3.1 Empirical Size under $H_0$

Under the null hypothesis, our data generating process (DGP) is a GARCH(1,1) model:

$$ \begin{align*} y_i=\sigma_i\epsilon_i\quad\;\; \text{and}\quad \;\;\sigma_i^2=\omega_0+\alpha_0 y_{i-1}^2+\beta_0 \sigma_{i-1}^2,\;\;\;1\leq i<N. \end{align*} $$

For the distribution of the innovation term, we have two settings: (i) $\epsilon _i$ follows a standard normal distribution $\mathcal {N} (0,1)$ , and (ii) $\epsilon _i$ follows the skewed t distribution of Hansen (Reference Hansen1994) with degrees of freedom $\nu $ and skewness $\lambda $ . We use $\nu =10$ and $\lambda =-0.15$ , which are taken from Patton (Reference Patton2013) and are close to the typical values implied by empirical financial returns.

We consider all three cases of GARCH(1,1). In the stationary case, the parameter values are set to $(\omega _0, \alpha _0, \beta _0)^\top = (0.014, 0.084, 0.905)^\top $ . These values are taken from Francq and Zakoian (Reference Francq and Zakoian2019) (Table 7.4, S&P 500) and reflect real data of stationary financial returns. In the explosive case, the parameter values are set to $(\omega _0, \alpha _0, \beta _0)^\top = (0.014, 0.084, 1.000)^\top $ , which violates the strict stationarity condition of GARCH(1,1). In the “boundary” case, the parameter values depend on the distribution of the innovation term: they are set to $(\omega _0, \alpha _0, \beta _0)^\top = (0.014, 0.084, 0.9219)^\top $ for the standard normal distribution, and $(\omega _0, \alpha _0, \beta _0)^\top = (0.014, 0.084, 0.9238)^\top $ for the skewed t distribution.

To implement the test based on Theorem 2.1(ii), we require a weight function that satisfies Assumption 2.4. We investigate the impact of different shapes of the weight function by specifying it as $w(t)=(t(t-1))^\kappa $ , where $\kappa = 0, 0.15, 0.25$ , and $0.35$ . From Theorem 2.1(i), we calculate the test statistic

(3.1) $$ \begin{align} T_N = \sup_{0 < t < 1} \dfrac{Z_N (t)}{w(t)}, \end{align} $$

using the steps detailed in the Appendix.

To obtain the critical values, we simulate two independent Brownian bridges $B_1(t)$ and $B_2(t)$ on a grid of 100,000 equally spaced points in $ \left [ 0, 1\right ] $ and calculate the quantity $ \sup _{0 < t < 1} \dfrac {1}{w(t)} \left ( B_1^2(t) + B_2^2(t)\right ) ^{1/2} $ . This is repeated 100,000 times, and we obtain the empirical percentile values of this quantity at the 90%, 95%, and 99%, which are corresponding to the critical values at the 10%, 5%, and 1% levels. The resulting critical values are shown in Table 1.

Table 1 Critical values.

We now perform the Monte Carlo simulation 10,000 times and record the computed test statistics. By comparing these simulated statistics with the critical values in Table 1, we obtain the empirical size under the null hypothesis. Table 2 reports the empirical sizes at the 5% significance level with a range of values of $\kappa $ for the three GARCH(1,1) cases. The differences between the three cases are marginal, with slightly higher sizes observed in the explosive case. Additionally, the empirical sizes under the normal and skewed t innovations are similar. Notably, the empirical size depends on the choice of $\kappa $ . Recall that $\kappa $ controls the shape of the weight function, which can be chosen arbitrarily between 0 and 0.5. Our simulation reveals that a smaller $\kappa $ is more conservative in rejection, while a larger $\kappa $ leads to a higher rejection percentage. Generally, our proposed test has reasonably good size control with $\kappa =0.15$ or $0.25$ , whereas it is under-sized for $\kappa =0$ and over-sized for $\kappa =0.35$ . Overall, we recommend $\kappa =0.15$ because its empirical size is typically close to or slightly below the theoretical levels, and there is no compromise in the empirical power with this choice of $\kappa $ , as shown in Section 3.2.

Table 2 Empirical size.

3.2 Empirical Power under $H_A$

We now turn to the analysis of the empirical power. The DGP under $H_A$ is similar to that under $H_0$ but with a change in one model parameter at time $k^*$ . We set the time of the change at $k^* = 0.5N$ , corresponding to the middle of the sample period.Footnote ¹

There are various possible scenarios under the alternative hypothesis; we consider the following representative scenarios. The parameter values of GARCH(1,1) before $k^*$ are set to be the same as those in the DGP of the three GARCH cases under the null. Since $\omega $ cannot be consistently estimated in the explosive regime of GARCH(1,1), we analyze a change in either $\alpha $ or $\beta $ in the simulation under $H_A$ . We consider both positive and negative changes in $\alpha $ or $\beta $ . Specifically, after the change at $k^*$ , the parameter values are as follows:

• • $H_{A,1}$ : a negative change in $\beta $ , $\beta_{\ast} = \beta _0 - 0.05$ ,

• • $H_{A,2}$ : a positive change in $\beta $ , $\beta_{\ast} = \beta _0 + 0.05$ ,

• • $H_{A,3}$ : a negative change in $\alpha $ , $\alpha _{\ast} = \alpha _0 - 0.05$ ,

• • $H_{A,4}$ : a positive change in $\alpha $ , $\alpha _{\ast} = \alpha _0 + 0.05$ .

Table 3 reports the empirical power for changes in the stationary, “boundary”, and explosive GARCH(1,1) cases. Several noticeable observations can be made. First, the empirical power increases with the sample size N under all scenarios, indicating the consistency of the proposed test. Second, our test has higher power when detecting changes in the explosive case than in the stationary and “boundary” cases. This can be attributed to the fact by that $\sigma _i^2$ and $y_i^2$ converge to infinity at a much faster rate (see Theorem 6.1), and the terms that are not in the limit are exponentially small. Thus, changes in the explosive case can be detected more easily. Third, the power is slightly lower under the skewed t innovation than under the normal innovation. This could be because the QMLE becomes the MLE for the normal innovation, and the derivatives are calculated exactly. Lastly, the choice of $\kappa $ has a nontrivial but not substantial impact on the power. Choosing $\kappa =0.15$ can provide satisfactory and balanced performance across a variety of scenarios.

Table 3 Empirical power.

4.1 Prototype Application

In the prototype application, we focus on the daily returns of the five individual stocks analyzed by Francq and Zakoïan (Reference Francq and Zakoïan2012), with the same sample period they selected. The five stocks are Icagen (Nasdaq:ICGN), Monarch Community Bancorp (Nasdaq:MCBF), KV Pharmaceutical (NYSE:KV-A), Community Bankers Trust (AMEX:BTC), and China MediaExpress (Nasdaq:CCME). The sample period is generally between 2007 and 2011, with KV-A from 2006 and CCME from 2009. These five stocks provide a useful demonstration because the nonstationary (explosive) assumption cannot be rejected for four of them at any reasonable significance level; see Table 8 in Francq and Zakoïan (Reference Francq and Zakoïan2012). Our change-point test can provide further insights into the stability of the two parameters $(\alpha , \beta )$ of GARCH(1,1) models for those five stocks.

Table 4 presents the results of the parameter estimation, the nonstationarity test of Francq and Zakoïan (Reference Francq and Zakoïan2012), and our change-point test for the five individual stock returns. We reproduce the parameter estimation and the nonstationary test results reported in Francq and Zakoïan (Reference Francq and Zakoïan2012). Regarding our change-point test, we reveal a change in the two parameters of the GARCH(1,1) model for the stock MCBF, which indicates the rejection of $H_0^*$ at the 5% significance level.

Table 4 Test results of change point.

We further use a change-point estimator

(4.1) $$ \begin{align} \hat{k}_N= \lfloor N \hat{t}^* \rfloor, \quad \text{ where } \; \hat{t}^* = \underset{0 < t < 1}{\arg\max}\dfrac{Z_N (t)}{w(t)}, \end{align} $$

to find the time of the change. It shows that the change date is on 9 February 2009 for MCBF. By splitting the samples, we reestimate the two parameters before and after the change. The estimates are $\hat {\alpha } = 0.118, \hat {\beta }=0.886$ before the change and $\hat {\alpha } = 0.013, \hat {\beta }=0.981$ after the change. This result suggests that, in this particular case, the ARCH effect is weaker after the change, while the GARCH effect becomes stronger.

4.2 Extensive Application

We repeat our prototype analysis on the universe of U.S. stocks in the CRSP database. We select all stocks with at least 2 years of records between 4 January 2011 and 30 December 2022 in the NYSE, AMEX, and NASDAQ stock markets, resulting in a total of 10,620 stocks. We choose this period to focus on the most recent decade. We note that not all stocks have observations covering the full period, because some stocks had an initial public offering (IPO) after 4 January 2011, or were delisted before 30 December 2022. The nonstationarity test of Francq and Zakoïan (Reference Francq and Zakoïan2012) suggests that there are 631 stocks for which the null hypothesis of nonstationarity (explosivity) cannot be rejected at the 1% significance level.

We apply our change-point test to the daily returnsFootnote ² of 10,620 stocks, regardless of whether they are in a stationary or explosive regime. We find evidence of change points in $(\alpha , \beta )$ for 3,416 stocks at the 5% level. Using the change-point estimator defined in (4.1), we further find the time of the change for stocks that are rejected by our test. Figure 1 shows a histogram of the times of detected changes. As can be seen, the largest number of changes occurred in first half of 2020 when the COVID-19 pandemic begun to affect the United States economy. Thus, our test is able to detect change points that align with a well-known market event, providing validation using a large dataset.

Figure 1 Histogram of times of changes in $(\alpha , \beta )$ for 3,416 U.S. stocks.

6.1 The Properties of $\sigma _i^2$ as $i\to \infty $

The properties of $\sigma _i^2$ depend on the sign of $E\log (\alpha _i\epsilon _0^2+\beta _0)$ . We consider the following cases:

(6.1) $$ \begin{align} E\log(\alpha_i\epsilon_0^2+\beta_0)<0, \end{align} $$

(6.2) $$ \begin{align} E\log(\alpha_i\epsilon_0^2+\beta_0)=0, \end{align} $$

and

(6.3) $$ \begin{align} E\log(\alpha_i\epsilon_0^2+\beta_0)>0. \end{align} $$

The case (6.1) is known as the stationary case, (6.2) as the “boundary” case, and (6.3) as the explosive case.

In the case (6.1), there is a unique stationary non anticipative sequence satisfying

(6.4) $$ \begin{align} x_i=h_i\epsilon_i,\quad\quad h_i^2=\omega_0 +\alpha_0x_{i-1}^2+\beta_0h_{i-1}^2,\quad\;\;-\infty<i<\infty. \end{align} $$

The next theorem shows that $\sigma _i^2$ is close to $h_i^2$ , as $i\to \infty $ , if (6.1) holds. We also show that in the “boundary” and explosive cases $\sigma _i^2\to \infty $ in probability but with different rates. Thus, $\sigma _i^2$ is explosive under (6.2) as well as (6.3).

Theorem 6.1. We assume that Assumptions 2.1 and 2.2 are satisfied. (i) If (6.1) holds, then there exist $\bar {\kappa }>0$ and $0<\bar {\rho }<1$ such that

(6.5) $$ \begin{align} E|\sigma_i^2-h_i^2|^{\bar{\kappa}}=O\left(\bar{\rho}^i\right). \end{align} $$

(ii) If (6.2) holds and

(6.6) $$ \begin{align} E|\log(\alpha_0\epsilon_0^2+\beta_0)|^{\bar{\nu}}<\infty\;\;\;\;\;\text{with some}\;\;\;\;\bar{\nu}>2, \end{align} $$

then for all $1/\bar {\nu }<\xi <1/2$

(6.7) $$ \begin{align} P\left\{\sigma_i^2<\exp \left(c_1i^{\xi}\right)\right\}\leq c_2\left(i^{\xi-1/2}+i^{1-\xi\bar{\nu}} \right) \end{align} $$

with some $c_1$ and $c_2$ . (iii) If (6.3) holds, then

$$ \begin{align*}\exp(-S(i-1))\sigma_i^2\to R\;\;\;\text{a.s.}, \end{align*} $$

where

$$ \begin{align*}R=\omega_0\sum_{k=0}^{\infty}\exp(-S(k))+\sigma_0^2(\alpha_0\epsilon_{0}^2+\beta_0)\geq \omega_0, \quad\quad P\{R=0\}=0, \end{align*} $$

and

(6.8) $$ \begin{align} S(j)=\sum_{k=1}^j\log (\alpha_0\epsilon_k^2+\beta_0),\quad\quad S(0)=0. \end{align} $$

For any ${\mathcal m}<\bar {{\mathcal m}}=E(\log (\alpha _0\epsilon _0^2+\beta _0),$ we have

(6.9) $$ \begin{align} \exp(-{\mathcal m} i)\sigma_i^2\stackrel{P}\to\infty. \end{align} $$

If (6.6) also holds, then for all $1/\bar {\nu }<\xi <1/2$

(6.10) $$ \begin{align} P\left\{ \sigma_i^2<\exp(\bar{ {\mathcal m}} i-c_1 i^\xi)\right\}\leq c_2\left( i^{1-\xi\bar{\nu}}+\exp(-c_3i \right)) \end{align} $$

with some $c_1, c_2$ , and $c_3$ .

Proof. We begin by proving Theorem 6.1(i). The recursion in (2.1) yields

$$ \begin{align*}\sigma_i^2=\omega_0+(\alpha_0\epsilon_i^2+\beta_0)\sigma_{i-1}^2, \end{align*} $$

which can be solved explicitly

(6.11) $$ \begin{align} \sigma_i^2&=\omega_0\sum_{\ell=1}^i\prod_{j=1}^{\ell-1}(\alpha_0\epsilon_{i-j}^2+\beta_0) +\sigma_0^2\prod_{\ell=0}^{i-1}(\alpha_0\epsilon_{\ell}^2+\beta_0). \end{align} $$

The explicit solution of the equation (6.4) is given by

$$ \begin{align*}h_i^2=\omega_0\sum_{\ell=1}^\infty\prod_{k=1}^{\ell-1}(\alpha_0\epsilon_{i-k}^2+\beta_0) \end{align*} $$

$(\prod _\emptyset =1$ ). If (6.1) holds, then there exists a $\bar {\kappa }>0$ such that

(6.12) $$ \begin{align} \bar{\rho}=E (\alpha_0\epsilon_0^2+\beta_0)^{\bar{\kappa}}<1 \end{align} $$

(cf. Berkes et al., Reference Berkes, Horváth and Kokoszka2003; Francq and Zakoian, Reference Francq and Zakoian2004). We can assume that $0<\bar {\kappa }<1$ , so by Jensen’s inequality

$$ \begin{align*} E\left(\sum_{\ell=i+1}^\infty\prod_{j=1}^{\ell-1}(\alpha_0\epsilon_{i-j}^2+\beta_0)\right)^{\bar{\kappa}} \leq \sum_{\ell=i+1}^\infty E\left(\prod_{j=1}^{\ell-1}(\alpha_0\epsilon_{i-j}^2+\beta_0)\right)^{\bar{\kappa}}\leq \sum_{\ell=i+1}^\infty \bar{\rho}^{\ell-1}{,} \end{align*} $$

and

$$ \begin{align*} E\left(\prod_{\ell=0}^{i-1}(\alpha_0\epsilon_{\ell}^2+\beta_0)\right)^{\bar{\kappa}}\leq \frac{\bar{\rho}^{i-1}}{1-\bar{\rho}}. \end{align*} $$

Hence, (6.5) in Theorem 6.1(i) is proven.

We now proceed with the proof of Theorem 6.1(ii). It follows from (6.11) that

(6.13) $$ \begin{align} \sigma_i^2&\geq \omega_0\sum_{\ell=1}^i\prod_{j=1}^{\ell-1}(\alpha_0\epsilon_{i-j}^2+\beta_0)\\ &\geq \omega_0\max_{1\leq \ell\leq i}\prod_{j=1}^{\ell-1}(\alpha_0\epsilon_{i-j}^2+\beta_0) \notag\\ &=\omega_0\exp\left(\max_{1\leq \ell\leq i}\sum_{j=1}^{\ell-1}\log(\alpha_0\epsilon_{i-j}^2+\beta_0)\right).\notag \end{align} $$

Using Assumption 2.1, we have

$$ \begin{align*} P\left\{\max_{1\leq \ell\leq i}\sum_{j=1}^{\ell-1}\log(\alpha_0\epsilon_{i-j}^2+\beta_0)<c_1i^{\xi} \right\} =P\left\{\max_{1\leq \ell\leq i-1}S(\ell)< c_1i^{\xi}\right\}, \end{align*} $$

where $S(\ell )$ is defined in (6.8). Let ${\mathcal m}_1^2=E(\log (\alpha _0\epsilon _0^2+\beta _0)^2{)}$ . Using the Komlós–Major–Tusnády approximation (cf. Theorem 2.6.7 in Csörgo and Révész, Reference Csörgo and Révész1981), there is a Wiener process $\{W(x), x\geq 0\}$ such that

$$ \begin{align*} P\left\{ \max_{0\leq x\leq i}|S(\lfloor x\rfloor)-{\mathcal m}_1W(x)|>x^\xi\right\}\leq c_3i^{1-\xi\bar{\nu}}. \end{align*} $$

By the scale transformation of $W(x)$ and Theorem 1.5.1 in Csörgő and Révész (1981), we have

$$ \begin{align*} P\left\{ \sup_{0\leq x \leq i}W(x)\leq c_4i^\xi \right\}=P\left\{|{\mathcal N}|\leq c_4 i^{\xi-1/2}\right\}\leq c_5 i^{\xi-1/2}, \end{align*} $$

where ${\mathcal N}$ is a standard normal random variable, completing the proof of (6.6) and (6.7) in Theorem 6.1(ii).

Lastly, we turn to the proof of Theorem 6.1(iii). The formula in (6.11) can be written as

(6.14) $$ \begin{align} \sigma_i^2 =\omega_0\sum_{\ell=1}^i\exp(S(i-1)-S(i-\ell))+\sigma_0^2(\alpha_0\epsilon_{0}^2+\beta_0)\exp(S(i-1)), \end{align} $$

where $S(j)$ is defined in (6.8). Thus, we obtain

$$ \begin{align*}\sigma_i^2=\exp(S(i-1))\left(\omega_0\bar{R}_i+\sigma_0^2(\alpha_0\epsilon_0^2+\beta_0)\right), \end{align*} $$

where

$$ \begin{align*}\bar{R}_i=\sum_{k=0}^{i-1}\exp(-S(k)). \end{align*} $$

According to the strong law of large numbers

$$ \begin{align*}\lim_{k\to\infty}\frac{1}{k}S(k)= E\log(\alpha_0\epsilon_0^2+\beta_0)>0\quad\quad \text{a.s.} \end{align*} $$

and therefore

$$ \begin{align*}\lim_{k\to\infty}\bar{R}_k=\bar{R}= \sum_{\ell=0}^{\infty}\exp(-S(\ell)), \quad\;\;\;\text{a.s.}\quad \end{align*} $$

and $P\{\bar {R}<\infty \}=1$ . We again use (6.13). By the Komlós–Major–Tusnády approximation (cf. Theorem 2.6.7 in Csörgő and Révész, 1981), we can define a Wiener process $\{W(x), x\geq 0\}$ such that

$$ \begin{align*} P\left\{ \max_{0\leq x \leq i} |S(\lfloor x\rfloor)-({\mathcal m}_1W(x)+\bar{{\mathcal m}}x) |>i^\xi \right\}\leq c_4 i^{1-\xi\bar{\nu}}. \end{align*} $$

By the scale transformation of the Wiener process, we have

$$ \begin{align*} \sup_{0\leq x\leq i}({\mathcal m}_1W(x)+\bar{{\mathcal m}}x)\stackrel{{\mathcal D}}{=}\frac{{\mathcal m}_1^2}{\bar{{\mathcal m}}}\sup_{0\leq u\leq i\bar{{\mathcal m}}^2/{\mathcal m}_1^2}(W(u)+u). \end{align*} $$

Applying again the Komlós–Major–Tusnády approximation, we can define a Poisson process $\{{\mathcal P}(x), x\geq 0\}$ such that

(6.15) $$ \begin{align} P\left\{ \sup_{1\leq x\leq i}|(W(x)+x)-{\mathcal P}(x)|>i^\xi \right\}\leq c_5 i^{1-\xi\bar{\nu}}. \end{align} $$

It can be seen that

$$ \begin{align*}\sup_{0\leq x \leq i}{\mathcal P}(x)={\mathcal P}(i). \end{align*} $$

Thus, we can use (6.15) to show that

$$ \begin{align*} P\left\{\max_{1\leq \ell\leq i}S(\ell)<c_6 x^\xi\right\}&\leq P\left\{W(i)+i\leq 2c_6x^\xi\right\}+c_7 i^{1-\xi\bar{\nu}}\\ &\leq P\left\{ {\mathcal N}<-i +2c_6x^\xi\right\}\\ &\leq 2 c_7 i^{1-\xi\bar{\nu}}+c_8\exp(-i^2/3), \end{align*} $$

where ${\mathcal N}$ is a standard normal random variable, completing the proof of (6.8) and (6.9) in Theorem 6.1(iii).

Remark 6.1. It follows from Assumption 2.1 and (6.5) that

$$ \begin{align*}E|y_i^2-x_i^2|^{\bar{\kappa}}=O\left(\bar{\rho}^i\right). \end{align*} $$

If $F(u)$ denotes the distribution function of $\epsilon _0^2$ and

$$ \begin{align*}|F(u)-F(v)|\leq c_1|u-v|^{c_2}\;\;\;\;\;\text{in a {neighborhood} of 0,} \end{align*} $$

then

$$ \begin{align*}P\left\{\epsilon_0^2\leq i^{-c_3/c_2}\right\}\leq c_1i^{-c_3}, \end{align*} $$

and therefore $y_i^2$ increases to $\infty $ at an exponential rate.

6.2 Proofs of Theorems 2.1 and 2.1

Proof of Theorem 2.1.

We show in Sections 6.3 and 6.4 that we can define Gaussian processes $\{\mathbf {W}_N(x), x\geq 0\}$ such that

(6.16) $$ \begin{align} \max_{x\in [1, N-1]}\left(\frac{N}{x(N-x)}\right)^\nu\left\| \bar{{\mathcal r}}_{ x }(\hat{\boldsymbol{\theta}}_N)-\boldsymbol \Gamma_N(x) \right\|=O_P(1) \end{align} $$

with some $\nu <1/2$ ,

$$ \begin{align*}\boldsymbol \Gamma_N(x)=\mathbf{W}_N(x)-\frac{x}{N}\mathbf{W}_N(N),\;\;\;0\leq x \leq N, \end{align*} $$

$E\mathbf {W}_N(x)=\mathbf {0}$ , and $E\mathbf {W}_N(x)\mathbf {W}^\top _N(y)=\min (x,y)\mathbf {D}$ . Checking the covariance functions, one can verify that

(6.17) $$ \begin{align} \left\{N^{-1/2}\mathbf{D}^{-1/2}\boldsymbol \Gamma_N(Nt), 0\leq t \leq 1\right\}\stackrel{{\mathcal D}}{=}\left\{\mathbf{B}(t), 0\leq t \leq 1 \right\}, \end{align} $$

(6.18) $$ \begin{align} \mathbf{B}(t)=\left(B_1(t), B_2(t)\right)^\top, \end{align} $$

where $\{B_1(t), 0\leq t \leq 1\}$ and $\{B_2(t), 0\leq t \leq 1\}$ are independent Brownian bridges. The approximation in (6.16) can be rewritten as

(6.19) $$ \begin{align} N^{1/2-\nu}\sup_{1/(N+1)\leq t \leq 1-1/(N+1)}&\frac{1}{(t(1-t))^{\nu}}\left\|N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)-N^{-1/2}\boldsymbol \Gamma_N(Nt) \right\|\\ &=O_P(1){,} \notag \end{align} $$

which implies that

(6.20) $$ \begin{align} \sup_{1/(N+1)\leq t \leq 1-1/(N+1)}&\frac{1}{(t(1-t))^{1/2}}\left\|N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)-N^{-1/2}\boldsymbol \Gamma_N(Nt) \right\| =O_P(1). \end{align} $$

We can show that (6.19) and (6.20) yield

(6.21) $$ \begin{align} \sup_{0<t<1}\frac{1}{w(t)}\frac{1}{N^{1/2}}\left\{\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N){\mathbf{D}}^{-1}\left(\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)\right)^\top\right\}^{1/2}\stackrel{{\mathcal D}}{\to}\sup_{0<t<1}\frac{1}{w(t)}\left\|\mathbf{B}(t)\right\|. \end{align} $$

Using Assumption 2.4(i) and the approximation in (6.19), we get for all $0<\delta <1/2$ that

(6.22) $$ \begin{align} \sup_{\delta\leq t\leq 1-\delta}\frac{1}{w(t)}\frac{1}{N^{1/2}}\left\{\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N){\mathbf{D}}^{-1}\left(\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)\right)^\top\right\}^{1/2}\stackrel{{\mathcal D}}{\to}\sup_{\delta\leq t\leq 1-\delta}\frac{1}{w(t)}\|\mathbf{B}(t)\|. \end{align} $$

If $I(w,c)<\infty $ with some $c>0$ , then

(6.23) $$ \begin{align} \lim_{t\to 0}\frac{t^{1/2}}{w(t)}=0 \end{align} $$

(cf. Section 4.1 in Csörgő and Horváth, 1993). We have, by (6.20), that

$$ \begin{align*} \sup_{1/(N+1)<t\leq \delta}&\frac{1}{w(t)}\left\| N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)-N^{-1/2}\boldsymbol \Gamma_N(Nt)\right\|\\ &=\sup_{1/(N+1)<t\leq \delta}\frac{t^{1/2}}{w(t)}\frac{1}{t^{1/2}}\left\| N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)-N^{-1/2}\boldsymbol \Gamma_N(Nt)\right\|\\ &=\sup_{1/(N+1)<t\leq \delta}\frac{t^{1/2}}{w(t)}O_P(1). \end{align*} $$

Hence, we conclude that

$$ \begin{align*} \lim_{\delta\to 0}\limsup_{N\to \infty}P\left\{\sup_{1/(N+1)\leq t\leq \delta}\frac{1}{w(t)}\left\| N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)-N^{-1/2}\boldsymbol \Gamma_N(Nt) \right\|>u \right\}=0 \end{align*} $$

for all $u>0$ and by symmetry

$$ \begin{align*} \lim_{\delta\to 0}\limsup_{N\to \infty}P\!\left\{\!\sup_{1-\delta\leq t\leq 1-1/(N+1)}\!\frac{1}{w(t)}\!\left\| N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)-N^{-1/2}\boldsymbol \Gamma_N(Nt) \right\|\!>\!u\! \right\}\!=\!0. \end{align*} $$

Thus, we get

$$ \begin{align*} \sup_{1/(N+1)\leq t\leq 1-1/(N+1)}\frac{1}{w(t)}\left\| N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)-N^{-1/2}\boldsymbol \Gamma_N(Nt) \right\|=o_P(1). \end{align*} $$

Using (6.23) and the definition of $\bar {{\mathcal r}}_{\lfloor (N+1)t\rfloor }$ , we obtain

$$ \begin{align*}\sup_{0<t\leq 1/(N+1)}\frac{1}{w(t)}\left\| N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor} (\hat{\boldsymbol{\theta}}_N) \right\|=o_P(1) \end{align*} $$

and

$$ \begin{align*}\sup_{N/(N+1)\leq t<1}\frac{1}{w(t)}\left\| N^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor} (\hat{\boldsymbol{\theta}}_N) \right\|=o_P(1). \end{align*} $$

Because of (6.17) and (6.18)

$$ \begin{align*} \sup_{0<t\leq \delta}\frac{1}{w(t)}\left\|N^{-1/2}\mathbf{D}^{-1/2}\boldsymbol \Gamma_N(Nt) \right\|\;\stackrel{{\mathcal D}}{=}\; \sup_{0<t\leq \delta}\frac{1}{w(t)}\left\|\mathbf{B}(t)\right\| \end{align*} $$

and

$$ \begin{align*}\lim_{\delta\to 0}\sup_{0<t\leq \delta}\frac{1}{w(t)}\left\|\mathbf{B}(t)\right\|=c_1\quad\quad a.s. \end{align*} $$

(cf. Section 4.1 in Csörgő and Horváth, 1993). By symmetry,

$$ \begin{align*} \sup_{1-\delta \leq t<1}\frac{1}{w(t)}\left\|N^{-1/2}\mathbf{D}^{-1/2}\boldsymbol \Gamma_N(Nt) \right\|\;\stackrel{{\mathcal D}}{=}\; \sup_{1-\delta \leq t<1}\frac{1}{w(t)}\left\|\mathbf{B}(t)\right\|, \end{align*} $$

and

$$ \begin{align*}\lim_{\delta\to 0}\sup_{1-\delta\leq t<1}\frac{1}{w(t)}\left\|\mathbf{B}(t)\right\|=c_2\quad\quad a.s. \end{align*} $$

completing the proof of the first part of Theorem 2.1.

According to the law of the iterated logarithm,

(6.24) $$ \begin{align} \frac{1}{(2\log \log N)^{1/2}}\sup_{1/(N+1)\leq t \leq 1-1/(N+1)}\frac{\|\mathbf{B}(t)\|}{(t(1-t))^{1/2}}\stackrel{P}{\to}1, \end{align} $$

Csörgő and Horváth (1993) proved that

(6.25) $$ \begin{align} A_{N,1}=\sup_{1/(N+1)\leq t \leq (\log N)^4/N}\frac{\|\mathbf{B}(t)\|}{(t(1-t))^{1/2}}=O_P((\log \log \log N)^{1/2}), \end{align} $$

(6.26) $$ \begin{align} A_{N,2}=\sup_{ 1-(\log N)^4/N\leq t \leq 1-1/(N+1)}\frac{\|\mathbf{B}(t)\|}{(t(1-t))^{1/2}}=O_P((\log \log \log N)^{1/2}). \end{align} $$

Now putting together (6.20) and (6.24)–(6.26), we obtain

(6.27) $$ \begin{align} \frac{1}{(2\log \log N)^{1/2}}\sup_{1/(N+1)\leq t \leq 1-1/(N+1)}\frac{1}{(t(1-t))^{1/2}} \left\|N^{-1/2}\mathbf{D}^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)\right\|\stackrel{P}{\to} 1, \end{align} $$

(6.28) $$ \begin{align} A_{N,3}=\sup_{1/(N+1)\leq t \leq (\log N)^4/N}&\frac{1}{(t(1-t))^{1/2}}\left\|N^{-1/2}\mathbf{D}^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)\right\|\\ &=O_P((\log \log \log N)^{1/2}) \notag{,} \end{align} $$

and

(6.29) $$ \begin{align} A_{N,4}=\sup_{ 1-(\log N)^4/N\leq t \leq 1-1/(N+1)}&\frac{1}{(t(1-t))^{1/2}}\left\|N^{-1/2}\mathbf{D}^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}(\hat{\boldsymbol{\theta}}_N)\right\|\\ &=O_P((\log \log \log N)^{1/2}).\notag \end{align} $$

It follows from (6.24) to (6.29) that

$$ \begin{align*} a(\log N)\max\left\{A_{N,1}, A_{N,2} \right\}-b(\log N)\;\stackrel{P}{\to}-\infty \end{align*} $$

and

$$ \begin{align*} a(\log N)\max\left\{A_{N,3}, A_{N,4} \right\}-b(\log N)\;\stackrel{P}{\to}-\infty \end{align*} $$

resulting in

$$ \begin{align*} \lim_{N\to \infty}&P\Bigg\{\sup_{1/(N+1)\leq t \leq 1-1/(N+1)}\frac{1}{(t(1-t))^{1/2}} \left\|N^{-1/2}\mathbf{D}^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}\right\|\\ &\hspace{2cm}=\sup_{(\log N)^4/(N+1)\leq t \leq 1-(\log N)^4/(N+1)}\frac{1}{(t(1-t))^{1/2}}\\ &\hspace{3cm}\left\|N^{-1/2}\mathbf{D}^{-1/2}\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}\right\| \Bigg\} =1. \end{align*} $$

Using (6.19), we conclude that

$$ \begin{align*} \sup_{(\log N)^4/(N+1)\leq t \leq 1-(\log N)^4/(N+1)}&\frac{1}{(t(1-t))^{1/2}} \left\|N^{-1/2}\mathbf{D}^{-1/2}\left(\bar{{\mathcal r}}_{\lfloor (N+1)t\rfloor}-\boldsymbol \Gamma_N(Nt)\right)\right\|\\ &=o_P((\log \log N)^{1/2}). \end{align*} $$

Csörgő and Horváth (1993) proved that

$$ \begin{align*} \lim_{N\to \infty}P\Bigg\{a(\log N)\sup_{(\log N)^4/(N+1)\leq t \leq 1-(\log N)^4/(N+1)}&\frac{\|\mathbf{B}(t)\|}{(t(1-t))^{1/2}}\leq x +b(\log N) \Bigg\}\\ &=\exp\left(-2e^{-x}\right), \end{align*} $$

and therefore the limit result in Theorem 2.1 follows directly from (6.17).

Proof of Theorem 2.2.

Since $\boldsymbol {\Theta }$ is compact and $\bar {\ell }_i(\boldsymbol {\theta })$ is continuous, the set $\tilde {\boldsymbol {\Theta }}$ of the possible limit points of $\hat {\boldsymbol {\theta }}_N$ is countable. It follows from the proofs in Sections 6.3 and 6.4 that

$$ \begin{align*}\frac{1}{N}\bar{{\mathcal r}}_{k^*}(\boldsymbol{\theta})\;\stackrel{P}{\to}\;{\mathcal a}_1(\boldsymbol{\theta}) \end{align*} $$

and

$$ \begin{align*}\frac{1}{N}\left(\bar{{\mathcal r}}_N(\boldsymbol{\theta})-\bar{{\mathcal r}}_{k^*}(\boldsymbol{\theta})\right)\;\stackrel{P}{\to}\;{\mathcal a}_2(\boldsymbol{\theta}), \end{align*} $$

uniformly in $\boldsymbol {\theta }\in \boldsymbol {\Theta }$ . We note that ${\mathcal a}_1(\boldsymbol {\theta })=\mathbf {0}$ if and only if the first two coordinates of $\boldsymbol {\theta }$ are $(\alpha _0,\beta _0)$ and similarly ${\mathcal a}_2(\boldsymbol {\theta })=\mathbf {0}$ if and only if the first two coordinates of $\boldsymbol {\theta }$ are $(\alpha _*,\beta _*)$ . Hence, for any $\tilde {\boldsymbol {\theta }}\in \tilde {\boldsymbol {\Theta }}$ ,

$$ \begin{align*}\max\left(\left\|{\mathcal a}_1(\tilde{\boldsymbol{\theta}})\right\|,\left\|{\mathcal a}_2(\tilde{\boldsymbol{\theta}})\right\|\right)>0. \end{align*} $$

We only need to show that

(6.30) $$ \begin{align} \inf\left\{\max\left(\left\|{\mathcal a}_1(\tilde{\boldsymbol{\theta}})\right\|,\left\|{\mathcal a}_2(\tilde{\boldsymbol{\theta}})\right\|\right):\;\tilde{\boldsymbol{\theta}}\in\tilde{\boldsymbol{\Theta}}\right\}>0. \end{align} $$

If the $\inf $ in (6.30) were 0, then we could construct a subsequence $\hat {\boldsymbol {\theta }}_{N_k}$ that would converge in probability to a point where the first two sets of coordinates are ${(}\alpha _0,\beta _0{)}$ and ${(}\alpha _*,\beta _*{)}$ , which contradicts Assumption 2.6.

6.3 Proof of (6.16) When $E\kern1pt\mathbf{log}\kern1pt (\alpha _0\epsilon _0^2+\beta _0)<0$

We use several results obtained by Berkes et al. (Reference Berkes, Horváth and Kokoszka2003) and Francq and Zakoian (Reference Francq and Zakoian2004). A survey on volatility processes with detailed proofs is given in Francq and Zakoian (Reference Francq and Zakoian2010). First, we define

$$ \begin{align*}\bar{h}^2_i(\boldsymbol{\theta})=\omega+\alpha x_{i-1}^2+\beta\bar{h}_{i-1}^2(\boldsymbol{\theta}), \end{align*} $$

the stationary version of $\bar {\sigma }^2_i(\boldsymbol {\theta })$ . Berkes et al. (Reference Berkes, Horváth and Kokoszka2003) proved that

$$ \begin{align*}\bar{h}^2_i(\boldsymbol{\theta})={\frak c}_0(\boldsymbol{\theta})+\sum_{\ell=1}^\infty{\frak c}_\ell(\boldsymbol{\theta}) x_{i-\ell}^2, \end{align*} $$

where the functions $\{{\frak c}_\ell (\boldsymbol {\theta }), 0\leq \ell <\infty \}$ are analytical, and

$$ \begin{align*}\sup\left\{|{\frak c}_\ell(\boldsymbol{\theta})|:\;\boldsymbol{\theta}\in\boldsymbol{\Theta}\right\}=O(\rho_1^\ell)\quad\;\;\text{with some}\quad\;\;0<\rho_1<1. \end{align*} $$

Hence, similarly to (6.5)

(6.31) $$ \begin{align} \sup\left\{|\bar{h}_i^2(\boldsymbol{\theta})-\bar{\sigma}_i^2(\boldsymbol{\theta})|:\;\boldsymbol{\theta}\in\boldsymbol{\Theta}\right\}=O(\rho_2^i)\quad\text{a.s.}\;\;\;\;\text{with some}\quad\;\;0<\rho_2<1. \end{align} $$

It is also known that

(6.32) $$ \begin{align} \|\hat{\boldsymbol{\theta}}_N-\boldsymbol{\theta}_0\|=O_P(N^{-1/2}) \end{align} $$

(cf. Berkes et al., Reference Berkes, Horváth and Kokoszka2003; Francq and Zakoian, Reference Francq and Zakoian2004). Furthermore, there exits a $\boldsymbol {\Theta }_0$ , a neighborhood of $\boldsymbol {\theta }_0$ , such that

(6.33) $$ \begin{align} \sup\left\{\|\nabla\bar{\sigma}_i^2(\boldsymbol{\theta})-\nabla\bar{h}_i^2(\boldsymbol{\theta})\|:\;\boldsymbol{\theta}\in\boldsymbol{\Theta}_0\right\}=O(\rho_3^i)\quad\text{a.s.}\;\;\;\;\text{with some}\quad\;\;0<\rho_3<1. \end{align} $$

It follows from the representation that $h_i^2$ is a Bernoulli shift; that is, there is a function $z: R^\infty \to R$ such that $h_i^2=z(\epsilon _{i-1}^2, \epsilon _{i-2}^2, \dots )$ . We show in the following lemma that $h^2_i$ is decomposable at an exponential rate. Let

$$ \begin{align*}h_{i,j}^2=\omega_0\sum_{\ell=1}^j\prod_{k=1}^{\ell-1}(\alpha_0\epsilon_{i-k}^2+\beta_0) +\omega_0\sum_{\ell=j+1}^\infty\prod_{k=1}^{\ell-1}(\alpha_0(\epsilon_{i,i-k,j}^*)^2+\beta_0), \quad\;\;j\geq 1, \end{align*} $$

and

$$ \begin{align*}h_{i,j}^2=\omega_0\sum_{\ell=1}^\infty\prod_{k=1}^{\ell-1}(\alpha_0(\epsilon_{i,i-k,j}^*)^2+\beta_0), \quad\;\;j\leq 0, \end{align*} $$

where $\{\epsilon _{i,k,j}^*, -\infty <i,k,j<\infty \}$ are independent copies of $\epsilon _0$ . We also define

$$ \begin{align*} x_{i,j}=\left\{ \begin{array}{@{}ll} \epsilon_ih_{i-1,j},\quad j\geq 1, \\ \epsilon_{i,i,j}^*h_{i-1,j},\quad j\leq 0. \end{array} \right. \end{align*} $$

In the next lemma, we establish the decomposable Bernoulli property of

$$ \begin{align*}\mathbf{g}_i=\left(\frac{1}{h_i^2}\frac{\partial \bar{h}_i(\boldsymbol{\theta}_0)}{\partial\alpha}, \frac{1}{h_i^2}\frac{\partial \bar{h}_i(\boldsymbol{\theta}_0)}{\partial\beta}, \frac{1}{h_i^2}\frac{\partial \bar{h}_i(\boldsymbol{\theta}_0)}{\partial\omega} \right)^\top, \end{align*} $$

and we approximate $\mathbf {g}_i$ by

$$ \begin{align*}\mathbf{g}_{i,j}=\left(\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}_{i,j}(\boldsymbol{\theta}_0)}{\partial\alpha}, \frac{1}{h_{i,j}^2}\frac{\partial \bar{h}_{i,j}(\boldsymbol{\theta}_0)}{\partial\beta}, \frac{1}{h_{i,j}^2}\frac{\partial \bar{h}_{i,j}(\boldsymbol{\theta}_0)}{\partial\omega} \right)^\top. \end{align*} $$

Lemma 6.1. If $H_0$ , Assumptions 2.1–2.3, and (6.1) are satisfied, then

(6.34) $$ \begin{align} \left(E|h^2_i-h_{i,j}^2|^{\bar{\kappa}}\right)^{1/\bar{\kappa}}\leq c_1\bar{\rho}^j,\quad\;\;1\leq j<\infty \end{align} $$

and

(6.35) $$ \begin{align} \left(E|x^2_i-x_{i,j}^2|^{\bar{\kappa}}\right)^{1/\bar{\kappa}}\leq c_2\bar{\rho}^j,\quad\;\;1\leq j<\infty, \end{align} $$

where $c_1, c_2>0$ , and $0<\bar {\rho }<1$ , $\bar {\kappa }>0$ are defined in (6.12). For all $\kappa _1>1$

(6.36) $$ \begin{align} \left(E\|\mathbf{g}_i-\mathbf{g}_{i,j}\|^{\kappa_1}\right)^{1/\kappa_1}\leq c_3\rho_1^j,\quad\;\;1\leq j<\infty, \end{align} $$

with some $c_3=c_3(\kappa _1)$ , and $0<\rho _1=\rho _1(\kappa _1)<1$ .

Proof. We can assume without loss of generality that $0<\bar {\kappa }<1$ , and therefore by Minkowski’s inequality,

(6.37) $$ \begin{align} E\left(\sum_{\ell=j+1}^\infty\prod_{m=1}^{\ell-1}(\alpha_0\epsilon_{i-m}^2+\beta_0)\right)^{\bar{\kappa}}\leq c_4\bar{\rho}^j. \end{align} $$

Using the representation of $h_i^2$ and the definition of $h_{i,j}^2$ , we obtain

$$ \begin{align*} |h_i^2-h^2_{i,j}|\leq \sum_{\ell=j+1}^\infty\prod_{m=1}^{\ell-1}(\alpha_0\epsilon_{i-m}^2+\beta_0) +\omega_0\sum_{\ell=j+1}^\infty\prod_{k=1}^{\ell-1}(\alpha_0(\epsilon_{i,i-k,j}^*)^2+\beta_0), \quad\;\;j\geq 1, \end{align*} $$

and thus (6.34) follows from (6.37). Observing that

$$ \begin{align*}E|x^2_i-x^2_{i,j}|^{\bar{\kappa}}= E|\epsilon_0|^{\bar{\kappa}}E|h_{i-1}^2-h^2_{i-1,j}|^{\bar{\kappa}}, \end{align*} $$

(6.35) is also proven.

It is also shown in Berkes et al. (Reference Berkes, Horváth and Kokoszka2003) (cf. also Francq and Zakoian, Reference Francq and Zakoian2010) that for all $\kappa _2>0$ ,

(6.38) $$ \begin{align} E\left(\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}\right)^{\kappa_2}\leq c_5(\kappa_2). \end{align} $$

Along the lines of the proof of (6.34), one can verify that there exists a $\kappa _3>0$ such that

(6.39) $$ \begin{align} E\left|\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}-\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial\alpha}\right|^{\kappa_3}\leq c_5\rho_3^j,\quad\text{with some}\;\;0<\rho_3<1. \end{align} $$

Let

$$ \begin{align*} A_{i,j}=\Bigg\{|h^2_i-h^2_{i,j}|>\bar{\rho}^{j/(2{\bar{\kappa}})}\;\;\;\text{or}\;\;\; \left|\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}-\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial\alpha}\right|>\rho_3^{j/(2{\kappa_3})} \Bigg\}. \end{align*} $$

Then, by Markov’s inequality,

(6.40) $$ \begin{align} P\{A_{i,j}\}\leq c_6\max(\bar{\rho}^{j/2}, \rho_3^{j/2}). \end{align} $$

Now

$$ \begin{align*} &\left|\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}-\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right|\\ &\hspace{.5cm}\leq \left( \left|\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}\right| +\left|\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right| \right)I\{ A_{i,j} \}+\frac{1}{\omega_0}\left|\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}-\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial\alpha}\right|I\{\bar{A}_{i,j}\}\\ &\hspace{2cm}+\frac{1}{\omega_0}\left|\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right| \left|\bar{h}_i^2(\boldsymbol{\theta}_0)-\bar{h}_{i,j}^2(\boldsymbol{\theta}_0)\right|I\{\bar{A}_{i,j}\}\\ &\hspace{.5cm}\leq \left( \left|\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}\right| +\left|\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right| \right)I\{ A_{i,j} \}+\frac{1}{\omega_0}\rho_3^{j/(2/{\kappa_3})}\\&\hspace{2cm} +\frac{1}{\omega_0}\left|\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right|\bar{\rho}^{j/(2/{\bar{\kappa}})}{,} \end{align*} $$

and therefore by Minkowski’s inequality, for any $\kappa _4\geq 2 $ , we have

$$ \begin{align*} &\left(\left|\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}-\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right|^{\kappa_4}\right)^{1/{\kappa_4}}\leq 2\left( E\left|\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}\right|^{\kappa_4}I\{ A_{i,j}\}\right)^{1/{\kappa_4}}\\ &\hspace{1cm}+ \frac{1}{\omega_0}\rho_3^{j/(2/{\kappa_3})}+\frac{1}{\omega_0}\bar{\rho}^{j/(2/{\bar{\kappa}})} \left(E\left|\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right|^{\kappa_4}\right)^{\kappa_4}. \end{align*} $$

Since by the Cauchy–Schwartz inequality

$$ \begin{align*}E\left\{\left|\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}\right|^{\kappa_4}I\{ A_{i,j}\}\right\} \leq \left\{E\left(\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}\right)^{2{\kappa_4}}\right\}^{1/2}\left\{ P\{ A_{i,j}\}\right\}^{1/2}, \end{align*} $$

we conclude that for all $\kappa _5\geq 2$ there is $0<\rho _5<1$ such that

(6.41) $$ \begin{align} \left(E\left|\frac{1}{h_i^2}\frac{\partial \bar{h}^2_i(\boldsymbol{\theta}_0)}{\partial\alpha}-\frac{1}{h_{i,j}^2}\frac{\partial \bar{h}^2_{i,j}(\boldsymbol{\theta}_0)}{\partial \alpha} \right|^{\kappa_5}\right)^{1/{\kappa_5}}\leq c_7 \rho^j_5. \end{align} $$

Hence, the decomposability of the first coordinate of $\mathbf {g}_i$ with an exponential rate is proven. The same argument can be applied to the other two coordinates.

Next, we introduce the stationary version of the likelihood function:

$$ \begin{align*}\bar{f}_i(\boldsymbol{\theta})=\log \bar{h}_{i}^2(\boldsymbol{\theta})+\frac{x_i^2}{\bar{h}_{i}^2(\boldsymbol{\theta})},\quad\quad-\infty<i<\infty, \end{align*} $$

$$ \begin{align*}\bar{{\mathcal f}}_k(\boldsymbol{\theta})=\sum_{i=1}^k\bar{f}_i(\boldsymbol{\theta}), \end{align*} $$

and

$$ \begin{align*}\bar{f}_{i,j}(\boldsymbol{\theta})=\log \bar{h}_{i,j}^2(\boldsymbol{\theta})+\frac{x_i^2}{\bar{h}_{i,j}^2(\boldsymbol{\theta})}. \end{align*} $$

Lemma 6.2. If $H_0$ , Assumptions 2.1–2.3, and (6.1) are satisfied, then

(6.42) $$ \begin{align} \left(\left\|\nabla(\bar{f}_i(\boldsymbol{\theta}_0) - \bar{f}_{i,j}(\boldsymbol{\theta}_0)\right\|^{\kappa/2}\right)^{2/\kappa}\leq c\rho^j \end{align} $$

and

(6.43) $$ \begin{align} \left(\left\|\nabla^2(\bar{f}_i(\boldsymbol{\theta}_0) - \bar{f}_{i,j}(\boldsymbol{\theta}_0) \right\|^{\kappa/2}\right)^{2/\kappa}\leq c\rho^j \end{align} $$

with some $0<\rho <1$ , and $\kappa>4$ is defined in Assumption 2.1.

Proof. The result in (6.42) is an immediate consequence of Lemma 6.1 since

$$ \begin{align*}\nabla\bar{f}_i(\boldsymbol{\theta}_0)=(1-\epsilon_i^2)\mathbf{g}_{i}. \end{align*} $$

Similar arguments give (6.43).

Let

$$ \begin{align*}\mathbf{H}=E\nabla^2\bar{f}_0(\boldsymbol{\theta}_0). \end{align*} $$

Berkes et al. (Reference Berkes, Horváth and Kokoszka2003) and Francq and Zakoian (Reference Francq and Zakoian2004) prove that $\mathbf {H}$ is not singular.

Lemma 6.3. If $H_0$ , Assumptions 2.1–2.3, and (6.1) are satisfied, then

$$ \begin{align*}N(\hat{\boldsymbol{\theta}}_N-\boldsymbol{\theta}_0)=-\mathbf{H}^{-1}\nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0)+O_P(1). \end{align*} $$

Proof. Berkes et al. (Reference Berkes, Horváth and Kokoszka2003) and Francq and Zakoian (Reference Francq and Zakoian2004) show that there exists a $\boldsymbol {\Theta }_0$ , a neighborhood of $\boldsymbol {\theta }_0$ , such that

(6.44) $$ \begin{align} \sup\left\{ \frac{1}{N}\left\| \nabla^3 \bar{{\mathcal f}}_N(\boldsymbol{\theta}) \right\|:\;\boldsymbol{\theta}\in\boldsymbol{\Theta}_0 \right\}=O_P(1) \quad\text{a.s.} \end{align} $$

By (6.31), we have

$$ \begin{align*}\|\nabla{{\mathcal L}}_N(\hat{\boldsymbol{\theta}}_N)-\nabla\bar{{\mathcal f}}_N(\hat{\boldsymbol{\theta}}_N)\|=O_P(1) \end{align*} $$

and since

$$ \begin{align*}\nabla\bar{{\mathcal L}}_N(\hat{\boldsymbol{\theta}}_N)=\mathbf{0}, \end{align*} $$

by a Taylor expansion, we get from (6.32) and (6.44) that

$$ \begin{align*} -\nabla\bar{{\mathcal L}}_N(\boldsymbol{\theta}_0)=\nabla\bar{{\mathcal f}}_N(\hat{\boldsymbol{\theta}}_N)-\nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0)+O_P(1) =\nabla^2\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0)(\hat{\boldsymbol{\theta}}_N-\boldsymbol{\theta}_0)+O_P(1). \end{align*} $$

Because of (6.42) and (6.43), we can use Lemma S2.1 of Aue et al. (Reference Aue, Hörmann, Horváth and Hušková2014) to conclude that

$$ \begin{align*}\|\nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0)\|=O_P(N^{1/2}) \end{align*} $$

and

$$ \begin{align*}\|\nabla^2\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0)-N\mathbf{H}\|=O_P(N^{1/2}). \end{align*} $$

Now the proof of Lemma 6.3 is complete.

Lemma 6.4. If $H_0$ , Assumptions 2.1–2.3, and (6.1) are satisfied, then

$$ \begin{align*}\max_{1\leq k \leq N-1}\left(\frac{N}{k(N-k)}\right)^\zeta\left\| \nabla\bar{{\mathcal L}}_k(\hat{\boldsymbol{\theta}}_N)-\left(\nabla\bar{{\mathcal f}}_k(\boldsymbol{\theta}_0) -\frac{k}{N} \nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0) \right) \right\|=O_P(1) \end{align*} $$

for any $\zeta>0$ .

Proof. It is enough to prove that

$$ \begin{align*}\max_{1\leq k \leq N-1}\left(\frac{N}{k(N-k)}\right)^\zeta \left\|\nabla\bar{{\mathcal f}}_k(\hat{\boldsymbol{\theta}}_N)-\left(\nabla\bar{{\mathcal f}}_k(\boldsymbol{\theta}_0) -\frac{k}{N} \nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0) \right) \right\|=O_P(1). \end{align*} $$

We follow the proof of Lemma 6.3. For any $\zeta $ , we choose $1/2<\zeta _1< 1/2+\zeta $ . Using again (6.43) and Lemma S2.1 of Aue et al. (Reference Aue, Hörmann, Horváth and Hušková2014), we have

(6.45) $$ \begin{align} \max_{1\leq k \leq N}\frac{1}{k^{\zeta_1}}\left\|\nabla^2\bar{{\mathcal f}}_k(\boldsymbol{\theta}_0)-k\mathbf{H} \right\|=O_P(1){.} \end{align} $$

Thus, we obtain

$$ \begin{align*} \max_{1\leq k\leq N}\frac{1}{k^\zeta}\left\|(\nabla^2\bar{{\mathcal f}}_k(\boldsymbol{\theta}_0)-k\mathbf{H})(\hat{\boldsymbol{\theta}}_N-\boldsymbol{\theta}_0)\right\|= O_P(1)\max_{1\leq k\leq N}k^{\zeta_1-\zeta}N^{-1/2}=O_P(1), \end{align*} $$

and similarly by Lemma 6.3

$$ \begin{align*} \max_{1\leq k\leq N}\frac{1}{k^\zeta}\left\|k\mathbf{H}\left[ (\hat{\boldsymbol{\theta}}_N-\boldsymbol{\theta}_0)-\left(-\mathbf{H} \frac{1}{N}\nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0)\right) \right]\right\|= O_P(1). \end{align*} $$

Our estimates imply that

$$ \begin{align*}\max_{1\leq k \leq N}\frac{1}{k^\zeta} \left\|\nabla\bar{{\mathcal f}}_k(\hat{\boldsymbol{\theta}}_N)-\left(\nabla\bar{{\mathcal f}}_k(\boldsymbol{\theta}_0) -\frac{k}{N} \nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0) \right) \right\|=O_P(1). \end{align*} $$

To prove the same estimate when we use $N-k$ instead of k in the weight function, we observe, as in (6.45), that

(6.46) $$ \begin{align} \max_{1\leq k \leq N}\frac{1}{(N-k)^{\zeta_1}}\left\|\nabla^2\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0)-\nabla^2\bar{{\mathcal f}}_k(\boldsymbol{\theta}_0)-(N-k)\mathbf{H} \right\|=OP(1) \end{align} $$

for all $\zeta _1>1/2$ . Using

$$ \begin{align*} &\max_{1\leq k \leq N_1} \left\|\nabla(\bar{{\mathcal f}}_N(\hat{\boldsymbol{\theta}}_N)-\bar{{\mathcal f}}_k(\hat{\boldsymbol{\theta}}_N))\right.\\&\qquad-\left.\left[\nabla(\bar{{\mathcal f}}_N({\boldsymbol{\theta}}_0)-\bar{{\mathcal f}}_k({\boldsymbol{\theta}}_0)) +\nabla(\bar{{\mathcal f}}_N({\boldsymbol{\theta}}_0)-\bar{{\mathcal f}}_k({\boldsymbol{\theta}}_0))(\hat{\boldsymbol{\theta}}_N-\boldsymbol{\theta}_0) \right] \right\|\\ &=O_P(1) \end{align*} $$

and (6.46), one can verify that

$$ \begin{align*}\max_{1\leq k \leq N-1}\frac{1}{(N-k)^\zeta} \left\|\nabla\bar{{\mathcal f}}_k(\hat{\boldsymbol{\theta}}_N)-\left(\nabla\bar{{\mathcal f}}_k(\boldsymbol{\theta}_0) -\frac{k}{N} \nabla\bar{{\mathcal f}}_N(\boldsymbol{\theta}_0) \right) \right\|=O_P(1). \end{align*} $$

The last lemma of this section is an immediate consequence of Lemma 6.2 and the approximation in Aue et al. (Reference Aue, Hörmann, Horváth and Hušková2014).

Lemma 6.5. If $H_0$ , Assumptions 2.1–2.3, and (6.1) are satisfied, then we can define independent Gaussian processes $\{\mathbf {W}_{N,1}(x), 0\leq x\leq N/2\}$ and $\{\mathbf {W}_{N,2}(x), 0\leq x\leq N/2\}$ such that

$$ \begin{align*}\max_{1\leq x\leq N/2}\frac{1}{x^\nu}\left\|\sum_{i=1}^{ x }\left(\frac{\partial \bar{f}_i(\boldsymbol{\theta}_0)}{\partial \alpha},\frac{\partial \bar{f}_i(\boldsymbol{\theta}_0)}{\partial \beta}\right)^\top-\mathbf{W}_{N,1}(x) \right\|=O_P(1){,} \end{align*} $$

and

$$ \begin{align*}\max_{N/2\leq x\leq N-1}\frac{1}{(N-x)^\nu}\left\|\sum^{N}_{i= x+1}\left(\frac{\partial \bar{f}_i(\boldsymbol{\theta}_0)}{\partial \alpha},\frac{\partial \bar{f}_i(\boldsymbol{\theta}_0)}{\partial \beta}\right)^\top-\mathbf{W}_{N,2}(N-x) \right\|=O_P(1){,} \end{align*} $$

with some $\nu <1/2$ , $E\mathbf {W}_{N,1}(x)=E\mathbf {W}_{N,2}(x)=\mathbf {0}$ , and $E\mathbf {W}_{N,1}(x)\mathbf {W}_{N,1}^\top (y)=E\mathbf {W}_{N,2}(x)\mathbf {W}_{N,2}^\top (y)=\min (x,y)\mathbf {D}$ .

6.4 Proof of (6.16) When $E\kern1pt\mathbf{log}\kern1pt (\alpha _0\epsilon _0^2+\beta _0)\geq 0$

We note that Jensen and Rahbek (Reference Jensen and Rahbek2004) prove their results under assumption (6.3). However, Theorem 6.1 implies that their arguments are also valid when (6.2) holds. Hence, in the section, we assume that

(6.47) $$ \begin{align} E\log(\alpha_0\epsilon_0^2+\beta_0)\geq 0. \end{align} $$

In Section 6.3, we showed that it is enough to consider the processes based on the stationary variables $x_i$ and $h_i^2$ . There is no stationary solution in the present case, so we need to work directly with $\bar {\sigma }_i^2(\boldsymbol {\theta })$ and $\bar {\ell }_i(\boldsymbol {\theta })$ . However, following the method of Jensen and Rahbek (Reference Jensen and Rahbek2004), we approximate $\nabla \bar {\ell }_i$ with a stationary sequence. Elementary arguments lead to

$$ \begin{align*} &\frac{\partial\bar{\ell}_i(\boldsymbol{\theta})}{\partial\alpha}=\left[ 1-\frac{y_i^2}{\bar{\sigma}^2_i(\boldsymbol{\theta})} \right]\bar{p}_{i,1}(\boldsymbol{\theta}), \quad \;\;\frac{\partial\bar{\ell}_i(\boldsymbol{\theta})}{\partial\beta}=\left[ 1-\frac{y_i^2}{\bar{\sigma}^2_i(\boldsymbol{\theta})} \right]\bar{p}_{i,2}(\boldsymbol{\theta}){,} \end{align*} $$

where

$$ \begin{align*}\bar{p}_{i,1}(\boldsymbol{\theta})=\frac{1}{\bar{\sigma}_i^2(\boldsymbol{\theta})} \frac{\partial \bar{\sigma}^2_i(\boldsymbol{\theta})}{\partial\alpha}, \quad \bar{p}_{i,2}(\boldsymbol{\theta})=\frac{1}{\bar{\sigma}_i^2(\boldsymbol{\theta})} \frac{\partial \bar{\sigma}^2_i(\boldsymbol{\theta})}{\partial\beta}. \end{align*} $$

Since we can estimate only $\alpha _0$ and $\beta _0$ , we use a special notation for the estimable parameters, $\bar {\boldsymbol {\theta }}_0=(\alpha _0, \beta _0)^\top $ . Let

$$ \begin{align*}z_{i,1}=\sum_{j=1}^\infty\epsilon_{i-j}^2\frac{1}{\beta_0}\prod_{k=1}^j\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}\;\;\;\;\;\text{and}\;\;\;\;\; \end{align*} $$

Lemma 6.6. If $H_0$ , Assumptions 2.1–2.3, and (6.47) are satisfied, then

(6.48) $$ \begin{align} \sup_{1/\bar{\omega}\leq \omega \leq \bar{\omega}}\left| \bar{p}_{i,1}(\bar{\boldsymbol{\theta}}_0,\omega)-z_{i,1} \right|=O(\rho^i)\quad\;\;\text{a.s.} \end{align} $$

and

(6.49) $$ \begin{align} \sup_{1/\bar{\omega}\leq \omega \leq \bar{\omega}}\left|\bar{p}_{i,2}(\bar{\boldsymbol{\theta}}_0,\omega)- z_{i,2} \right|=O(\rho^i)\quad\;\;\text{a.s.} \end{align} $$

with some $0<\rho <1$ .

Proof. First, we note that

$$ \begin{align*} \bar{\sigma}_i^2(\boldsymbol{\theta})&=\omega\sum_{k=0}^{i-1}\beta^k+\alpha\sum_{\ell=1}^i\beta^{\ell-1}y^2_{i-\ell}+\beta^i\sigma_0^2, \end{align*} $$

which gives the representations

$$ \begin{align*} \bar{\sigma}_i^2(\boldsymbol{\theta}_0)&=\omega_0\sum_{k=0}^{i-1}\beta_0^k+\alpha_0\sum_{\ell=1}^i\beta_0^{\ell-1}y^2_{i-\ell}+\beta_0^i\sigma_0^2 \end{align*} $$

and

$$ \begin{align*} \bar{\sigma}_i^2(\bar{\boldsymbol{\theta}}_0, \omega)&=\omega\sum_{k=0}^{i-1}\beta_0^k+\alpha_0\sum_{\ell=1}^i\beta_0^{\ell-1}y^2_{i-\ell}+\beta_0^i\sigma_0^2. \end{align*} $$

We get, by Theorem 6.1(ii) and (iii), that there exist $c_1>0, c_2>0$ such that

(6.50) $$ \begin{align} \exp(-c_1i^{c_2})\max_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}|\bar{\sigma}_i^2(\bar{\boldsymbol{\theta}}_0, \omega)-\bar{\sigma}_i^2(\boldsymbol{\theta}_0)|=o(1)\;\;\text{a.s.} \end{align} $$

and

(6.51) $$ \begin{align} \max_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}\left|\frac{\bar{\sigma}_i^2(\bar{\boldsymbol{\theta}}_0, \omega)}{\bar{\sigma}_i^2(\boldsymbol{\theta}_0)}-1\right| =O(\rho_1^i) \end{align} $$

with some $0<\rho _1<1$ . Using the recursion, we have

$$ \begin{align*} \bar{p}_{i,1}(\boldsymbol{\theta})=\frac{1}{\bar{\sigma}_i^2(\boldsymbol{\theta})}\sum_{j=1}^{i}\beta^{j-1} y^2_{i-j}=\sum_{j=1}^i\beta^{j-1}\frac{y_{i-j}^2}{\bar{\sigma}_{i-j}^2(\boldsymbol{\theta})}\prod_{k=1}^j\frac{\bar{\sigma}_{i-k}^2(\boldsymbol{\theta})}{\bar{\sigma}_{i-k+1}^2(\boldsymbol{\theta})} \end{align*} $$

and therefore,

$$ \begin{align*}\bar{p}_{i,1}(\bar{\boldsymbol{\theta}}_0,\omega)=\sum_{j=1}^i\frac{1}{\beta_0}\frac{y_{i-j}^2}{\bar{\sigma}_{i-j}^2(\bar{\boldsymbol{\theta}}_0,\omega)} \prod_{k=1}^j\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\bar{\sigma}_{i-k+1}^2(\bar{\boldsymbol{\theta}}_0,\omega)}. \end{align*} $$

Since

$$ \begin{align*} \frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\bar{\sigma}_{i-k+1}^2(\bar{\boldsymbol{\theta}}_0,\omega)} =\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\omega+\alpha_0y_{i-k}^2 +\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}< \frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}< 1, \end{align*} $$

we obtain

$$ \begin{align*} &\sum_{j=1}^{i/2}\beta^{j-1}_0\left|\frac{y_{i-j}^2}{\bar{\sigma}_{i-j}^2(\bar{\boldsymbol{\theta}}_0,\omega)}-\epsilon_{i-j}^2\right| \prod_{k=1}^j\frac{\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\bar{\sigma}_{i-k+1}^2(\bar{\boldsymbol{\theta}}_0,\omega)}\\&\quad \leq \frac{1}{\beta_0}\sum_{j=1}^{i/2}\epsilon_{i-j}^2\left|\frac{\sigma_{i-j}^2}{\bar{\sigma}_{i-j}^2(\bar{\boldsymbol{\theta}}_0,\omega)}-1\right| =O(\rho_2^i)\;\;\;\text{a.s.} \end{align*} $$

on account of (6.51) and

(6.52) $$ \begin{align} \max_{1\leq i <\infty}i^{-1/\kappa}\epsilon_i^2=O_P(1)\;\;\;\text{a.s.} \end{align} $$

By applying Minkowski’s inequality, for any $\kappa _1>2$ , we have

(6.53) $$ \begin{align} \left(E\left(\sum_{j=i/2}^i \prod_{k=1}^j\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\bar{\sigma}_{i-k+1}^2(\bar{\boldsymbol{\theta}}_0,\omega)}\right)^{\kappa_1} \right)^{1/\kappa_{1}} &\leq \left(E\sum_{j=i/2}^i \left(\prod_{k=1}^j\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}\right)^{\kappa_1} \right)^{1/\kappa_{1}}\\ &\leq \sum_{j=i/2}^i\left(E\left(\prod_{k=1}^j\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}\right)^{\kappa_1}\right)^{1/\kappa_{1}}\notag\\ &=\sum_{j=i/2}^i\rho_3^j.\notag \end{align} $$

Hence, the Borel–Cantelli lemma implies that

$$ \begin{align*}\sum_{j=i/2}^i \prod_{k=1}^j\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\bar{\sigma}_{i-k+1}^2(\bar{\boldsymbol{\theta}}_0,\omega)}=O(\rho_4^i)\;\;\;\text{a.s.} \end{align*} $$

It follows from the recursion and (6.51) that

$$ \begin{align*}\frac{y_{i-j}^2}{\bar{\sigma}_{i-j}^2(\bar{\boldsymbol{\theta}}_0,\omega)}=\epsilon_{i-j}^2\frac{\sigma_{i-j}^2}{\bar{\sigma}_{i-j}^2(\bar{\boldsymbol{\theta}}_0,\omega)} \leq \epsilon_{i-j}\sup_{1\leq \ell<\infty}\sup_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}\frac{\sigma_{\ell}^2}{\bar{\sigma}_{\ell}^2(\bar{\boldsymbol{\theta}}_0,\omega)} \end{align*} $$

and

$$ \begin{align*}\sup_{1\leq \ell<\infty}\sup_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}\frac{\sigma_{\ell}^2}{\bar{\sigma}_{\ell}^2(\bar{\boldsymbol{\theta}}_0,\omega)} =O(1)\;\;\;\text{a.s.} \end{align*} $$

Thus, we conclude that

(6.54) $$ \begin{align} \sup_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}\left|p_{i,1}(\bar{\theta}_0,\omega)-\sum_{j=1}^i\frac{1}{\beta_0}\epsilon_{i-j}^2 \prod_{k=1}^j\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\bar{\sigma}_{i-k+1}^2(\bar{\boldsymbol{\theta}}_0,\omega)}\right| =O(\rho_5^i)\;\;\;\text{a.s.} \end{align} $$

Elementary argument leads to

(6.55) $$ \begin{align} &\hspace{0cm}\left|\prod_{k=1}^{j}\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\omega+(\alpha_0\epsilon_{i-k}^2+\beta_0)\bar{\sigma}^2_{i-k}(\bar{\boldsymbol{\theta}}_0,\omega)} - \prod_{k=1}^{j}\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0} \right|\\ &\quad\leq \sum_{k=1}^{j}\left|\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\omega+(\alpha_0\epsilon_{i-k}^2+\beta_0)\bar{\sigma}^2_{i-k}(\bar{\boldsymbol{\theta}}_0,\omega)} -\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0} \right|\notag\\ &\quad\leq \beta_0\omega\sum_{k=1}^j\frac{1}{\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}. \notag \end{align} $$

Using (6.50) and (6.52), we get that

(6.56) $$ \begin{align} &\hspace{0cm}\sup_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}\sum_{j=1}^{i/2}\epsilon_{i-j}^2\left|\prod_{k=1}^{j} \frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\omega+(\alpha_0\epsilon_{i-k}^2+\beta_0)\bar{\sigma}^2_{i-k}(\bar{\boldsymbol{\theta}}_0,\omega)} - \prod_{k=1}^{j}\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0} \right|\\ &\quad\stackrel{\text{ a.s.}}{=}O(1) \sum_{j=1}^{i/2}\epsilon_{i-j}^2\sum_{k=1}^{j}\exp(-S(i-k))\notag\\ &\quad\stackrel{\text{ a.s.}}{=}O(\rho_6^i){,}\notag \end{align} $$

with some $0<\rho _6<1$ . Similarly to (6.53)

$$ \begin{align*} \sum_{j=i/2}^{i}\epsilon_{i-j}^2\prod_{k=1}^{j}\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}\stackrel{\text{ a.s.}}{=}O(\rho_7^i){,}\notag \end{align*} $$

and

$$ \begin{align*} \sup_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}\sum_{j=i/2}^{i}\epsilon_{i-j}^2\prod_{k=1}^{j} \frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)}{\omega+(\alpha_0\epsilon_{i-k}^2+\beta_0)\bar{\sigma}^2_{i-k}(\bar{\boldsymbol{\theta}}_0,\omega)} \stackrel{\text{ a.s.}}{=}O(\rho_7^i),\notag \end{align*} $$

with $0<\rho _7<1$ , completing the proof of (6.48). Since

$$ \begin{align*} \bar{p}_{i,2}(\bar{\boldsymbol{\theta}}_0,\omega)=\sum_{j=1}^i\frac{1}{\beta_0}\prod_{k=1}^j\frac{\beta_0\bar{\sigma}_{i-k}^2(\bar{\boldsymbol{\theta}}_0,\omega)} {\omega_0+(\alpha_0\epsilon_{i-k}^2+\beta_0)\bar{\sigma}^2_{i-k}(\bar{\boldsymbol{\theta}}_0,\omega)}, \end{align*} $$

the proof of (6.49) is nearly the same but simpler than that of (6.49).

Let

$$ \begin{align*} z_{i,j,1}&=\sum_{\ell=1}^j\epsilon_{i-\ell}^2\frac{1}{\beta_0}\prod_{k=1}^\ell\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0} +\sum_{\ell=j+1}^\infty\epsilon_{i,i-\ell,j}^2\frac{1}{\beta_0}\\ &\quad\times \left(\prod_{k=1}^j\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}\right)\left(\prod_{r=j+1}^\ell\frac{\beta_0}{\alpha_0\epsilon_{i,i-r,j}^2+\beta_0}\right){,} \end{align*} $$

and

$$ \begin{align*}z_{i,j,2}=\sum_{\ell=1}^j\frac{1}{\beta_0}\prod_{k=1}^\ell\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0} \!+\!\sum_{\ell=j+1}^\infty\frac{1}{\beta_0} \left(\prod_{k=1}^j\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}\!\right)\!\left(\prod_{r=j+1}^\ell\frac{\beta_0}{\alpha_0\epsilon_{i,i-r,j}^2+\beta_0}\right)\!{,} \end{align*} $$

where $\{\epsilon _{i,j,k}, -\infty <i,j,k<\infty \}$ are independent copies of $\epsilon _0$ .

Lemma 6.7. If $H_0$ , Assumptions 2.1–2.3, and (6.47) are satisfied, then

(6.57) $$ \begin{align} \left(E\left[(1-\epsilon_i^2)(z_{i,1}-z_{i,j,1})\right]^\kappa\right)^{1/\kappa}\leq c\rho^j \end{align} $$

and

(6.58) $$ \begin{align} \left(E\left[(1-\epsilon_i^2)(z_{i,2}-z_{i,j,2})\right]^\kappa\right)^{1/\kappa}\leq c\rho^j, \end{align} $$

where $0<\rho <1$ , and $\kappa $ is defined in Assumption 2.1.

Proof. By Assumption 2.1 and Minkowski’s inequality, we have

$$ \begin{align*} &\left( E\left[(1-\epsilon_i^2)\sum_{\ell=j+1}^\infty\epsilon_{i,i-\ell,j}^2\frac{1}{\beta_0} \left(\prod_{k=1}^j\frac{\beta_0}{\alpha_0\epsilon_{i-k}^2+\beta_0}\right)\left(\prod_{r=j+1}^\ell\frac{\beta_0}{\alpha_0\epsilon_{i,i-r,j}^2+\beta_0}\right) \right]^\kappa\right)^{1/\kappa}\\ &\hspace{1cm}\leq \left(E(1-\epsilon_0^2)^\kappa\right)\frac{1}{\beta_0}\sum_{\ell=j+1}^\infty\left(E\left[\frac{\beta_0}{\alpha_0\epsilon_{}^2+\beta_0}\right]^\kappa \right)^{1/\kappa}\leq c_1\rho^j_1 \end{align*} $$

with some $0<\rho _1<1$ , proving (6.57). The same argument gives (6.58).

Let

$$ \begin{align*} \bar{\mathbf{J}}_i(\boldsymbol{\theta})= \begin{pmatrix} \frac{\partial^2 \bar{\ell}_i(\boldsymbol{\theta})}{\partial \alpha^2}, & \frac{\partial^2 \bar{\ell}_i(\boldsymbol{\theta})}{\partial \alpha\partial\beta} \\ \frac{\partial^2 \bar{\ell}_i(\boldsymbol{\theta})}{\partial \alpha\partial\beta}, & \frac{\partial^2 \bar{\ell}_i(\boldsymbol{\theta})}{\partial \beta^2} \end{pmatrix} \end{align*} $$

be the matrix of the second-order partial derivatives of $\bar {\ell }_i(\boldsymbol {\theta })$ . Similarly to Lemma 6.6, it can be approximated with a matrix $\mathbf {Q}_i$ of the form $\mathbf {Q}_i=\mathbf {q}(\epsilon _i,\epsilon _{i-1}, \dots )$ . The sequence $\mathbf {Q}_i$ is a decomposable Bernoulli shift with $\mathbf {Q}_{i,j}=\mathbf {q}(\epsilon _i,\epsilon _{i-1},\dots , \epsilon _{i-1}, \epsilon _{i, j-i-1, j}, \epsilon _{i, j-i-2, j},\dots )$ .

Lemma 6.8. If $H_0$ , Assumptions 2.1–2.3, and (6.47) are satisfied, then

(6.59) $$ \begin{align} \sup_{1/\bar{\omega}\leq \omega\leq \bar{\omega}}\left\| \bar{\mathbf{J}}_i(\bar{\boldsymbol{\theta}}_0,\omega) -\mathbf{Q}_i \right\|=O(\rho^i)\;\;\;\text{a.s.} \end{align} $$

and

(6.60) $$ \begin{align} \left(E\left\|\mathbf{Q}_i-\mathbf{Q}_{i,j}\right\|^\kappa\right)^{1/\kappa}\leq c \rho^j, \end{align} $$

where $0<\rho <1$ , and $\kappa $ is defined in Assumption 2.1.

The rest of the proof is a repetition of the calculations in Section 6.3, since we have showed again that the partial derivatives can be approximated with decomposable Bernoulli shifts with exponential rates. Thus, we get the following analogue of Lemma 6.4.

Lemma 6.9. If $H_0$ , Assumptions 2.1–2.3, and (6.47) are satisfied, then

$$ \begin{align*} \max_{1\leq k \leq N-1}&\left( \frac{N}{k(N-k)} \right)^\zeta\left\|\vphantom{\sum_{i=1}^N} \bar{{\mathcal r}}_k(\hat{\boldsymbol{\theta}}_N)\right.\\&\quad\left.-\left(\sum_{i=1}^k(1-\epsilon_i^2)(z_{i,1},z_{i,2})^\top -\frac{k}{N}\sum_{i=1}^N(1-\epsilon_i^2)(z_{i,1},z_{i,2})^\top \right) \right\|\\ &=O_P(1) \end{align*} $$

for any $\zeta>0$ .

Lemma 6.10. If $H_0$ , Assumptions 2.1–2.3, and (6.47) are satisfied, then we can define independent Gaussian processes $\{\mathbf {W}_{N,1}(x), 0\leq x\leq N/2\}$ and $\{\mathbf { W}_{N,2}(x), 0\leq x\leq N/2\}$ such that

$$ \begin{align*}\max_{1\leq x\leq N/2}\frac{1}{x^\nu}\left\|\sum_{i=1}^{\lfloor x\rfloor}(1-\epsilon_i^2)(z_{i,1},z_{i,2})^\top-\mathbf{W}_{N,1}(x) \right\|=O_P(1){,} \end{align*} $$

and

$$ \begin{align*}\max_{N/2\leq x\leq N-1}\frac{1}{(N-x)^\nu}\left\|\sum^{N}_{i=\lfloor x\rfloor+1}(1-\epsilon_i^2)(z_{i,1},z_{i,2})^\top-\mathbf{W}_{N,2}(N-x) \right\|=O_P(1){} \end{align*} $$

with some $\nu <1/2$ , $E\mathbf {W}_{N,1}(x)=E\mathbf {W}_{N,2}(x)=\mathbf {0}$ , and $E\mathbf {W}_{N,1}(x)\mathbf {W}_{N,1}^\top (y)=E\mathbf {W}_{N,2}(x)\mathbf {W}_{N,2}^\top (y)=\min (x,y)E(1-\epsilon _0^2)^2 E[(z_{i,1},z_{i,2})^\top (z_{i,1},z_{i,2})^\top ]$ .