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Limit Theory for Moderate Deviation from Integrated GARCH Processes

Yubo Tao111I would like to thank the co-editor, an associate editor and the referees for helping improve the paper. All possible errors are mine. Yubo Tao, School of Economics, Singapore Management University, 90 Stamford Road, Singapore 178903. Email: yubo.tao.2014@phdecons.smu.edu.sg. 90 Stamford Rd, Singapore Management University
Abstract

This paper develops the limit theory of the GARCH(1,1) process that moderately deviates from IGARCH process towards both stationary and explosive regimes. The GARCH(1,1) process is defined by equations ut=σt​εtu_{t}=\sigma_{t}\varepsilon_{t}, σt2=ω+αn​ut−12+βn​σt−12\sigma_{t}^{2}=\omega+\alpha_{n}u_{t-1}^{2}+\beta_{n}\sigma_{t-1}^{2} and αn+βn\alpha_{n}+\beta_{n} approaches to unity as sample size goes to infinity. The asymptotic theory extends Berkes et al. [2005] by allowing the parameters to have a slower rate of convergence. The results can be applied to unit root test for processes with mildly-integrated GARCH innovations (e.g. Boswijk [2001], Cavaliere & Taylor [2007, 2009]) and deriving limit theory of estimators for models involving mildly-integrated GARCH processes (e.g. Jensen & Rahbek [2004], Francq & Zakoïan [2012, 2013]).

keywords:
Central Limit Theorem , Limiting Process , Localization , Explosive GARCH , Volatility Process
MSC:
[2010] 62M10 , 91B84
††journal: Statistics and Probability Letters

1 Introduction

The model considered in this paper is a GARCH(1,1) process:

(Return Process) ut=σt​εt,\displaystyle u_{t}=\sigma_{t}\varepsilon_{t},
(Volatility Process) σt2=ω+αn​ut−12+βn​σt−12,ω>0​, ​αn≥0​, and ​βn≥0,\displaystyle\sigma_{t}^{2}=\omega+\alpha_{n}u_{t-1}^{2}+\beta_{n}\sigma_{t-1}^{2},\quad\omega>0\text{, }\alpha_{n}\geq 0\text{, and }\beta_{n}\geq 0,

where {εt}t=0n\{\varepsilon_{t}\}_{t=0}^{n} is a sequence of independent identically distributed (i.i.d) variables such that E​ε0=0E\varepsilon_{0}=0 and E​ε02=1E\varepsilon_{0}^{2}=1.

Unlike conventional GARCH(1,1) process, the innovation process considered in this paper is a mildly-integrated GARCH process whose key parameters, αn\alpha_{n} and βn\beta_{n}, are changing with the sample size, viz.

αn=O​(n−p),βn=1+O​(n−q), where ​p,q∈(0,1),\alpha_{n}=O(n^{-p}),\quad\beta_{n}=1+O(n^{-q}),\text{ where }p,q\in(0,1),

and

γn=αn+βn−1=O​(n−κ),κ=min⁡{p,q}.\gamma_{n}=\alpha_{n}+\beta_{n}-1=O(n^{-\kappa}),\quad\kappa=\min\{p,q\}.

The limiting process of this GARCH process is first derived in Berkes et al. [2005] by imposing the assumption κ∈(1/2,1)\kappa\in(1/2,1). Extending their results, we obtain the limiting process that applies to parameter values that covers the whole range of (0,1)(0,1). This is a non-trivial extension because when the process deviates further from the integrated GARCH process, the approximation errors in Berkes et al. [2005] diverges and thus a different normalization is needed.

2 Main Results

The main results are summarized in the following one proposition and three theorems. The first proposition modifies the additive representation for σt2\sigma_{t}^{2} in Berkes et al. [2005] to accommodate κ∈(0,1)\kappa\in(0,1). Based on the proposition, we establish three theorems to describe the asymptotic behaviours of σt2\sigma_{t}^{2} and utu_{t} under the cases γn⪋0\gamma_{n}\lesseqqgtr 0 respectively.

To establish the additive representation of σt2\sigma_{t}^{2}, we make the following assumptions on the distribution of the innovations {εt}t=0n\{\varepsilon_{t}\}_{t=0}^{n} and the convergence rate of the GARCH coefficients, αn\alpha_{n} and βn\beta_{n}.

Assumption 1.

{εt}t=0n\{\varepsilon_{t}\}_{t=0}^{n} is an i.i.d sequence with E​ε02=1E\varepsilon_{0}^{2}=1 and E​|ε0|4+δ<∞E|\varepsilon_{0}|^{4+\delta}<\infty, for some δ>0\delta>0.

Assumption 2.

αn​log⁡log⁡n→0\alpha_{n}\log\log n\rightarrow 0, n​αn→∞n\alpha_{n}\rightarrow\infty and βn→1\beta_{n}\rightarrow 1.

Assumption 1 imposes a non-degeneracy condition on the distribution of εt2\varepsilon_{t}^{2} and thus ensures its applicability to the central limit theorem. Assumption 2 bounds the convergence rate of αn\alpha_{n} so that the normalized sequence could converge to a proper limit. Based on these assumptions, we obtain a modified additive representation for σt2\sigma_{t}^{2} in Proposition 1 on the top of Berkes et al. [2005].

Proposition 1 (Additive Representation).

Under Assumption 1 and 2, we have the additive representation for σt2\sigma_{t}^{2} as

σt2\displaystyle\sigma_{t}^{2} =σ02​tt/2​et​γn​(1+αnt​∑j=1tξt−j+Rt(1))+ω​[1+∑j=1ttj/2​ej​γnt​(1+αnt​∑i=1jξt−i+Rt,j(2))​(1+Rt,j(3))]\displaystyle=\sigma_{0}^{2}t^{t/2}e^{\sqrt{t}\gamma_{n}}\left(1+\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{j=1}^{t}\xi_{t-j}+R_{t}^{(1)}\right)+\omega\left[1+\sum_{j=1}^{t}t^{j/2}e^{\frac{j\gamma_{n}}{\sqrt{t}}}\left(1+\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}+R_{t,j}^{(2)}\right)\left(1+R_{t,j}^{(3)}\right)\right]

where ξt=εt2−1\xi_{t}=\varepsilon_{t}^{2}-1 and the remainder terms satisfy

|Rt(1)|=Op​(αn2+γn2),\displaystyle\left|R_{t}^{(1)}\right|=O_{p}\left(\alpha_{n}^{2}+\gamma_{n}^{2}\right), max1≤j≤t⁡|Rt,j(2)|=Op​(αn2)\displaystyle\quad\max\limits_{1\leq j\leq t}\left|R_{t,j}^{(2)}\right|=O_{p}\left(\alpha_{n}^{2}\right)
max1≤j≤t⁡1j​log⁡log⁡j​|Rt,j(2)|=Op​(αn2t),\displaystyle\max\limits_{1\leq j\leq t}\dfrac{1}{j\log\log j}\left|R_{t,j}^{(2)}\right|=O_{p}\left(\dfrac{\alpha_{n}^{2}}{t}\right), max1≤j≤t⁡1j​|Rt,j(3)|=Op​(αn2+γn2t)\displaystyle\quad\max\limits_{1\leq j\leq t}\dfrac{1}{j}\left|R_{t,j}^{(3)}\right|=O_{p}\left(\dfrac{\alpha_{n}^{2}+\gamma_{n}^{2}}{t}\right)
Remark 1.

The key difference between our results and Berkes et al. [2005] is the convergence rate of the approximation errors. In Berkes et al. [2005], the approximation error |Rt(p)||R_{t}^{(p)}|, ∀p={1,2,3}\forall p=\{1,2,3\} is of order t​(αn2+γn2)t(\alpha_{n}^{2}+\gamma_{n}^{2}) or t​αn2t\alpha_{n}^{2} asymptotically. Hence, these errors are negligible only when κ∈(1/2,1)\kappa\in(1/2,1). We relax this restrictive assumption by normalizing the original terms with t\sqrt{t}. Under this new normalization, all the approximation errors remains negligible when κ∈(0,1)\kappa\in(0,1).

To formulate the theorems below, I introduce the following notations. For 0<t1<t2<⋯<tN<10<t_{1}<t_{2}<\cdots<t_{N}<1 define k​(m)=⌊n​tm⌋k(m)=\lfloor nt_{m}\rfloor, 1≤m≤N1\leq m\leq N. Further, we need the assumptions for relative convergence rate between αn\alpha_{n} and γn\gamma_{n} to regulate the asymptotic behaviours of returns and volatilities for near-stationary case.

Assumption 3.

|γn|αn​n1/4→∞\dfrac{\sqrt{\lvert\gamma_{n}\rvert}}{\alpha_{n}n^{1/4}}\rightarrow\infty, while |γn|3αn​n1/4→0\dfrac{\sqrt{\lvert\gamma_{n}\rvert^{3}}}{\alpha_{n}n^{1/4}}\rightarrow 0, as n→∞n\rightarrow\infty.

Assumption 3 imposes a rate condition on the localized parameters αn\alpha_{n} and γn\gamma_{n}. This condition is less restrictive than that in Berkes et al. [2005] in the sense that instead of requiring |γn|3/2/αn\lvert\gamma_{n}\rvert^{3/2}/\alpha_{n} to converge to 0, we allow it to diverge slowly at a rate of n1/4n^{1/4}. The relaxation of the assumption also attributes to the change of the normalization.

Theorem 1 (Near-stationary Case).

Suppose γn<0\gamma_{n}<0, then under Assumption 1-3, the random variables

2​|γn|3αn​k​(m)1/4​1E​ξ02​(σk​(m)2ω​k​(m)k​(m)/2−∑j=1k​(m)−1ej​γnk​(m))→𝑑𝒩​(0,1).\dfrac{\sqrt{2\lvert\gamma_{n}\rvert^{3}}}{\alpha_{n}k(m)^{1/4}}\dfrac{1}{\sqrt{E\xi_{0}^{2}}}\left(\dfrac{\sigma_{k(m)}^{2}}{\omega{k(m)}^{k(m)/2}}-\sum_{j=1}^{k(m)-1}e^{\frac{j\gamma_{n}}{\sqrt{k(m)}}}\right)\xrightarrow{d}\mathcal{N}(0,1).

In addition, the random variables

(|γn|ω​k​(m)(k​(m)+1)/2)1/2​uk​(m)\left(\dfrac{\lvert\gamma_{n}\rvert}{\omega{k(m)}^{(k(m)+1)/2}}\right)^{1/2}u_{k(m)}

are asymptotically independent, each with the asymptotic distribution equals to that of ε0\varepsilon_{0}.

Theorem 2 (Integrate Case).

Suppose γn=0\gamma_{n}=0, then under Assumption 1 and 2, the volatility has the asymptotic distribution

k​(m)1/2n3/2​αn​1E​ξ02​(σk​(m)2ω​k​(m)k​(m)/2−k​(m))→𝑑∫0tmx​𝑑W​(x).\dfrac{k(m)^{1/2}}{n^{3/2}\alpha_{n}}\dfrac{1}{\sqrt{E\xi_{0}^{2}}}\left(\dfrac{\sigma_{k(m)}^{2}}{\omega{k(m)}^{k(m)/2}}-k(m)\right)\xrightarrow{d}\int_{0}^{t_{m}}xdW(x).

In addition, the random variables

(ω​k​(m)k​(m)/2+1)−1/2​uk​(m)\left(\omega{k(m)}^{k(m)/2+1}\right)^{-1/2}u_{k(m)}

are asymptotically independent, each with the asymptotic distribution equals to that of ε0\varepsilon_{0}.

Similar to the near-stationary case, we have to impose additional assumption on the relative speed of converging to zero between αn\alpha_{n} and γn\gamma_{n}.

Assumption 4.

γn/αn→0\gamma_{n}/\alpha_{n}\rightarrow 0, as n→∞n\rightarrow\infty.

Theorem 3 (Near-explosive Case).

Suppose γn>0\gamma_{n}>0, then under Assumption 1, 2 and 4, the volatility has the asymptotic distribution

γn​e−k​(m)​γnαn​k​(m)​1E​ξ02​(σk​(m)2ω​k​(m)k​(m)/2−∑j=1k​(m)−1ej​γnk​(m))⇒W​(tm).\dfrac{\gamma_{n}e^{-\sqrt{k(m)}\gamma_{n}}}{\alpha_{n}\sqrt{k(m)}}\dfrac{1}{\sqrt{E\xi_{0}^{2}}}\left(\dfrac{\sigma_{k(m)}^{2}}{\omega k(m)^{k(m)/2}}-\sum_{j=1}^{k(m)-1}e^{\frac{j\gamma_{n}}{\sqrt{k(m)}}}\right)\Rightarrow W(t_{m}).

In addition, the random variables

(γn​e−k​(m)​γnω​k​(m)(k​(m)+1)/2)1/2​uk​(m)\left(\dfrac{\gamma_{n}e^{-\sqrt{k(m)}\gamma_{n}}}{\omega k(m)^{(k(m)+1)/2}}\right)^{1/2}u_{k(m)}

are asymptotically independent, each with the asymptotic distribution equals to that of ε0\varepsilon_{0}.

Remark 2.

As one may notice, the rate of convergence for both volatility process and return process in all three cases decreases to 0 asymptotically. These seemingly awkward results are reasonable in the sense that the convergence rate is a part of the normalization which reflects the order of the process. In other words, when we compute a partial sum of XXs in form of ∑i=1nai​Xi\sum_{i=1}^{n}a_{i}X_{i}, the normalization just plays the role of aia_{i} which is usually required to decrease to 0 for applying a central limit theorem.

3 Proofs

In this section, I present detailed proofs for all the propositions and the theorems listed in the previous section. For readers’ convenience, I provide a roadmap for understanding the proofs of the theorems. In general, the proofs are done in three steps:

Step 1: We decompose the volatility process into 4 components, σk,s2\sigma_{k,s}^{2}, s=1,⋯,4s=1,\cdots,4, by expanding the multiplicative form provided in Proposition 1.

Step 2: We show the first 3 volatility components are negligible after normalization, and the last term converges to a proper limit by using Cramer-Wold device and Liapounov central limit theorem or Donsker’s theorem.

Step 3: We figure out a normalization to make the normalized volatility converges to 1. Then, applying this normalization to the return process, we complete the proof.

Proof of Proposition 1.

First, note the GARCH(1,1) model can be written into the following multiplicative form:

σt2\displaystyle\sigma_{t}^{2} =σ02​∏i=1t(βn+αn​εt−i2)+ω​[1+∑j=1t−1∏i=1j(βn+αn​εt−i2)]\displaystyle=\sigma_{0}^{2}\prod_{i=1}^{t}\left(\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}\right)+\omega\left[1+\sum_{j=1}^{t-1}\prod_{i=1}^{j}\left(\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}\right)\right]
=σ02​tt/2​∏i=1t(βn+αn​εt−i2)t+ω​[1+tt/2​∑j=1t−1∏i=1j(βn+αn​εt−i2)t].\displaystyle=\sigma_{0}^{2}t^{t/2}\prod_{i=1}^{t}\dfrac{\left(\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}\right)}{\sqrt{t}}+\omega\left[1+t^{t/2}\sum_{j=1}^{t-1}\prod_{i=1}^{j}\dfrac{\left(\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}\right)}{\sqrt{t}}\right].

Note that

max1≤i≤t⁡|βn+αn​εt−i2−1|t≤|γn|t+αn​max1≤i≤t⁡|εt−i2−1|t=|γn|t+αn​max1≤i≤t−1⁡|εi2−1|t.\max\limits_{1\leq i\leq t}\dfrac{\left|\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}-1\right|}{\sqrt{t}}\leq\dfrac{|\gamma_{n}|}{\sqrt{t}}+\alpha_{n}\max\limits_{1\leq i\leq t}\dfrac{|\varepsilon_{t-i}^{2}-1|}{\sqrt{t}}=\dfrac{|\gamma_{n}|}{\sqrt{t}}+\alpha_{n}\max\limits_{1\leq i\leq t-1}\dfrac{|\varepsilon_{i}^{2}-1|}{\sqrt{t}}.

Then by Assumption 1 and Chow & Teicher [2012], we have the almost sure convergence of

max1≤j≤t−1⁡|εi2−1|=O​(t).\max_{1\leq j\leq t-1}|\varepsilon_{i}^{2}-1|=O(\sqrt{t}).

Therefore, the term above is

max1≤i≤t⁡|βn+αn​εt−i2−1|t=op​(1).\max\limits_{1\leq i\leq t}\dfrac{\left|\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}-1\right|}{\sqrt{t}}=o_{p}(1).

Now consider the sequence of events

An={max1≤i≤t⁡|βn+αn​εt−i2−1|t≤12}.A_{n}=\left\{\max\limits_{1\leq i\leq t}\dfrac{\lvert\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}-1\rvert}{\sqrt{t}}\leq\dfrac{1}{2}\right\}.

From the previous result we know limn→∞P​(An)=1\lim\limits_{n\rightarrow\infty}P(A_{n})=1. Then by Taylor expansion, |log⁡(1+x)−x|≤2​x2\lvert\log(1+x)-x\rvert\leq 2x^{2}, |x|≤1/2\lvert x\rvert\leq 1/2 on the event AnA_{n}, which implies

|Rt,j(3)|\displaystyle\left|R_{t,j}^{(3)}\right| =|∑i=1jlog⁡(βn+αn​εt−i2)t−∑i=1j(γn+αn​ξt−i)t|\displaystyle=\left|\sum_{i=1}^{j}\log\dfrac{\left(\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}\right)}{\sqrt{t}}-\sum_{i=1}^{j}\dfrac{\left(\gamma_{n}+\alpha_{n}\xi_{t-i}\right)}{\sqrt{t}}\right|
=|∑i=1jlog⁡(γn+αn​ξt−i+1)t−∑i=1j(γn+αn​ξt−i)t|\displaystyle=\left|\sum_{i=1}^{j}\log\dfrac{\left(\gamma_{n}+\alpha_{n}\xi_{t-i}+1\right)}{\sqrt{t}}-\sum_{i=1}^{j}\dfrac{\left(\gamma_{n}+\alpha_{n}\xi_{t-i}\right)}{\sqrt{t}}\right|
≤∑i=1j|log⁡(γn+αn​ξt−it+1)−(γn+αn​ξt−i)t|\displaystyle\leq\sum_{i=1}^{j}\left|\log\left(\dfrac{\gamma_{n}+\alpha_{n}\xi_{t-i}}{\sqrt{t}}+1\right)-\dfrac{\left(\gamma_{n}+\alpha_{n}\xi_{t-i}\right)}{\sqrt{t}}\right|
≤2​∑i=1j(γn+αn​ξt−i)2t≤4​j​γn2t+4​αn2​∑i=1jξt−i2t.\displaystyle\leq 2\sum_{i=1}^{j}\dfrac{\left(\gamma_{n}+\alpha_{n}\xi_{t-i}\right)^{2}}{t}\leq\dfrac{4j\gamma_{n}^{2}}{t}+\dfrac{4\alpha_{n}^{2}\sum_{i=1}^{j}\xi_{t-i}^{2}}{t}.

By Assumption 1 and law of large numbers (LLN), we know

max1≤j≤t⁡1j​|∑i=1jξt−i2|∼max1≤j≤t⁡1j​|∑i=1jξi2|=Op​(1).\max\limits_{1\leq j\leq t}\dfrac{1}{j}\left|\sum_{i=1}^{j}\xi_{t-i}^{2}\right|\sim\max\limits_{1\leq j\leq t}\dfrac{1}{j}\left|\sum_{i=1}^{j}\xi_{i}^{2}\right|=O_{p}(1).

Then by the equation above, we have

max1≤i≤j⁡1j​|Rt,j(3)|=Op​(γn2+αn2t).\max\limits_{1\leq i\leq j}\dfrac{1}{j}\lvert R_{t,j}^{(3)}\rvert=O_{p}\left(\dfrac{\gamma_{n}^{2}+\alpha_{n}^{2}}{t}\right).

Now by direct plugging into the key multiplicative term we care about, we have

∏i=1j(βn+αn​εt−i2)t\displaystyle\prod_{i=1}^{j}\dfrac{\left(\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}\right)}{\sqrt{t}} =exp⁡{∑i=1jlog⁡(βn+αn​εt−i2t)}\displaystyle=\exp\left\{\sum_{i=1}^{j}\log\left(\dfrac{\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}}{\sqrt{t}}\right)\right\}
=exp⁡{j​γnt}​exp⁡{αn​∑i=1jξt−it}​exp⁡{Rt,j(3)}\displaystyle=\exp\left\{\dfrac{j\gamma_{n}}{\sqrt{t}}\right\}\exp\left\{\dfrac{\alpha_{n}\sum_{i=1}^{j}\xi_{t-i}}{\sqrt{t}}\right\}\exp\left\{R_{t,j}^{(3)}\right\}
=ej​γnt​exp⁡{αn​∑i=1jξt−it}​(1+Rt,j(3)).\displaystyle=e^{\frac{j\gamma_{n}}{\sqrt{t}}}\exp\left\{\dfrac{\alpha_{n}\sum_{i=1}^{j}\xi_{t-i}}{\sqrt{t}}\right\}\left(1+R_{t,j}^{(3)}\right).

Further, note {ξt}t=1n\{\xi_{t}\}_{t=1}^{n} is an i.i.d sequence with E​ξ02<∞E\xi_{0}^{2}<\infty, then we know

max1≤j≤t⁡|∑i=1jξt−i|=Op​(t),\max\limits_{1\leq j\leq t}\left|\sum_{i=1}^{j}\xi_{t-i}\right|=O_{p}(\sqrt{t}),

which implies

max1≤j≤t⁡|αnt​∑i=1jξt−i|=Op​(αn)=op​(1).\max\limits_{1\leq j\leq t}\left|\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}\right|=O_{p}(\alpha_{n})=o_{p}(1).

Similarly, we define the sequence of events

Bn={max1≤j≤t⁡|αnt​∑i=1jξt−i|≤12},B_{n}=\left\{\max\limits_{1\leq j\leq t}\left|\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}\right|\leq\dfrac{1}{2}\right\},

which is known to have the property limn→∞P​(Bn)=1\lim\limits_{n\rightarrow\infty}P(B_{n})=1. Then by Taylor expansion, |exp⁡(x)−(1+x)|≤e​x2/2\lvert\exp(x)-(1+x)\rvert\leq\sqrt{e}x^{2}/2 when |x|≤1/2\lvert x\rvert\leq 1/2, on the event BnB_{n}

|Rt,j(2)|=|exp⁡{αnt​∑i=1jξt−i}−(1+αnt​∑i=1jξt−i)|≤e2​(αnt​∑i=1jξt−i)2=Op​(αn2),\left|R_{t,j}^{(2)}\right|=\left|\exp\left\{\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}\right\}-\left(1+\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}\right)\right|\leq\dfrac{\sqrt{e}}{2}\left(\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}\right)^{2}=O_{p}\left(\alpha_{n}^{2}\right),

and by law of iterated logarithm, we know

max1≤j≤t⁡1j​log⁡log⁡j​(αnt​∑i=1jξt−i)2=Op​(αn2t).\max\limits_{1\leq j\leq t}\dfrac{1}{j\log\log j}\left(\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}\right)^{2}=O_{p}\left(\dfrac{\alpha_{n}^{2}}{t}\right).

Combining the results above, we have thus showed that

∏i=1j(βn+αn​εt−i2t)=ej​γnt​(1+αnt​∑i=1jξt−i+Rt,j(2))​(1+Rt,j(3)).\prod_{i=1}^{j}\left(\dfrac{\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}}{\sqrt{t}}\right)=e^{\frac{j\gamma_{n}}{\sqrt{t}}}\left(1+\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{j}\xi_{t-i}+R_{t,j}^{(2)}\right)\left(1+R_{t,j}^{(3)}\right).

Lastly, by the equation above, we know

∏i=1t(βn+αn​εt−i2t)\displaystyle\prod_{i=1}^{t}\left(\dfrac{\beta_{n}+\alpha_{n}\varepsilon_{t-i}^{2}}{\sqrt{t}}\right) =et​γnt​(1+αnt​∑i=1tξt−i+Op​(αn2))​(1+Op​(γn2+αn2))\displaystyle=e^{\frac{t\gamma_{n}}{\sqrt{t}}}\left(1+\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{t}\xi_{t-i}+O_{p}(\alpha_{n}^{2})\right)\left(1+O_{p}(\gamma_{n}^{2}+\alpha_{n}^{2})\right)
=et​γn​(1+αnt​∑i=1tξt−i+Op​(γn2+αn2)),\displaystyle=e^{\sqrt{t}\gamma_{n}}\left(1+\dfrac{\alpha_{n}}{\sqrt{t}}\sum_{i=1}^{t}\xi_{t-i}+O_{p}(\gamma_{n}^{2}+\alpha_{n}^{2})\right),

and this establishes Rt(1)R_{t}^{(1)}. ∎

Proof of Theorem 1.

First, we focus on the volatilities. Denote k=⌊n​t⌋k=\lfloor nt\rfloor, 0<t≤10<t\leq 1,

σk2\displaystyle\sigma_{k}^{2} =ω+σ02​kk/2​ek​γn​(1+αnk​∑j=1kξk−j+Rk(1))+ω​kk/2​∑j=1k−1ej​γnk​(1+αnk​∑i=1jξk−i+Rk,j(2))​Rk,j(3)\displaystyle=\omega+\sigma_{0}^{2}k^{k/2}e^{\sqrt{k}\gamma_{n}}\left(1+\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{j=1}^{k}\xi_{k-j}+R_{k}^{(1)}\right)+\omega k^{k/2}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\left(1+\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{i=1}^{j}\xi_{k-i}+R_{k,j}^{(2)}\right)R_{k,j}^{(3)}
+ω​kk/2​∑j=1k−1ej​γnk​Rk,j(2)+ω​kk/2​∑j=1k−1ej​γnk​(1+αnk​∑i=1jξk−i)\displaystyle\ \ \ +\omega k^{k/2}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}R_{k,j}^{(2)}+\omega k^{k/2}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\left(1+\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{i=1}^{j}\xi_{k-i}\right)
=ω+σk,12+σk,22+σk,32+σk,42.\displaystyle=\omega+\sigma_{k,1}^{2}+\sigma_{k,2}^{2}+\sigma_{k,3}^{2}+\sigma_{k,4}^{2}.

For σk,12\sigma_{k,1}^{2}, note k−1/2​∑j=1kξk−jk^{-1/2}\sum_{j=1}^{k}\xi_{k-j} is asymptotically normal, then by Proposition 1,

αnk​∑j=1kξk−j+Rk(1)=op​(1),\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{j=1}^{k}\xi_{k-j}+R_{k}^{(1)}=o_{p}(1),

and this implies

|σk,12|\displaystyle\left\lvert\sigma_{k,1}^{2}\right\rvert =Op​(kk/2​ek​γn).\displaystyle=O_{p}\left(k^{k/2}e^{\sqrt{k}\gamma_{n}}\right).

For σk,22\sigma_{k,2}^{2}, note by Lemma 4.1 in Berkes et al. [2005], we have

∑j=1kj​ej​γnk∼k|γn|2​Γ​(2),\sum_{j=1}^{k}je^{\frac{j\gamma_{n}}{\sqrt{k}}}\sim\dfrac{k}{\lvert\gamma_{n}\rvert^{2}}\Gamma(2), (1)

and note that

max1≤j≤k−1⁡|αnk​∑i=1jξk−i+Rk,j(2)|=op​(1).\max\limits_{1\leq j\leq k-1}\left\lvert\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{i=1}^{j}\xi_{k-i}+R_{k,j}^{(2)}\right\rvert=o_{p}(1). (2)

Then by equation (1), (2) and Proposition 1 we have

|σk,22|\displaystyle\left\lvert\sigma_{k,2}^{2}\right\rvert =|ω​kk/2​∑j=1k−1j​ej​γnk​(1+αnk​∑i=1jξk−i+Rk,j(2))​1j​Rk,j(3)|\displaystyle=\left\lvert\omega k^{k/2}\sum_{j=1}^{k-1}je^{\frac{j\gamma_{n}}{\sqrt{k}}}\left(1+\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{i=1}^{j}\xi_{k-i}+R_{k,j}^{(2)}\right)\dfrac{1}{j}R_{k,j}^{(3)}\right\rvert
=Op​(1)​ω​kk/2​αn2+γn2k​k|γn|2\displaystyle=O_{p}(1)\omega k^{k/2}\dfrac{\alpha_{n}^{2}+\gamma_{n}^{2}}{k}\dfrac{k}{\lvert\gamma_{n}\rvert^{2}}
=Op​(kk/2​(αn2+γn2)γn2).\displaystyle=O_{p}\left(\dfrac{k^{k/2}\left(\alpha_{n}^{2}+\gamma_{n}^{2}\right)}{\gamma_{n}^{2}}\right).

For σk,32\sigma_{k,3}^{2}, similarly, by Proposition 1 and Lemma 4.1 in Berkes et al. [2005], we have

|σk,32|\displaystyle\left\lvert\sigma_{k,3}^{2}\right\rvert =|ω​kk/2​∑j=1k−1ej​γnk​Rk,j(2)|\displaystyle=\left\lvert\omega k^{k/2}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}R_{k,j}^{(2)}\right\rvert
=Op​(1)​ω​kk/2​αn2k​∑j=1k−1j​ej​γk​log⁡log⁡j\displaystyle=O_{p}(1)\omega k^{k/2}\dfrac{\alpha_{n}^{2}}{k}\sum_{j=1}^{k-1}je^{\frac{j\gamma}{\sqrt{k}}}\log\log j
=Op​(kk/2​(αn2​log⁡log⁡k)γn2).\displaystyle=O_{p}\left(\dfrac{k^{k/2}\left(\alpha_{n}^{2}\log\log k\right)}{\gamma_{n}^{2}}\right).

Lastly, for σk,42\sigma_{k,4}^{2}, by Lemma 4.1 in 1 we have

σk,42\displaystyle\sigma_{k,4}^{2} =ω​kk/2​∑j=1k−1ej​γnk+ω​kk/2​αnk​∑j=1k−1ej​γnk​∑i=1jξk−i\displaystyle=\omega k^{k/2}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}+\omega k^{k/2}\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}
=Op​(kk/2​k1/2|γn|)+ω​kk/2​αnk​∑j=1k−1ej​γnk​∑i=1jξk−i.\displaystyle=O_{p}\left(\dfrac{k^{k/2}k^{1/2}}{\lvert\gamma_{n}\rvert}\right)+\omega k^{k/2}\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}.

Therefore, we only have to consider the last term in the above equation. Define

τm=k​(m)−1/4​∑j=1k​(m)−1ej​γnk​(m)​ξk​(m)−j,1≤m≤N,\tau_{m}=k(m)^{-1/4}\sum_{j=1}^{k(m)-1}e^{\frac{j\gamma_{n}}{\sqrt{k(m)}}}\xi_{k(m)-j},\quad 1\leq m\leq N,

and

τm∗=k​(m)−1/2​∑j=1k​(m)−1ej​γnk​(m)​∑i=1jξk​(m)−i,1≤m≤N.\tau_{m}^{*}=k(m)^{-1/2}\sum_{j=1}^{k(m)-1}e^{\frac{j\gamma_{n}}{\sqrt{k(m)}}}\sum_{i=1}^{j}\xi_{k(m)-i},\quad 1\leq m\leq N.

Then by Cramer-Wold device (Theorem 29.4 of Billingsley [1995]), we have

∑m=1Nμm​τm\displaystyle\sum_{m=1}^{N}\mu_{m}\tau_{m} =∑i=1k​(1)−1∑m=1Nμmk​(m)1/4​e(k​(m)−i)​γnk​(m)+∑i=k​(1)k​(2)−1∑m=2Nμmk​(m)1/4​e(k​(m)−i)​γnk​(m)\displaystyle=\sum_{i=1}^{k(1)-1}\sum_{m=1}^{N}\dfrac{\mu_{m}}{k(m)^{1/4}}e^{\frac{(k(m)-i)\gamma_{n}}{\sqrt{k(m)}}}+\sum_{i=k(1)}^{k(2)-1}\sum_{m=2}^{N}\dfrac{\mu_{m}}{k(m)^{1/4}}e^{\frac{(k(m)-i)\gamma_{n}}{\sqrt{k(m)}}}
+⋯+∑i=k​(N−1)k​(N)−1μNk​(N)1/4​e(k​(N)−i)​γnk​(N)\displaystyle+\cdots+\sum_{i=k(N-1)}^{k(N)-1}\dfrac{\mu_{N}}{k(N)^{1/4}}e^{\frac{(k(N)-i)\gamma_{n}}{\sqrt{k(N)}}}
=S1+S2+⋯+SN.\displaystyle=S_{1}+S_{2}+\cdots+S_{N}.

Observe that

E​S12\displaystyle ES_{1}^{2} =E​ξ02​(∑i=1k​(1)−1∑m=1Nk​(m)−1/4​μm​e(k​(m)−i)​γnk​(m))2\displaystyle=E\xi_{0}^{2}\left(\sum_{i=1}^{k(1)-1}\sum_{m=1}^{N}k(m)^{-1/4}\mu_{m}e^{\frac{(k(m)-i)\gamma_{n}}{\sqrt{k(m)}}}\right)^{2}
=E​ξ02​∑m=1Nμm2k​(m)​∑i=1k​(1)−1e2​(k​(m)−i)​γnk​(m)+E​ξ02​∑1≤m≠l≤N(k​(m)​k​(l))−1/4​μm​μl​∑i=1k​(1)−1e(k​(m)−i)​γnk​(m)+(k​(l)−i)​γnk​(l)\displaystyle=E\xi_{0}^{2}\sum_{m=1}^{N}\dfrac{\mu_{m}^{2}}{\sqrt{k(m)}}\sum_{i=1}^{k(1)-1}e^{\frac{2(k(m)-i)\gamma_{n}}{\sqrt{k(m)}}}+E\xi_{0}^{2}\sum_{1\leq m\neq l\leq N}\left(k(m)k(l)\right)^{-1/4}\mu_{m}\mu_{l}\sum_{i=1}^{k(1)-1}e^{\frac{(k(m)-i)\gamma_{n}}{\sqrt{k(m)}}+\frac{(k(l)-i)\gamma_{n}}{\sqrt{k(l)}}}
=E​ξ02​μ12k​(1)​∑i=1k​(1)−1e2​(k​(1)−i)​γnk​(1)+E​ξ02​∑m=2Nμm2k​(m)​∑i=1k​(1)−1e2​(k​(m)−i)​γnk​(m)\displaystyle=E\xi_{0}^{2}\dfrac{\mu_{1}^{2}}{\sqrt{k(1)}}\sum_{i=1}^{k(1)-1}e^{\frac{2(k(1)-i)\gamma_{n}}{\sqrt{k(1)}}}+E\xi_{0}^{2}\sum_{m=2}^{N}\dfrac{\mu_{m}^{2}}{\sqrt{k(m)}}\sum_{i=1}^{k(1)-1}e^{\frac{2(k(m)-i)\gamma_{n}}{\sqrt{k(m)}}}
+E​ξ02​∑1≤m≠l≤N(k​(m)​k​(l))−1/4​μm​μl​∑i=1k​(1)−1e(k​(m)−i)​γnk​(m)+(k​(l)−i)​γnk​(l)\displaystyle\ \ \ +E\xi_{0}^{2}\sum_{1\leq m\neq l\leq N}\left(k(m)k(l)\right)^{-1/4}\mu_{m}\mu_{l}\sum_{i=1}^{k(1)-1}e^{\frac{(k(m)-i)\gamma_{n}}{\sqrt{k(m)}}+\frac{(k(l)-i)\gamma_{n}}{\sqrt{k(l)}}}
=E​ξ02​μ12k​(1)​∑i=1k​(1)−1e2​i​γnk​(1)+E​ξ02​∑m=2Nμm2k​(m)​e2​(k​(m)−k​(1))​γnk​(m)​∑i=1k​(1)−1e2​i​γnk​(m)\displaystyle=E\xi_{0}^{2}\dfrac{\mu_{1}^{2}}{\sqrt{k(1)}}\sum_{i=1}^{k(1)-1}e^{\frac{2i\gamma_{n}}{\sqrt{k(1)}}}+E\xi_{0}^{2}\sum_{m=2}^{N}\dfrac{\mu_{m}^{2}}{\sqrt{k(m)}}e^{\frac{2(k(m)-k(1))\gamma_{n}}{\sqrt{k(m)}}}\sum_{i=1}^{k(1)-1}e^{\frac{2i\gamma_{n}}{\sqrt{k(m)}}}
+E​ξ02​∑1≤m≠l≤N(k​(m)​k​(l))−1/4​μm​μl​e(k​(m)−k​(1))​γnk​(m)+(k​(l)−k​(1))​γnk​(l)​∑i=1k​(1)−1ei​γnk​(m)+i​γnk​(l)\displaystyle\ \ \ +E\xi_{0}^{2}\sum_{1\leq m\neq l\leq N}(k(m)k(l))^{-1/4}\mu_{m}\mu_{l}e^{\frac{(k(m)-k(1))\gamma_{n}}{\sqrt{k(m)}}+\frac{(k(l)-k(1))\gamma_{n}}{\sqrt{k(l)}}}\sum_{i=1}^{k(1)-1}e^{\frac{i\gamma_{n}}{\sqrt{k(m)}}+\frac{i\gamma_{n}}{\sqrt{k(l)}}}
∼E​ξ02​μ12​12​|γn|+E​ξ02​∑m=2Nμm2​e2​(k​(m)−k​(1))​γnk​(m)​12​|γn|\displaystyle\sim E\xi_{0}^{2}\mu_{1}^{2}\dfrac{1}{2\lvert\gamma_{n}\rvert}+E\xi_{0}^{2}\sum_{m=2}^{N}\mu_{m}^{2}e^{\frac{2(k(m)-k(1))\gamma_{n}}{\sqrt{k(m)}}}\dfrac{1}{2\lvert\gamma_{n}\rvert}
+E​ξ02​∑1≤m≠l≤Nμm​μl(k​(m)+k​(l))​|γn|​e(k​(m)−k​(1))​γnk​(m)+(k​(l)−k​(1))​γnk​(l)\displaystyle\ \ \ +E\xi_{0}^{2}\sum_{1\leq m\neq l\leq N}\dfrac{\mu_{m}\mu_{l}}{(\sqrt{k(m)}+\sqrt{k(l)})\lvert\gamma_{n}\rvert}e^{\frac{(k(m)-k(1))\gamma_{n}}{\sqrt{k(m)}}+\frac{(k(l)-k(1))\gamma_{n}}{\sqrt{k(l)}}}
=E​ξ02​μ12​12​|γn|+o​(1|γn|),\displaystyle=E\xi_{0}^{2}\mu_{1}^{2}\dfrac{1}{2\lvert\gamma_{n}\rvert}+o\left(\dfrac{1}{\lvert\gamma_{n}\rvert}\right),

we then have

E​(∑m=1Nμm​τm)2\displaystyle E\left(\sum_{m=1}^{N}\mu_{m}\tau_{m}\right)^{2} =(∑m=1Nμm2)​E​ξ0​12​|γn|+o​(1|γn|).\displaystyle=\left(\sum_{m=1}^{N}\mu_{m}^{2}\right)E\xi_{0}\dfrac{1}{2\lvert\gamma_{n}\rvert}+o\left(\dfrac{1}{\lvert\gamma_{n}\rvert}\right).

Observe also that, for some cic_{i}, 1≤i≤k​(N)−11\leq i\leq k(N)-1, we have

∑m=1Nμm​τm=∑i=1k​(N)−1ci​ξi,\sum_{m=1}^{N}\mu_{m}\tau_{m}=\sum_{i=1}^{k(N)-1}c_{i}\xi_{i},

and by Jensen’s inequality, we know for some δ>0\delta>0,

|ci|2+δ\displaystyle\lvert c_{i}\rvert^{2+\delta} =|k​(1)−1/4​μ1​e(k​(1)−i)​γnk​(1)+k​(2)−1/4​μ1​e(k​(2)−i)​γnk​(2)+⋯+k​(N)−1/4​μ1​e(k​(N)−i)​γnk​(N)|2+δ\displaystyle=\left|k(1)^{-1/4}\mu_{1}e^{\frac{(k(1)-i)\gamma_{n}}{\sqrt{k(1)}}}+k(2)^{-1/4}\mu_{1}e^{\frac{(k(2)-i)\gamma_{n}}{\sqrt{k(2)}}}+\cdots+k(N)^{-1/4}\mu_{1}e^{\frac{(k(N)-i)\gamma_{n}}{\sqrt{k(N)}}}\right|^{2+\delta}
≤C1​(N)​[|μ1|2+δk​(1)1/2+δ/4​e(k​(1)−i)​(2+δ)​γnk​(1)+⋯+|μN|2+δk​(N)1/2+δ/4​e(k​(N)−i)​(2+δ)​γnk​(N)].\displaystyle\leq C_{1}(N)\left[\dfrac{\lvert\mu_{1}\rvert^{2+\delta}}{k(1)^{1/2+\delta/4}}e^{\frac{(k(1)-i)(2+\delta)\gamma_{n}}{\sqrt{k(1)}}}+\cdots+\dfrac{\lvert\mu_{N}\rvert^{2+\delta}}{k(N)^{1/2+\delta/4}}e^{\frac{(k(N)-i)(2+\delta)\gamma_{n}}{\sqrt{k(N)}}}\right].

This implies that

∑i=1k​(N)−1|ci|2+δ∼C1​(N)​|μ1|2+δ​1k​(1)δ/4​(2+δ)​|γn|+O​(1k​(2)δ/4​|γn|)=o​(1|γn|).\sum_{i=1}^{k(N)-1}\lvert c_{i}\rvert^{2+\delta}\sim C_{1}(N)\lvert\mu_{1}\rvert^{2+\delta}\dfrac{1}{k(1)^{\delta/4}(2+\delta)\lvert\gamma_{n}\rvert}+O\left(\dfrac{1}{k(2)^{\delta/4}\lvert\gamma_{n}\rvert}\right)=o\left(\dfrac{1}{\lvert\gamma_{n}\rvert}\right).

Now we can easily check the Liapounov’s condition, where

(∑i=1k​(N)−1|ci|2+δ​E​|ξi|2+δ)1/(2+δ)(∑i=1k​(N)−1ci2​E​ξi2)1/2=o​(|γn|1/2−1/(2+δ))=op​(1).\dfrac{\left(\sum_{i=1}^{k(N)-1}\lvert c_{i}\rvert^{2+\delta}E\lvert\xi_{i}\rvert^{2+\delta}\right)^{1/(2+\delta)}}{\left(\sum_{i=1}^{k(N)-1}c_{i}^{2}E\xi_{i}^{2}\right)^{1/2}}=o\left(\lvert\gamma_{n}\rvert^{1/2-1/(2+\delta)}\right)=o_{p}(1).

Then by Liapounov central limit theorem (Theorem 27.3, p.362 of Billingsley [1995]), we have

2​|γn|​[τ1,τ2,⋯,τN]→𝑑E​ξ02​[η1,η2,⋯,ηN],\sqrt{2\lvert\gamma_{n}\rvert}\left[\tau_{1},\tau_{2},\cdots,\tau_{N}\right]\xrightarrow{d}\sqrt{E\xi_{0}^{2}}\left[\eta_{1},\eta_{2},\cdots,\eta_{N}\right],

where η1,η2,⋯,ηN\eta_{1},\eta_{2},\cdots,\eta_{N} are independent standard normal random variables.

Now we have to check the relationship between τm\tau_{m} and τm∗\tau_{m}^{*}. Note by k−1/2​(eγnk−1)−1=(γn+o​(1))−1k^{-1/2}\left(e^{\frac{\gamma_{n}}{\sqrt{k}}}-1\right)^{-1}=\left(\gamma_{n}+o(1)\right)^{-1}, we have

1k​∑j=ik−1ej​γnk−|γn|−1​ei​γnk\displaystyle\dfrac{1}{\sqrt{k}}\sum_{j=i}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}-\lvert\gamma_{n}\rvert^{-1}e^{\frac{i\gamma_{n}}{\sqrt{k}}} =1k​ek​γnk−ei​γnkeγnk−1−|γn|−1​ei​γnk\displaystyle=\dfrac{1}{\sqrt{k}}\dfrac{e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{i\gamma_{n}}{\sqrt{k}}}}{e^{\frac{\gamma_{n}}{\sqrt{k}}}-1}-\lvert\gamma_{n}\rvert^{-1}e^{\frac{i\gamma_{n}}{\sqrt{k}}}
=(γn+o​(1))−1​(ek​γnk−ei​γnk)−|γn|−1​ei​γnk\displaystyle=\left(\gamma_{n}+o(1)\right)^{-1}\left(e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{i\gamma_{n}}{\sqrt{k}}}\right)-\lvert\gamma_{n}\rvert^{-1}e^{\frac{i\gamma_{n}}{\sqrt{k}}}
=(γn−1+O​(1))​ek​γnk−ei​γnk​O​(1).\displaystyle=\left(\gamma_{n}^{-1}+O(1)\right)e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{i\gamma_{n}}{\sqrt{k}}}O(1).

Then, we know

E​[2​|γn|3​τm∗−2​|γn|​τm]2\displaystyle E\left[\sqrt{2\lvert\gamma_{n}\rvert^{3}}\tau_{m}^{*}-\sqrt{2\lvert\gamma_{n}\rvert}\tau_{m}\right]^{2} =2​|γn|3k​E​[1k​∑i=1k−1(∑j=ik−1ej​γk)​ξk−i−|γn|−1​∑i=1k−1ei​γnk​ξk−i]2\displaystyle=\dfrac{2\lvert\gamma_{n}\rvert^{3}}{\sqrt{k}}E\left[\dfrac{1}{\sqrt{k}}\sum_{i=1}^{k-1}\left(\sum_{j=i}^{k-1}e^{\frac{j\gamma}{\sqrt{k}}}\right)\xi_{k-i}-\lvert\gamma_{n}\rvert^{-1}\sum_{i=1}^{k-1}e^{\frac{i\gamma_{n}}{\sqrt{k}}}\xi_{k-i}\right]^{2}
=2​|γn|3k​E​ξ02​∑i=1k−1(1k​∑j=ik−1ej​γnk−|γn|−1​ei​γnk)2\displaystyle=\dfrac{2\lvert\gamma_{n}\rvert^{3}}{\sqrt{k}}E\xi_{0}^{2}\sum_{i=1}^{k-1}\left(\dfrac{1}{\sqrt{k}}\sum_{j=i}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}-\lvert\gamma_{n}\rvert^{-1}e^{\frac{i\gamma_{n}}{\sqrt{k}}}\right)^{2}
∼2​|γn|3k​E​ξ02​(k​γn−2​ek​γn+k2​|γ|−2​γn−1​ek​γn​k|γ|)\displaystyle\sim\dfrac{2\lvert\gamma_{n}\rvert^{3}}{\sqrt{k}}E\xi_{0}^{2}\left(k\gamma_{n}^{-2}e^{\sqrt{k}\gamma_{n}}+\dfrac{\sqrt{k}}{2\lvert\gamma\rvert}-2\gamma_{n}^{-1}e^{\sqrt{k}\gamma_{n}}\dfrac{\sqrt{k}}{\lvert\gamma\rvert}\right)
=2​E​ξ02​O​(k​|γn|​ek​γn)+op​(1)\displaystyle=2E\xi_{0}^{2}O\left(\sqrt{k}\lvert\gamma_{n}\rvert e^{\sqrt{k}\gamma_{n}}\right)+o_{p}(1)
=op​(1),\displaystyle=o_{p}(1),

where the last equality comes from the well known limits of x​e−xxe^{-x},

limx→∞xex=limx→∞1ex=0andlimx→0xex=0.\lim\limits_{x\rightarrow\infty}\dfrac{x}{e^{x}}=\lim\limits_{x\rightarrow\infty}\dfrac{1}{e^{x}}=0\quad\text{and}\quad\lim\limits_{x\rightarrow 0}\dfrac{x}{e^{x}}=0.

Therefore, we have

2​|γn|3​[τ1∗,τ2∗,⋯,τN∗]→𝑑E​ξ02​[η1,η2,⋯,ηN],\sqrt{2\lvert\gamma_{n}\rvert^{3}}\left[\tau_{1}^{*},\tau_{2}^{*},\cdots,\tau_{N}^{*}\right]\xrightarrow{d}\sqrt{E\xi_{0}^{2}}\left[\eta_{1},\eta_{2},\cdots,\eta_{N}\right],

Now combine the results above, we have, for each k=⌊n​tm⌋k={\lfloor nt_{m}\rfloor}, m=1,⋯,Nm=1,\cdots,N

2​|γn|3αn​k1/4​1E​ξ02​(σk2ω​kk/2−∑j=1k−1ej​γnk)→𝑑𝒩​(0,1).\displaystyle\dfrac{\sqrt{2\lvert\gamma_{n}\rvert^{3}}}{\alpha_{n}k^{1/4}}\dfrac{1}{\sqrt{E\xi_{0}^{2}}}\left(\dfrac{\sigma_{k}^{2}}{\omega{k}^{k/2}}-\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\right)\xrightarrow{d}\mathcal{N}(0,1).

Now, for returns, we know from the above result that

|γn|​σk2ω​k(k+1)/2−1=Op​(αn​n1/4|γn|)=op​(1).\displaystyle\dfrac{\lvert\gamma_{n}\rvert\sigma_{k}^{2}}{\omega{k}^{(k+1)/2}}-1=O_{p}\left(\dfrac{\alpha_{n}n^{1/4}}{\sqrt{\lvert\gamma_{n}\rvert}}\right)=o_{p}(1).

Therefore, by the return equation, we have

(|γn|ω​k(k+1)/2)1/2​uk=(|γn|​σk2ω​k(k+1)/2)1/2​εk∼εk.\left(\dfrac{\lvert\gamma_{n}\rvert}{\omega{k}^{(k+1)/2}}\right)^{1/2}u_{k}=\left(\dfrac{\lvert\gamma_{n}\rvert\sigma_{k}^{2}}{\omega{k}^{(k+1)/2}}\right)^{1/2}\varepsilon_{k}\sim\varepsilon_{k}.

∎

Proof of Theorem 2.

Similar to Theorem 1, when γn=0\gamma_{n}=0, the volatility admits the decomposition. Then, for σk,12\sigma_{k,1}^{2}, by central limit theorem, we know

αnk​∑j=1kξk−j=Op​(αn)=op​(1)\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{j=1}^{k}\xi_{k-j}=O_{p}(\alpha_{n})=o_{p}(1)

which, combining with Proposition 1, implies that

|σk,12|=Op​(kk/2)\left\lvert\sigma_{k,1}^{2}\right\rvert=O_{p}\left(k^{k/2}\right)

For σk,22\sigma_{k,2}^{2}, note that we have established equation (2), then by Proposition 1, we have

|σk,22|=Op​(kk/2​αn2).\left\lvert\sigma_{k,2}^{2}\right\rvert=O_{p}\left(k^{k/2}\alpha_{n}^{2}\right).

For σk,32\sigma_{k,3}^{2}, by Proposition 1 we have

|σk,32|=Op​(kk/2​αn2).\left\lvert\sigma_{k,3}^{2}\right\rvert=O_{p}(k^{k/2}\alpha_{n}^{2}).

Lastly, for σk,42\sigma_{k,4}^{2}, note by Lemma 5.1 in Berkes et al. [2005], for k=⌊n​t⌋k=\lfloor nt\rfloor, t∈(0,1)t\in(0,1) , we have

1n3/2​∑j=1⌊n​t⌋−1∑i=1jξk−i→𝑑E​ξ02​∫0tx​𝑑W​(x),\dfrac{1}{n^{3/2}}\sum_{j=1}^{\lfloor nt\rfloor-1}\sum_{i=1}^{j}\xi_{k-i}\xrightarrow{d}\sqrt{E\xi_{0}^{2}}\int_{0}^{t}xdW(x),

where W​(x)W(x) is a Wiener process.

Therefore, for k​(m)=⌊n​tm⌋k(m)=\lfloor nt_{m}\rfloor, m=1,⋯,Nm=1,\cdots,N, we have

k​(m)1/2n3/2​αn​(σk​(m)2ω​k​(m)k​(m)/2−k​(m))=1n3/2​∑j=1⌊n​tm⌋−1∑i=1jξk​(m)−i+op​(1)→𝑑E​ξ02​∫0tmx​𝑑W​(x).\displaystyle\dfrac{k(m)^{1/2}}{n^{3/2}\alpha_{n}}\left(\dfrac{\sigma_{k(m)}^{2}}{\omega{k(m)}^{k(m)/2}}-k(m)\right)=\dfrac{1}{n^{3/2}}\sum_{j=1}^{\lfloor nt_{m}\rfloor-1}\sum_{i=1}^{j}\xi_{k(m)-i}+o_{p}(1)\xrightarrow{d}\sqrt{E\xi_{0}^{2}}\int_{0}^{t_{m}}xdW(x).

Further, note the results above implies that

σk​(m)2ω​k​(m)k​(m)/2+1−1=Op​((nk)3/2​αn)=op​(1).\dfrac{\sigma_{k(m)}^{2}}{\omega{k(m)}^{k(m)/2+1}}-1=O_{p}\left(\left(\dfrac{n}{k}\right)^{3/2}\alpha_{n}\right)=o_{p}(1).

Hence, by return equation, we obtain

(1ω​k​(m)k​(m)/2+1)1/2​uk​(m)=(σk​(m)2ω​k​(m)k​(m)/2+1)1/2​εk​(m)→𝑑εk​(m).\left(\dfrac{1}{\omega{k(m)}^{k(m)/2+1}}\right)^{1/2}u_{k(m)}=\left(\dfrac{\sigma_{k(m)}^{2}}{\omega{k(m)}^{k(m)/2+1}}\right)^{1/2}\varepsilon_{k(m)}\xrightarrow{d}\varepsilon_{k(m)}.

∎

Proof of Theorem 3.

Similar to proof of Theorem 1, when γn>0\gamma_{n}>0, the volatility admits the additive representation. For σk,12\sigma_{k,1}^{2}, similar to that in Theorem 1,

|σk,12|=Op​(kk/2​ek​γn).\left\lvert\sigma_{k,1}^{2}\right\rvert=O_{p}\left(k^{k/2}e^{\sqrt{k}\gamma_{n}}\right).

For σk,22\sigma_{k,2}^{2}, by Proposition 1 and equation (2), we have the relation

|σk,22|\displaystyle\left\lvert\sigma_{k,2}^{2}\right\rvert =|ω​kk/2​∑j=1k−1j​ej​γnk​(1+op​(1))​1j​Rk,j(3)|\displaystyle=\left\lvert\omega k^{k/2}\sum_{j=1}^{k-1}je^{\frac{j\gamma_{n}}{\sqrt{k}}}(1+o_{p}(1))\dfrac{1}{j}R_{k,j}^{(3)}\right\rvert
=Op​(1)​ω​kk/2​(αn2+γn2)​ek​γnk−eγnkeγnk−1\displaystyle=O_{p}(1)\omega k^{k/2}(\alpha_{n}^{2}+\gamma_{n}^{2})\dfrac{e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{\gamma_{n}}{\sqrt{k}}}}{e^{\frac{\gamma_{n}}{\sqrt{k}}}-1}
<Op​(kk/2​(αn2+γn2)​k​ek​γnγn),\displaystyle<O_{p}\left(k^{k/2}\left(\alpha_{n}^{2}+\gamma_{n}^{2}\right)\dfrac{\sqrt{k}e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\right),

where the last inequality comes from the fact that

ek​γnk−eγnkeγnk−1<ek​γnγn/k.\dfrac{e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{\gamma_{n}}{\sqrt{k}}}}{e^{\frac{\gamma_{n}}{\sqrt{k}}}-1}<\dfrac{e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}/\sqrt{k}}.

For σk,32\sigma_{k,3}^{2}, by Proposition 1, we have

|σk,32|\displaystyle\left\lvert\sigma_{k,3}^{2}\right\rvert =|ω​kk/2​∑j=1k−1ej​γnk​(j​log⁡log⁡j)​1j​log⁡log⁡j​Rk,j(2)|\displaystyle=\left\lvert\omega k^{k/2}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\left(j\log\log j\right)\dfrac{1}{j\log\log j}R_{k,j}^{(2)}\right\rvert
=Op​(1)​ω​kk/2​(k​log⁡log⁡k)​αn2k​ek​γnk−eγnkeγnk−1\displaystyle=O_{p}(1)\omega k^{k/2}\left(k\log\log k\right)\dfrac{\alpha_{n}^{2}}{k}\dfrac{e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{\gamma_{n}}{\sqrt{k}}}}{e^{\frac{\gamma_{n}}{\sqrt{k}}}-1}
<Op​(kk/2​(αn2​log⁡log⁡k)​k​ek​γnγn).\displaystyle<O_{p}\left(k^{k/2}\left(\alpha_{n}^{2}\log\log k\right)\dfrac{\sqrt{k}e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\right).

Lastly, for σk,42\sigma_{k,4}^{2}, we have

σk,42=ω​kk/2​∑j=1k−1ej​γnk+ω​kk/2​αnk​∑j=1k−1ej​γnk​∑i=1jξk−i.\sigma_{k,4}^{2}=\omega k^{k/2}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}+\omega k^{k/2}\dfrac{\alpha_{n}}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}.

Now, we introduce the following lemma to assist the proof.

Lemma 1.

If Assumption 1 and 2 hold, then

γn2k​e−2​k​γn​E​(1k​∑j=1k−1ej​γnk​∑i=1jξk−i−ek​γnγn​∑i=1k−1ξi)2→0.\dfrac{\gamma_{n}^{2}}{k}e^{-2\sqrt{k}\gamma_{n}}E\left(\dfrac{1}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}-\dfrac{e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\sum_{i=1}^{k-1}\xi_{i}\right)^{2}\rightarrow 0.

Then by Lemma 1, we have

γn​e−k​γnk​αn​(σk,42ω​kk/2−∑j=1k−1ej​γnk)\displaystyle\dfrac{\gamma_{n}e^{-\sqrt{k}\gamma_{n}}}{\sqrt{k}\alpha_{n}}\left(\dfrac{\sigma_{k,4}^{2}}{\omega k^{k/2}}-\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\right) =γn​e−k​γnk​1k​∑j=1k−1ej​γnk​∑i=1jξk−i+op​(1)=1k​∑i=1k−1ξi+op​(1).\displaystyle=\dfrac{\gamma_{n}e^{-\sqrt{k}\gamma_{n}}}{\sqrt{k}}\dfrac{1}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}+o_{p}(1)=\dfrac{1}{\sqrt{k}}\sum_{i=1}^{k-1}\xi_{i}+o_{p}(1).

Therefore, by Donsker’s theorem, we obtain that, for k​(m)=⌊n​tm⌋k(m)=\lfloor nt_{m}\rfloor, tm∈(0,1)t_{m}\in(0,1) and m=1,2,⋯,Nm=1,2,\cdots,N,

γn​e−k​(m)​γnk​(m)​αn​1E​ξ02​(σk​(m)2ω​k​(m)k​(m)/2−∑j=1k​(m)−1ej​γnk​(m))⇒W​(tm),\dfrac{\gamma_{n}e^{-\sqrt{k(m)}\gamma_{n}}}{\sqrt{k(m)}\alpha_{n}}\dfrac{1}{\sqrt{E\xi_{0}^{2}}}\left(\dfrac{\sigma_{k(m)}^{2}}{\omega k(m)^{k(m)/2}}-\sum_{j=1}^{k(m)-1}e^{\frac{j\gamma_{n}}{\sqrt{k(m)}}}\right)\Rightarrow W(t_{m}),

where W​(t)W(t) is a finite dimensional Wiener process.

Further, note that

γnk​e−k​γn​(∑j=1k−1ej​γnk−k​ek​γnγn)=o​(1),\dfrac{\gamma_{n}}{\sqrt{k}}e^{-\sqrt{k}\gamma_{n}}\left(\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}-\dfrac{\sqrt{k}e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\right)=o(1),

then by the result above we know

γn​e−k​(m)​γnk​(m)​(σk​(m)2ω​k​(m)k​(m)/2−∑j=1k​(m)−1ej​γnk​(m))=Op​(αn)=op​(1).\dfrac{\gamma_{n}e^{-\sqrt{k(m)}\gamma_{n}}}{\sqrt{k(m)}}\left(\dfrac{\sigma_{k(m)}^{2}}{\omega k(m)^{k(m)/2}}-\sum_{j=1}^{k(m)-1}e^{\frac{j\gamma_{n}}{\sqrt{k(m)}}}\right)=O_{p}(\alpha_{n})=o_{p}(1).

Hence, by return equantion, we derive

(γn​e−k​γnω​k(k+1)/2)1/2​uk=(γn​e−k​γnω​k(k+1)/2​σk2)1/2​εk∼εk.\left(\dfrac{\gamma_{n}e^{-\sqrt{k}\gamma_{n}}}{\omega k^{(k+1)/2}}\right)^{1/2}u_{k}=\left(\dfrac{\gamma_{n}e^{-\sqrt{k}\gamma_{n}}}{\omega k^{(k+1)/2}}\sigma_{k}^{2}\right)^{1/2}\varepsilon_{k}\sim\varepsilon_{k}.

∎

Proof of Lemma 1. Note that

1k​∑j=1k−1ej​γnk​∑i=1jξk−i=1k​∑i=1k−1(∑j=ik−1ej​γnk)​ξk−iand∑i=1k−1ξi=∑i=1k−1ξk−i,\dfrac{1}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}=\dfrac{1}{\sqrt{k}}\sum_{i=1}^{k-1}\left(\sum_{j=i}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\right)\xi_{k-i}\quad\text{and}\quad\sum_{i=1}^{k-1}\xi_{i}=\sum_{i=1}^{k-1}\xi_{k-i},

Then,

E​(1k​∑j=1k−1ej​γnk​∑i=1jξk−i−ek​γnγn​∑i=1k−1ξi)2\displaystyle E\left(\dfrac{1}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}-\dfrac{e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\sum_{i=1}^{k-1}\xi_{i}\right)^{2} =E​ξ02​∑i=1k−1(1k​∑j=ik−1ej​γnk−k​ek​γnγn)2\displaystyle=E\xi_{0}^{2}\sum_{i=1}^{k-1}\left(\dfrac{1}{\sqrt{k}}\sum_{j=i}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}-\dfrac{\sqrt{k}e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\right)^{2}
=E​ξ02k​∑i=1k−1(ek​γnk−ei​γnkeγnk−1−k​ek​γnγn)2.\displaystyle=\dfrac{E\xi_{0}^{2}}{k}\sum_{i=1}^{k-1}\left(\dfrac{e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{i\gamma_{n}}{\sqrt{k}}}}{e^{\frac{\gamma_{n}}{\sqrt{k}}}-1}-\dfrac{\sqrt{k}e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\right)^{2}.

Note by Taylor expansion,

|ek​γnk−ei​γnkeγnk−1−k​ek​γnγn|≤C1​(k​ei​γnkγn+ek​γn),\left\lvert\dfrac{e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{i\gamma_{n}}{\sqrt{k}}}}{e^{\frac{\gamma_{n}}{\sqrt{k}}}-1}-\dfrac{\sqrt{k}e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\right\rvert\leq C_{1}\left(\dfrac{\sqrt{k}e^{\frac{i\gamma_{n}}{\sqrt{k}}}}{\gamma_{n}}+e^{\sqrt{k}\gamma_{n}}\right),

which implies that

∑i=1k−1(ek​γnk−ei​γnkeγnk−1−k​ek​γnγn)2\displaystyle\sum_{i=1}^{k-1}\left(\dfrac{e^{\frac{k\gamma_{n}}{\sqrt{k}}}-e^{\frac{i\gamma_{n}}{\sqrt{k}}}}{e^{\frac{\gamma_{n}}{\sqrt{k}}}-1}-\dfrac{\sqrt{k}e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\right)^{2} ≤2​C12​(∑i=1k−1k​e2​i​γnkγn2+k​e2​k​γn)\displaystyle\leq 2C_{1}^{2}\left(\sum_{i=1}^{k-1}\dfrac{ke^{\frac{2i\gamma_{n}}{\sqrt{k}}}}{\gamma_{n}^{2}}+ke^{2\sqrt{k}\gamma_{n}}\right)
=O​(1)​(kγn2​e2​k​γnk−e2​γnke2​γnk−1+k​e2​k​γn)\displaystyle=O(1)\left(\dfrac{k}{\gamma_{n}^{2}}\dfrac{e^{\frac{2k\gamma_{n}}{\sqrt{k}}}-e^{\frac{2\gamma_{n}}{\sqrt{k}}}}{e^{\frac{2\gamma_{n}}{\sqrt{k}}}-1}+ke^{2\sqrt{k}\gamma_{n}}\right)
=O​(1)​(kγn3​e2​k​γn+k​e2​k​γn).\displaystyle=O(1)\left(\dfrac{k}{\gamma_{n}^{3}}e^{2\sqrt{k}\gamma_{n}}+ke^{2\sqrt{k}\gamma_{n}}\right).

Now we can see that

γn2k​e−2​k​γn​E​(1k​∑j=1k−1ej​γnk​∑i=1jξk−i−ek​γnγn​∑i=1k−1ξi)2\displaystyle\dfrac{\gamma_{n}^{2}}{k}e^{-2\sqrt{k}\gamma_{n}}E\left(\dfrac{1}{\sqrt{k}}\sum_{j=1}^{k-1}e^{\frac{j\gamma_{n}}{\sqrt{k}}}\sum_{i=1}^{j}\xi_{k-i}-\dfrac{e^{\sqrt{k}\gamma_{n}}}{\gamma_{n}}\sum_{i=1}^{k-1}\xi_{i}\right)^{2} =O​(1)​γn2k​e−2​k​γn​E​ξ02k​(kγn3​e2​k​γn+k​e2​k​γn)=op​(1).\displaystyle=O(1)\dfrac{\gamma_{n}^{2}}{k}e^{-2\sqrt{k}\gamma_{n}}\dfrac{E\xi_{0}^{2}}{k}\left(\dfrac{k}{\gamma_{n}^{3}}e^{2\sqrt{k}\gamma_{n}}+ke^{2\sqrt{k}\gamma_{n}}\right)=o_{p}(1).

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