggr) \,du \int_{\frac{i_{k}-1}{n}}^{\frac{i_{k}}{n}} K_{\alpha_{k}}\biggl( \frac{\lfloor nt^{l}_{k}\rfloor}{n},u\biggr)\,du\bigl(\xi_{i_{1},i_{2},\ldots ,i_{d}}^{n} \bigr)^{2}. \end{aligned}$$
Consider now the inner sum. By the mean value theorem, we have
$$\begin{aligned}& n\sum_{i_{k}=1}^{n} \int_{\frac{i_{k}-1}{n}}^{\frac{i_{k}}{n}} K_{\alpha_{k}}\biggl( \frac{\lfloor nt^{j}_{k}\rfloor}{n},u\biggr) \,du \int_{\frac{i_{k}-1}{n}}^{\frac{i_{k}}{n}} K_{\alpha_{k}}\biggl( \frac{\lfloor nt^{l}_{k}\rfloor}{n},u\biggr)\,du \\& \quad = \frac{1}{n} \sum _{i_{k}=1}^{n} K_{\alpha_{k}}\biggl( \frac{\lfloor nt^{j}_{k}\rfloor}{n},u_{i,k}^{n}\biggr) K_{\alpha_{k}}\biggl( \frac{\lfloor nt^{l}_{k}\rfloor}{n},u_{i,l}^{n}\biggr) \end{aligned}$$
(3.1)
for some \(u_{i,k}^{n},u_{i,l}^{n}\in(\frac{i_{k}-1}{n},\frac{i_{k}}{n}]\). Since the kernel \(K_{\alpha_{k}}(t,\cdot)\) is continuous and decreasing we see that (3.1) is equal to
$$ \frac{1}{n} \sum_{i_{k}=1}^{n} K_{\alpha_{k}}\biggl(\frac{\lfloor nt^{j}_{k}\rfloor}{n},u_{i}^{n}\biggr) K_{\alpha_{k}}\biggl(\frac{\lfloor nt^{l}_{k}\rfloor}{n},u_{i}^{n}\biggr) $$
(3.2)
for some
$$ u_{i}^{n}\in \bigl[\min\bigl(u_{i,k}^{n},u_{i,l}^{n} \bigr), \max \bigl(u_{i,k}^{n},u_{i,l}^{n} \bigr) \bigr]\subseteq \biggl(\frac{i_{k}-1}{n}, \frac {i_{k}}{n} \biggr]. $$
On the other hand, we observe that the kernel \(K_{H}\) with \(\frac{1}{2}< H<1\) is continuous with respect to both arguments and the maps \(t\mapsto\frac{\lfloor nt\rfloor}{n}\) converge uniformly to the identity map in \([0,T]\). So (3.2) is a Riemann type sum. Thus, combining with (2.3), we see that (3.1) converges to
$$ \int_{0}^{1}K_{\alpha_{k}}\bigl(t^{j}_{k},u \bigr)K_{\alpha_{k}}\bigl(t^{l}_{k},u\bigr)\,du. $$
As a consequence, we see that \((\sigma^{n})^{2}\) converges to
$$ \sum_{j,l=1}^{N}a_{j}a_{l} \prod_{k=1}^{d} \int_{0}^{1}K_{\alpha_{k}}\bigl(t^{j}_{k},u \bigr)K_{\alpha_{k}}\bigl(t^{l}_{k},u\bigr)\,du = E \Biggl( \sum_{j=1}^{N}a_{j}B^{\alpha}_{t^{j}} \Biggr)^{2}. $$
Let us now write \(Y^{n}\) as
$$\begin{aligned} Y^{n} =& \sum_{j=1}^{N}a_{j}Z^{n}_{t^{j}} \\ =& \sum_{k=1}^{d}\sum _{i_{k}=1}^{n} n^{\frac{d}{2}} \xi_{i_{1},i_{2},\ldots,i_{d}}^{n} \sum_{j=1}^{N}a_{j} \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}}\cdots \int_{\frac {i_{1}-1}{n}}^{\frac{i_{1}}{n}} K_{\alpha_{1}}\biggl( \frac{\lfloor nt^{j}_{1}\rfloor}{n},u_{1}\biggr) \\ &{}\times\cdots\times K_{\alpha _{d}}\biggl( \frac{\lfloor nt^{j}_{d}\rfloor}{n},u_{d}\biggr) \,du_{1}\cdots \,du_{d} \\ :=&\sum_{k=1}^{d}\sum _{i_{k}=1}^{n} Y^{n}_{i_{1},i_{2},\ldots,i_{d}}. \end{aligned}$$
Then, it remains to prove that the following Lindeberg condition is satisfied:
$$ \lim_{n\rightarrow\infty} \sum_{k=1}^{d} \sum_{i_{k}=1}^{n} E \bigl[\bigl(Y^{n}_{i_{1},i_{2},\ldots,i_{d}} \bigr)^{2} 1_{\{|Y^{n}_{i_{1},i_{2},\ldots ,i_{d}}|>\varepsilon\}}|\mathcal{P}(i_{1}-1,i_{2}-1, \ldots,i_{d}-1) \bigr]=0 $$
(3.3)
for all \(\varepsilon>0\). By the Cauchy-Schwarz inequality and the fact that the kernel \(K_{H}(t,s)\) with \(\frac{1}{2}< H<1\) is increasing in t and decreasing in s, we have
$$\begin{aligned} \bigl(Y^{n}_{i_{1},i_{2},\ldots,i_{d}}\bigr)^{2} =&n^{d}\bigl( \xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr)^{2} \\ &{}\times\Biggl(\sum _{j=1}^{N}a_{j} \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}}\cdots \int_{\frac {i_{1}-1}{n}}^{\frac{i_{1}}{n}} K_{\alpha_{1}}\biggl( \frac{\lfloor nt^{j}_{1}\rfloor}{n},u_{1}\biggr)\cdots K_{\alpha _{d}}\biggl( \frac{\lfloor nt^{j}_{d}\rfloor}{n},u_{d}\biggr) \,du_{1}\cdots \,du_{d} \Biggr)^{2} \\ \leq& n^{d}\bigl(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr)^{2} A \biggl( \int_{\frac{i_{1}-1}{n}}^{\frac{i_{1}}{n}} K_{\alpha_{1}}(1,u_{1}) \,du_{1} \times \cdots \times \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}} K_{\alpha_{d}}(1,u_{d}) \,du_{d} \biggr)^{2} \\ \leq&\bigl(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr)^{2} A \int_{\frac{i_{1}-1}{n}}^{\frac{i_{1}}{n}} K^{2}_{\alpha_{1}}(1,u_{1}) \,du_{1} \times \cdots \times \int_{\frac{i_{d}-1}{n}}^{\frac{i_{d}}{n}} K^{2}_{\alpha_{d}}(1,u_{d}) \,du_{d} \\ \leq&\bigl(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr)^{2}A \prod _{k=1}^{d} \int_{0}^{\frac{1}{n}} K^{2}_{\alpha_{k}}(1,u) \,du \\ =&\bigl(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr)^{2}A \delta^{n}, \end{aligned}$$
where \(A:=(\sum_{j=1}^{N}a_{j})^{2}\) and \(\delta^{n}:=\prod_{k=1}^{d} \int_{0}^{\frac{1}{n}} K^{2}_{\alpha_{k}}(1,u)\,du\). So we get
$$ \bigl\{ \bigl|Y_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr|>\varepsilon \bigr\} = \bigl\{ \bigl(Y^{n}_{i_{1},i_{2},\ldots,i_{d}}\bigr)^{2}>\varepsilon^{2} \bigr\} \subseteq \bigl\{ \bigl(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n} \bigr)^{2}A\delta^{n}> \varepsilon^{2} \bigr\} . $$
Consequently, we obtain
$$\begin{aligned}& E \bigl[\bigl(Y^{n}_{i_{1},i_{2},\ldots,i_{d}}\bigr)^{2} 1_{\{|Y^{n}_{i_{1},i_{2},\ldots ,i_{d}}|>\varepsilon\}}|\mathcal{P}(i_{1}-1,i_{2}-1, \ldots,i_{d}-1) \bigr] \\& \quad \leq E \bigl[\bigl(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n}\bigr)^{2}A \delta^{n} 1_{\{(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n})^{2}A\delta^{n}> \varepsilon^{2} \}} |\mathcal{P}(i_{1}-1,i_{2}-1, \ldots,i_{d}-1) \bigr] \\& \quad \leq CA\delta^{n}E \bigl[ 1_{\{(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n})^{2}A\delta^{n}> \varepsilon^{2} \}} | \mathcal{P}(i_{1}-1,i_{2}-1,\ldots,i_{d}-1) \bigr] \end{aligned}$$
for all \(i_{k}=1,2,\ldots,n\), \(k=1,2,\ldots,d\), and that
$$\begin{aligned}& \sum_{k=1}^{d}\sum _{i_{k}=1}^{n} E \bigl[\bigl(Y^{n}_{i_{1},i_{2},\ldots,i_{d}} \bigr)^{2} 1_{\{|Y^{n}_{i_{1},i_{2},\ldots ,i_{d}}|>\varepsilon\}}|\mathcal{P}(i_{1}-1,i_{2}-1, \ldots,i_{d}-1) \bigr] \\& \quad \leq \sum_{k=1}^{d}\sum _{i_{k}=1}^{n} CA\delta^{n}E \bigl[ 1_{\{(\xi_{i_{1},i_{2},\ldots,i_{d}}^{n})^{2}A\delta^{n}> \varepsilon^{2} \}} |\mathcal{P}(i_{1}-1,i_{2}-1, \ldots,i_{d}-1) \bigr] \\& \quad \leq CA\delta^{n}\sum_{k=1}^{d} \sum_{i_{k}=1}^{n} E [ 1_{\{CA\delta^{n}> \varepsilon^{2} \}} ] \rightarrow0 \quad (n\rightarrow\infty) \end{aligned}$$
because \(\delta^{n}\rightarrow0\), implies \(1_{\{CA\delta^{n}> \varepsilon^{2} \}}\rightarrow0\).

Thus, the Lindeberg condition (3.3) holds and the proof of Theorem 2.1 is now complete. □

Declarations

Acknowledgements

This work was supported by National Natural Science Foundation-Mathematical Tianyuan Foundation of China (11526117), National Natural Science Foundation of China (11401010, 11401313, 11401093, 61503205), Zhejiang Provincial Natural Science Foundation (LQ16A010006) and the Natural Science Foundation of Ningbo Municipality (2015A610158).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Sciences, Ningbo University of Technology
(2)
Department of Statistics, Anhui Normal University

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