Asymptotic theory for quadratic variation of harmonizable fractional stable processes

TL;DR

This paper develops the asymptotic theory for the quadratic variation of harmonizable fractional stable processes, revealing non-ergodic limits and convergence to Lévy-driven Rosenblatt variables under specific conditions.

Contribution

It provides new asymptotic results for quadratic variation of harmonizable fractional stable processes, including non-ergodic limits and weak convergence to Rosenblatt distributions.

Findings

Law of large numbers with non-ergodic limit

Weak convergence to Lévy-driven Rosenblatt variable

Conditions on Hurst parameter and stability index

Abstract

In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional $\al$-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a L\'evy-driven Rosenblatt random variable when the Hurst parameter satisfies $H\in (1/2,1)$ and $\al(1-H)<1/2$. This result complements the asymptotic theory for fractional stable processes investigated in e.g. \cite{BHP19,BLP17,BP17,BPT20,LP18,MOP20}.

Equations

$$X_t={\int }_{\mathbb{R}}\frac{\exp (its)-1}{is}|s{|}^{1-H-1/\alpha }d{L}_s,$$

$$\sum _{j=0}^{n-1}\Vert X_j-X_{j-1}{\Vert }^2,$$

$$\frac{1}{n}\sum _{j=0}^{n-1}\Vert X_j-X_{j-1}{\Vert }^2\stackrel{\mathbb{P}}{⟶}U:=2{\int }_{\mathbb{R}}|s{|}^{-2H-2/\alpha }\left(1-\cos (s)\right)d([L^1{]}_s+[L^2{]}_s),$$

$$n^{2-2H}\left(\frac{1}{n}\sum _{j=0}^{n-1}\Vert X_j-X_{j-1}{\Vert }^2-U\right)$$

$$\displaystyle\qquad\quad\stackrel{{\scriptstyle\raisebox{-1.0pt}{{\tiny d}}}}{{\longrightarrow}}2{\textrm{Re}}\Big{(}\int_{\mathbb{R}^{2}}\frac{1-\exp(i(s-u))}{i(s-u)}|su|^{1-H-1/\alpha}1_{\{u<s\}}\,d\overline{L}_{u}\,dL_{s}\Big{)}.$$

$$\frac{1}{n^{2/\alpha }}\sum _{j=0}^{n-1}\Vert Y_j{\Vert }^2\stackrel{d}{⟶}Z,$$

$$\nu \circ (x\mapsto x\exp (i\theta ){)}^{-1}=\nu \forall \theta \in (0,2\pi ).$$

$${\int }_{\mathbb{R}}f(s)d{L}_s$$

$$\displaystyle\quad:=\Big{(}\int_{\mathbb{R}}\text{Re}(f(s))\,dL^{1}_{s}-\int_{\mathbb{R}}\text{Im}(f(s))\,dL^{2}_{s}\Big{)}+i\Big{(}\int_{\mathbb{R}}\text{Re}(f(s))\,dL^{2}_{s}+\int_{\mathbb{R}}\text{Im}(f(s))\,dL^{1}_{s}\Big{)},$$

$$\int_{\mathbb{R}}\int_{\mathbb{R}}|f(s,t)|^{\alpha}\Big{[}1+\log_{+}\Big{(}\frac{|f(s,t)|}{\psi(s)\psi(t)}\Big{)}\Big{]}\,ds\,dt<\infty,$$

$$\Big{\|}\int_{\mathbb{R}}g(s)\,dL_{s}\Big{\|}^{2}=2\text{Re}\Big{(}\int_{\mathbb{R}^{2}}\overline{g}(u)g(s)1_{\{u<s\}}\,d\overline{L}_{u}\,dL_{s}\Big{)}+\int_{\mathbb{R}}\|g(s)\|^{2}\,d([L^{1}]_{s}+[L^{2}]_{s}).$$

$$Y_t{\overline{Y}}_t-Y_w{\overline{Y}}_w=$$

$$=$$

$$Y_{\infty}\overline{Y}_{\infty}=2\text{Re}\Big{(}\int_{\mathbb{R}}\overline{Y}_{s-}\,dY_{s}\Big{)}+[Y,\overline{Y}]_{\infty}.$$

$$\|Y_{\infty}\|^{2}=Y_{\infty}\overline{Y}_{\infty}=2\text{Re}\Big{(}\int_{\mathbb{R}^{2}}\overline{g}(u)g(v)1_{\{u<v\}}\,d\overline{L}_{u}\,dL_{v}\Big{)}+\int_{\mathbb{R}}\|g(s)\|^{2}\,d([L^{1}]_{s}+[L^{2}]_{s}),$$

$${\int }_{{\mathbb{R}}^2}{h}_n(u,v)d{Z}_u^{1}d{Z}_v^2\stackrel{\mathbb{P}}{⟶}{\int }_{{\mathbb{R}}^2}h(u,v)d{Z}_u^{1}d{Z}_v^2\text{as }n\to \infty .$$

$$X_j-X_{j-1}={\int }_{\mathbb{R}}\exp (ijs)r(s)d{L}_s\text{with}r(s):=\frac{1-\exp (-is)}{is}|s{|}^{1-H-1/\alpha }.$$

$$\Vert r(s)\Vert \le C\left(|s{|}^{1-H-1/\alpha }{1}_{\{|s|\le 1\}}+|s{|}^{-H-1/\alpha }{1}_{\{|s|>1\}}\right),$$

$$\Vert r{\Vert }_{\alpha }<\infty .$$

$$\Vert X_j-X_{j-1}{\Vert }^2=V_j+U,$$

$$U$$

$$V_j$$

$$\frac{1}{n}\sum _{j=0}^{n-1}\Vert X_j-X_{j-1}{\Vert }^2=U+\frac{1}{n}\sum _{j=0}^{n-1}{V}_j,$$

$$\frac{1}{n}\sum _{j=0}^{n-1}\exp (ij(s-u))=\frac{1-\exp (in(s-u))}{n(1-\exp (i(s-u)))}\to 0\text{as }n\to \infty ,$$

$$\Vert \exp (ij(s-u))r(s)r(u)\Vert \le \Vert r(s)\Vert \Vert r(u)\Vert .$$

$$n^{-1}\sum _{j=0}^{n-1}{V}_j\stackrel{\mathbb{P}}{⟶}0,$$

$$h_n(s,u)$$

$$h(s,u)$$

$$n^{2-2H}\left(\frac{1}{n}\sum_{j=0}^{n-1}\|X_{j}-X_{j-1}\|^{2}-U\right)=n^{1-2H}\sum_{j=0}^{n-1}V_{j}=2\textrm{Re}\Big{(}\int_{\mathbb{R}^{2}}h_{n}(s,u)\,d\overline{L}_{u}\,dL_{s}\Big{)},$$

$${\int }_{{\mathbb{R}}^2}{h}_n(s,u)d{\overline{L}}_{u}d{L}_s\stackrel{d}{⟶}{\int }_{{\mathbb{R}}^2}h(s,u)d{\overline{L}}_{u}d{L}_s.$$

$$\gamma :=1-H-1/\alpha .$$

$$\psi (s):=c\left(|s{|}^{-r}\text{mathbbm}{1}_{\{|s|>1\}}+\text{mathbbm}{1}_{\{|s|\le 1\}}\right)$$

$$\Vert f(s,u)\Vert \le C|su{|}^{-r_1}\left(1_{[s-1,s)}(u)+|s-u{|}^{-r_2}{1}_{(-\infty ,s-1)}(u)\right),$$

Full text

Key words:  fractional processes, harmonizable processes, limit theorems, quadratic variation, stable Lévy motion.

AMS 2010 subject classifications: 60F05, 60F15, 60G22, 60G48, 60H05.

1 Introduction

Since the seminal work of Mandelbrot and Van Ness [20] fractional stochastic processes and fields have gained a lot of attention in the probabilistic and statistical literature. One of the key results of [20] was the introduction of the fractional Brownian motion, the most famous fractional stochastic process. The authors show that, up to scaling, fractional Brownian motion with Hurst parameter $H\in(0,1)$ is the unique self-similar centered Gaussian process with stationary increments. When the Gaussianity assumption is replaced by $\alpha$-stable distributions the aforementioned class becomes much richer: Rosinski [24] shows that it consists of (mixed) moving average $\alpha$-stable processes, $\alpha$-stable harmonizable processes and processes of a third kind (which lack an explicit description). The most well-known representative of the first class is the linear fractional stable motion (lfsm), whose probabilistic and statistical properties have been studied in numerous articles. We refer to [4, 5, 6, 7, 17, 22, 23] for a detailed exposition of limit theorems for functionals of lfsm, and to [1, 10, 12, 18, 19, 21] for statistical estimation of lfsm and related processes.

In the paper [9] the authors introduced the $\alpha$-stable harmonizable fractional motion, defined as

[TABLE]

where $i:=\sqrt{-1}$, $H\in(0,1)$ is the Hurst parameter and $L_{t}:=L_{t}^{1}+iL_{t}^{2}$, $t\in\mathbb{R}$, denotes an isotropic complex-valued $\alpha$-stable Lévy motion with $\alpha\in(0,2)$. The processes $(X_{t})_{t\in\mathbb{R}}$ is self-similar with index $H$, i.e. $(X_{t})_{t\in\mathbb{R}}\stackrel{{\scriptstyle d}}{{=}}(a^{-H}X_{at})_{t\in\mathbb{R}}$, and possesses stationary increments. Probabilistic properties of model (1.1) and their extensions have been studied in numerous articles, see e.g. [2, 8, 9, 11, 15] among others. However, to the best of our knowledge, asymptotic theory for statistics of harmonizable fractional $\alpha$-stable processes has not yet being investigated in the literature. In contrast to lfsm, harmonizable fractional $\alpha$-stable processes are not ergodic, which indicates potential difficulties when studying limit theorems.

This paper aims to derive asymptotic results for the quadratic statistic

[TABLE]

and our main statement is the following theorem:

Theorem 1.1**.**

Let $(X_{t})_{t\geq 0}$ be the $\alpha$-stable harmonizable fractional motion introduced in (1.1) with parameters $\alpha\in(0,2)$ and $H\in(0,1)$.

1. (i)

As $n\to\infty$,

[TABLE]

where $[L^{k}]$ denotes the quadratic variation of $L^{k}$ for $k=1,2$. 2. (ii)

For $H>1/2$ and $\alpha(1-H)<1/2$ we obtain the weak convergence

[TABLE]

Remark 1.2**.**

(A) The result of Theorem 1.1(i) is in sharp contrast to the setting of short memory processes. To make this comparison concrete, let us consider a sequence $(Y_{i})_{i\in\mathbb{N}}$ of independent $\alpha$-stable isotropic random variables with $\alpha\in(0,2)$. According to the Stable Central Limit Theorem, cf.  [13, Theorem 5.25],

[TABLE]

where $Z$ is an $(\alpha/2)$-stable random variable. That is, for independent $\alpha$-stable random variables, the quadratic variation $\sum_{j=0}^{n}\|Y_{j}\|^{2}$ is of the order $n^{\alpha/2}$, whereas $\sum_{j=0}^{n-1}\|X_{j}-X_{j-1}\|^{2}$ is of the order $n$, cf. Theorem 1.1(i). This difference may be viewed as an indication of the strong dependence between the random variables $(X_{j}-X_{j-1})_{j\in\mathbb{N}}$.

(B) We believe that the randomness of the limit in (1.3) is connected to the non-ergodicity of the process $X$.

(C) The weak convergence result of (1.4) is similar in spirit to the Gaussian case. Indeed, the quadratic variation of the fractional Brownian motion with Hurst parameter $H$ is asymptotically distributed according to the Rosenblatt random variable when $H\in(3/4,1)$ with convergence rate $n^{2H-2}$; the classical Rosenblatt random variable possesses the same representation as the right hand side of (1.4) where $L$ is replaced by a complex-valued Gaussian measure. We also recall that there is no convergence regime of the form (1.4) in the setting of lfsm (cf. [4, 5]).

(D) The proof of Theorem 1.1 heavily relies on the quadratic form of the statistic. Extensions to more general functional forms and statistical estimation of the process $X$ are therefore far from obvious. $\diamond$

The rest of the paper is structured as follows. In Chapter 2, we discuss the integration theory and a dominated convergence result we will use in our proof. Chapter 3 is devoted to the proof of Theorem 1.1.

Notation

Throughout the paper all random variables are defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. By $\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}$ we denote convergence in probability, and $\stackrel{{\scriptstyle\raisebox{-1.0pt}{{\tiny d}}}}{{\longrightarrow}}$ denotes convergence in distribution. For a complex number $x\in\mathbb{C}$, $\overline{x}$ denotes its complex conjugate and $\|x\|$ its norm. Furthermore, we denote by $\text{Re}(x)$ (resp. $\text{Im}(x)$) the real (resp. imaginary) part of $x$.

2 Some background

In this section we will collect some important results on stochastic integration theory. A complex-valued random variable $Z$ is called isotropic if it holds that $\exp(i\theta)Z\stackrel{{\scriptstyle d}}{{=}}Z$ for any $\theta\in(0,2\pi)$. Similarly, a stochastic process $(Z_{t})_{t\in\mathbb{R}}$ is isotropic when any linear combination $\sum_{j=1}^{d}a_{j}Z_{t_{j}}$ is isotropic. It is easy to see that a complex-valued Lévy process $L$ with characteristic triplet $(0,0,\nu)$, where $\nu$ denotes the Lévy measure, is isotropic if and only if

[TABLE]

In particular, $L^{1}$ and $L^{2}$ both have the same symmetric $\alpha$-stable distribution. For any measurable function $f:\mathbb{R}\to\mathbb{C}$ we define

[TABLE]

whenever the four real-valued integrals on the right-hand side exist. We note that the integral $\int_{\mathbb{R}}f(s)\,dL_{s}$ is well defined if and only if $\int_{\mathbb{R}}\|f(s)\|\,dL^{1}_{s}$ is well defined, which holds if $\int_{\mathbb{R}}\|f(s)\|^{\alpha}\,ds<\infty$. The next result, which follows by [25, Theorem 2], gives a criterion for the existence of the double integral:

Proposition 2.1**.**

Let $\psi:\mathbb{R}\to(0,\infty)$ be a measure function satisfying $\int_{\mathbb{R}}\psi(s)^{\alpha}\,ds=1$, and suppose that $(Z_{t})_{t\in\mathbb{R}}$ is a two-dimensional symmetric $\alpha$-stable Lévy process with $Z_{t}=(Z^{1}_{t},Z^{2}_{t})$. The double integral $\int_{\mathbb{R}^{2}}f(u,v)\,dZ^{1}_{u}\,dZ^{2}_{v}$ of a measurable function $f:\mathbb{R}^{2}\to\mathbb{R}$ vanishing outside $\{(u,v)\in\mathbb{R}:u<v\}$ exists when the following condition holds:

[TABLE]

where $\log_{+}(x):=\log(x)$ for $x\geq 1$ and $\log_{+}(x):=0$ for $x<1$.

We remark that [25, Theorem 2] is only stated in the case where $Z^{1}=Z^{2}$; however, the extension to $Z^{1}\neq Z^{2}$ follows directly using a polarization techniques similar to the polarization equality for real numbers $xy=((x+y)^{2}-(x-y)^{2})/4$. Proposition 2.4 extends directly to complex-valued functions and processes as well.

Remark 2.2**.**

Let $g:\mathbb{R}\to\mathbb{C}$ is measurable function with $\|g\|_{\alpha}^{\alpha}:=\int_{\mathbb{R}}\|g(x)\|^{\alpha}dx<\infty$ and set $f(u,v)=\overline{g}(u)g(v)1_{\{u<v\}}$ for all $u,v\in\mathbb{R}$. Then we have the following statements.

(A) The function $f$ satisfies condition (2.4), and hence the double integral $\int_{\mathbb{R}^{2}}f(u,v)\,d\overline{L}_{u}\,dL_{v}$ is well defined, cf. Proposition 2.1. Indeed, (2.4) follows by choosing $\psi(s)=\|g(s)\|/\|g\|_{\alpha}$.

(B) The following identity holds

[TABLE]

To show (2.5) we note that $Y_{t}:=\int_{-\infty}^{t}g(u)\,dL_{u}$ exists for all $t\in\mathbb{R}\cup\{\infty\}$. For all $w<t$ in $\mathbb{R}$ we have by integration by parts that

[TABLE]

where $[Z,\tilde{Z}]$ denotes the quadratic variation of two complex-valued semimartingales $Z$ and $\tilde{Z}$. If we let $t\to\infty$ and $w\to-\infty$ in (2.7) we obtain

[TABLE]

By reconizing the integral on the right-hand side of (2.8) as a double integral and calculating the quadratic variation $[Y,\overline{Y}]$ we derive the identity

[TABLE]

where we have used that $[L,\overline{L}]_{t}=[L^{1}]_{t}+[L^{2}]_{t}$. Equation (2.9) implies (2.5). $\diamond$

We will also use the dominated convergence result [14, Theorem 6.2(c)], which reads as follows.

Proposition 2.3**.**

Assume that $(h_{n})_{n\geq 1}$, $h$ and $g$ are deterministic measurable functions $\mathbb{R}^{2}\to\mathbb{R}$ supported on $\{(u,v)\in\mathbb{R}^{2}\!:u<v\}$ all satisfying condition (2.4). Suppose moreover that $(Z_{t})_{t\in\mathbb{R}}$ is a two-dimensional symmetric $\alpha$-stable Lévy process with $Z_{t}=(Z^{1}_{t},Z^{2}_{t})$. If $h_{n}(u,v)\to h(u,v)$ for Lebesgue almost all points $(u,v)\in\mathbb{R}^{2}$ and $\sup_{n}\|h_{n}\|\leq g$, then

[TABLE]

3 Proof of Theorem 1.1

Throughout this section all positive constants are denoted by $C$, although they may change from line to line. We start with the identity

[TABLE]

In particular, we have that

[TABLE]

which implies the statement

[TABLE]

By using the representations (3.1) and (2.5) on the function $g:\mathbb{R}\to\mathbb{C}$ given by $g(s)=\exp(ijs)r(s)$ we deduce the identity

[TABLE]

where

[TABLE]

3.1 Proof of convergence in (1.3)

By the decomposition (3.4) we have

[TABLE]

and thus we only need to prove that $n^{-1}\sum_{j=0}^{n-1}V_{j}\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$. We first note that

[TABLE]

for any $s-u\not\in\{2\pi m:~{}m\in\mathbb{Z}\}$, using the geometric series. On the other hand, we have

[TABLE]

In view of (3.3) and Remark 2.2(A), we can apply Proposition 2.3 to conclude that

[TABLE]

which implies (1.3).

3.2 Proof of weak convergence

We will divide the proof into several steps. We write $g_{n}(x):=(1-\exp(inx))/(1-\exp(ix))$ and define the following complex-valued functions

[TABLE]

By the representation (3.4) we have

[TABLE]

where the last equality follows from the identity in (3.7) and the definition of $h_{n}$. Thus, to show the limit theorem (1.4) it is enough to show the convergence

[TABLE]

The following subsections will be denoted to the proof of (3.13), and to this aim we define

[TABLE]

3.2.1 Existence of the limit

In this subsection we will show that the limiting integral $\int_{\mathbb{R}^{2}}h(s,u)\,d\overline{L}_{u}\,dL_{s}$ from (3.13) is well-defined by proving that $h$ satisfies condition (2.4). Set

[TABLE]

for all $s\in\mathbb{R}$, where $r$ is any real number satisfying $r>\alpha^{-1}$ and $c=c_{r}>0$ is chosen such that $\int_{\mathbb{R}}\psi(s)^{\alpha}\,ds=1$. To show condition (2.4) with respect to the function $\psi$ we will use the following lemma.

Lemma 3.1**.**

Let $f:\mathbb{R}^{2}\to\mathbb{C}$ be a complex-valued function satisfying the inequality

[TABLE]

where the numbers $r_{1},r_{2}$ satisfy the conditions $r_{1}\in(1/2,1)$ and $2r_{1}+r_{2}>2$. Then it holds that

[TABLE]

Proof.

According to condition (3.15) we have that $\|f(s,u)\|\leq f_{1}(s,u)+f_{2}(s,u)$ and hence we need to show the integrability of the functions $f_{1}$ and $f_{2}$. We have that

[TABLE]

since $r_{1}\in(1/2,1)$. Hence, we conclude

[TABLE]

and the latter is finite as $r_{1}\in(1/2,1)$.

Now we turn our attention to the function $f_{2}$. We use the substitution $s=xu$ to obtain the bound

[TABLE]

Indeed, the latter integral is finite since $r_{1}+r_{2}>2r_{1}+r_{2}-1>1$. Thus, we deduce

[TABLE]

due to assumptions $r_{1}\in(1/2,1)$ and $2r_{1}+r_{2}>2$. This completes the proof. ∎

Now, we proceed with the proof of existence. First of all, we observe that the function $\|h(s,u)\|^{\alpha}$ satisfies the condition (3.15) with

[TABLE]

It is easy to see that assumptions $r_{1}\in(1/2,1)$ and $2r_{1}+r_{2}>2$ are satisfied due to $H>1/2$ and $\alpha(1-H)<1/2$. On the other hand, for any $\varepsilon>0$ we readily deduce that

[TABLE]

When we choose $\varepsilon>0$ small enough, we conclude that the function

[TABLE]

also satisfies the conditions of Lemma 3.1. This proves the existence of the double integral $\int_{\mathbb{R}^{2}}h(s,u)\,d\overline{L}_{u}\,dL_{s}$, cf. Proposition 2.1.

3.2.2 Main decomposition

Now, we show the weak limit theorem in (1.4). For this purpose, we introduce the sets

[TABLE]

for some fixed $\delta\in(0,1)$. For $x\in\mathbb{R}_{+}$ we also use the notation $[x]:=2\pi j$ where $2\pi j\in\mathcal{Z}$ is the closest number to $x$. We obtain the following estimate for the function $g_{n}$:

[TABLE]

for any $a\in(0,1)$. We decomposition the double integral on the right-hand side of (3.12) as follows

[TABLE]

with

[TABLE]

In the following, we will prove that $Z_{n}^{(1)}$ is the dominating term and show its convergence, while we will prove that $\|Z_{n}^{(2)}\|+\|Z_{n}^{(3)}\|\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$. The following identity in distribution

[TABLE]

where $nA:=\{(ns,nu)\!:~{}(s,u)\in A\}$, will be used in our proof. The identity (3.25) follows from the self-similarity of $L$; more precisely, from the fact that $(L_{t/n})_{t\in\mathbb{R}}$ equals $(n^{-1/\alpha}L_{t})_{t\in\mathbb{R}}$ in finite dimensional distributions.

3.2.3 Term $Z_{n}^{(1)}$

We apply the distributional identity (3.25) to deduce that

[TABLE]

For any fixed $(s,u)$ with $s-u\not\in\mathcal{Z}$ we obtain the convergence

[TABLE]

Hence, we conclude that

[TABLE]

On the other hand, we have that

[TABLE]

As in the proof of existence we deduce that the function $f$ satisfies the condition (2.4). Thus,

[TABLE]

due to Proposition 2.3.

3.2.4 Term $Z_{n}^{(2)}$

We will use the approximation (3.20). First of all, we have that

[TABLE]

Note that $n^{1-2H}\to 0$ since $H>1/2$. On the other hand, the function $f(s,u):=|su|^{\gamma-1}1_{\{u<s\}}$ satisfies the condition

[TABLE]

where we use an estimate of the type (3.19) for the log term. For all $a\in(0,1)$ we have

[TABLE]

It is easy to show that the function $(s,u)\mapsto|su|^{\gamma-1}|(s-u)-[s-u]|^{-a}1_{((-\delta,\delta)^{c})^{2}}(s,u)1_{\{u<s\}}$ satisfies the condition (2.4) when $a\alpha<1$. Choosing $a=1/\alpha-\varepsilon$ we deduce that

[TABLE]

if we choose $\varepsilon>0$ small enough since $\alpha(1-H)<1/2$. We, therefore, conclude that there exists a measurable function $\phi$ satisfying (2.4) such that $|h_{n}(u,v)|\leq\phi(u,v)$ for all $(u,v)\in(-\delta,\delta)^{c}\times(-\delta,\delta)^{c}$. Since $h_{n}(u,v)\to 0$ Lebesgue almost surely, cf. (3.31)–(3.32), Proposition 2.3 yields $Z_{n}^{(2)}\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$.

3.2.5 Term $Z_{n}^{(3)}$

The proof is similar to the one in Subsection 3.2.4. For simplicity we only consider the integration region $(-\delta,\delta)\times(-\delta,\delta)^{c}$; the $(-\delta,\delta)^{c}\times(-\delta,\delta)$ case is analogue. For all $(u,s)\in(-\delta,\delta)\times(-\delta,\delta)^{c}$:

[TABLE]

We have $n^{1-2H}\to 0$ and the function $f(s,u):=|u|^{\gamma}|s|^{\gamma-1}1_{(-\delta,\delta)}(u)1_{(-\delta,\delta)^{c}}(s)1_{\{u<s\}}$ satisfies the condition (2.4). On the other hand, we have

[TABLE]

The bounds (3.33) and (3.34) imply the existence of a measurable function $f$ satisfying (2.4) such that $|h_{n}(u,v)|\leq f(u,v)$ for all $(u,v)\in(-\delta,\delta)\times(-\delta,\delta)^{c}$. Since a similar bound holds on $(-\delta,\delta)^{c}\times(-\delta,\delta)$ we conclude, cf. Proposition 2.3, that $Z_{n}^{(3)}\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$ since $h_{n}\to 0$ (cf.  (3.33)–(3.34)).

By the decomposition (3.21) and the convergence results of Subsections 3.2.3–3.2.5 we obtain the limit theorem (3.13). This completes the proof of (1.4) and of Theorem 1.1. ∎