1 Introduction and statistical framework

In this work we study the estimation of a change-point at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. More precisely, we observe a time series \(X_1,\ldots ,X_n\) possessing the structure

$$\begin{aligned} X_t=(\beta _0+\beta _1 g(t,t_0))X_{t-1}+e_t\quad (t=1,2,\ldots ),\quad \text {with}\quad X_0=e_0, \end{aligned}$$
(1.1)

where \(\{e_t\}_{t=0,1,\ldots }\) is a sequence of independent, identically distributed (i.i.d.) random variables with \(Ee_0=0\), \(0<Ee_0^2=\sigma ^2\), \(Ee_0^4<\infty \), \(\beta _0\), \(\beta _1\) are unknown parameters satisfying

$$\begin{aligned} |\beta _0|<1,\quad \beta _1=\beta _{1,n}\rightarrow 0,\quad |\beta _1|\sqrt{n}\rightarrow \infty \quad (n\rightarrow \infty ), \end{aligned}$$
(1.2)

and \(g(\cdot ,t_0)=g_n(\cdot ,t_0)\) is a (known) real function such that

$$\begin{aligned} g_n(t,t_0)=0\quad (t\le t_0)\quad \text {and}\quad g_n(t,t_0)> 0\quad (t>t_0) \end{aligned}$$
(1.3)

(see more detailed assumptions below). That is, we assume that the parameter \(\beta _0\) of the (stationary) AR(1)-process changes gradually at an unknown time-point \(t_0=t_{0,n} =\lfloor n\tau _0\rfloor \), with \(0<\tau _0<1\), \(\lfloor \,\cdot \,\rfloor \) denoting the integer part, and our aim is to provide an estimator for \(t_0\) making use of the observations \(X_1,\ldots ,X_n\) and under certain assumptions on the function \(g_n(\cdot ,t_0)\) to be specified below.

Remark 1

  1. (a)

    As in earlier works, we only study the case of a gradual change under “local alternatives” here, i.e., under \(\beta _{1,n}\rightarrow 0\), but \(\beta _{1,n}\sqrt{n}\rightarrow \infty \) as \(n\rightarrow \infty \) (cf., e.g., Dümbgen (1991), Jarušková (1998a), Hušková (1998a), Hušková (2001), or Hušková and Steinebach (2002)).

  2. (b)

    In case of the gradual change function \(g_n\) being unknown, it would be sufficient to have an estimating function \(\widehat{g}_n\) (say), which approximates \(g_n\) at a certain rate. For a more detailed discussion we refer to Remarks 5 and 7 below.

Note that, if \(g(\cdot ,t_0)\) is a bounded function, then

$$\begin{aligned} b:=\sup _{t\ge 1}|\beta _0 +\beta _1 g(t,t_0)|<1 \end{aligned}$$
(1.4)

for sufficiently large n and, by a repeated application of (1.1),

$$\begin{aligned} X_t=e_t +\sum _{j=1}^te_{t-j}\prod _{i=0}^{j-1} (\beta _0+\beta _1 g(t-i,t_0))\quad (t=1,2,\ldots ). \end{aligned}$$
(1.5)

Before we turn to formulating our main results, we give a brief account of related works, particularly concerning the detection of gradual changes in various dependent data sets. Most of the earlier papers on the change analysis in autoregressive processes deal with abrupt changes, either in the mean or in the autoregressive parameters, respectively in the variance of the error process. Picard (1985) proposed a procedure for testing changes in the covariance structure of an AR(p)-process based on a likelihood ratio approach and obtained the asymptotic distribution of the likelihood estimators of the change parameters. Gaussian-type likelihood ratio procedures for testing abrupt changes in autoregressive models were studied in Davis et al. (1995), where also the limit distribution of the test statistic was established. Gombay (2008) used efficient score vectors to develop statistics that are able to test changes in any of the parameters of a Gaussian AR(p)-model separately, or in any collection of them, and she also studied large sample properties of the change-point estimator.

Other results were based on partial sums of residual processes, see, e.g., Horváth (1993) for testing or Bai (1994) for proving consistency of the change-point estimator. Hušková et al. (2007) used an approach based on partial sums of weighted residuals and obtained asymptotic distributions for various max-type test statistics together with proving the consistency of the change-point estimator in an AR(p)-model. Moreover, bootstrap versions of the proposed tests were studied in Hušková et al. (2008).

A quasi-maximum likelihood method was used in Bai (2000) to analyze vector autoregressive models (VAR) possessing multiple structural changes. The approach developed in Davis et al. (1995) has been extended to VAR models by Dvořák (2015), who dealt with asymptotic tests for an abrupt change in the autoregressive parameters and the variance structure under various assumptions on the correlations of the errors. Kirch et al. (2015) extended the class of max-type change-point statistics considered in Hušková et al. (2007) to the VAR case and epidemic change alternatives and developed a new approach taking possible misspecification of the model into account.

Slama and Saggou (2017) considered a Bayesian analysis of a possible change in the parameters of an AR(p)-model and developed a test, which can detect a change in any of the parameters separately. Moreover, the posterior density of the change-point is given by using a Gibbs sampler. Many references on the change-point analysis in time series can also be found in a survey paper by Aue and Horváth (2013).

Concerning gradual changes in autoregression, Salazar (1982) studied a model similar to (1.1) from a Bayesian point of view. Under the assumption of normality of the error process and with some joint prior distribution of the change-point \(t_0\) and other parameters under consideration, he obtained a joint posterior distribution from which the marginal distribution of the change-point could be obtained via numerical integration. A similar, though not identical problem was solved by Venkatesan and Arumugam (2007), who considered an AR(p)-model with a gradual switch in the parameters over a finite interval. Here again, computation of the posterior distribution of the change-point requires the use of numerical integration.

He et al. (2008) derived a parameter constancy test in a stationary vector autoregressive model against the hypothesis that the parameters of the model change smoothly over time. Though model (1.1) could be considered a special case of the model studied in He et al. (2008), the authors treat other type of smooth functions and do not consider any estimator of the breaking point.

Our approach below is motivated by the previous work by Hušková, see, e.g., Hušková (1998a, 1998b, 1999, 2001), Jarušková (1998a, 1998b, 1999, 2001, 2002, 2003), or by Hušková and Steinebach (2000), Hušková and Steinebach (2002), respectively by Albin and Jarušková (2003). In the above cited papers a gradual-type change in the mean of a location model is considered and asymptotic tests for detecting the change together with limit properties of the estimator of the change-point are developed for various types of smoothly changing parameters. More specifically, the mentioned model can be written in the form

$$\begin{aligned} Y_t = \mu + \delta _n \,g \big ((t-t_0)/n\big ) + \epsilon _t, \quad t=1,\ldots , n, \end{aligned}$$
(1.6)

where \(\mu , \delta _n, t_0\) are unknown parameters, \(\epsilon _1,\ldots , \epsilon _n\) are i.i.d. errors, with zero mean and finite moments of order \(2+\Delta \), \(\Delta > 0\), and the function g satisfies the assumption

$$\begin{aligned} g(x) = 0,\quad (x\le 0), \quad g(x)> 0 \quad (x>0), \end{aligned}$$

together with other assumptions specified for the formulated problems.

Döring and Jensen (2015), and also Döring (2015a, 2015b), extended the methodology proposed by Hušková (1999) to regression models with independently distributed random regressors. Wang (2007) studied the same location model as Hušková (1999) with errors that exhibit long memory dependence, and Slabý (2001) considered a test based on ranks. Hlávka and Hušková (2017), motivated by gender differences observed in a real data set, proposed a two-sample gradual change test that leads to more precise results than the application of a procedure based on the standard two-sample t-test. Račkauskas and Tamulis (2013) studied epidemic changes in a location model, in which the transition between regimes is gradual.

Several authors studied smooth changes in other contexts. For example, Aue and Steinebach (2002) discuss an extension of Hušková’s (1999) approach to certain statistical models, which cover more general classes of stochastic processes satisfying an invariance principle (see also Kirch and Steinebach (2006), Steinebach (2000), Steinebach and Timmermann (2011) or Timmermann (2014), Timmermann (2015)).

Vogt and Dette (2015) developed a nonparametric method to estimate a smooth change-point in a locally stationary framework and established the rate of convergence of the change-point estimator. Their procedure allows to deal with a wide variety of stochastic characteristics including the mean, covariances and higher moments.

Hoffmann et al. (2018) and Hoffmann and Dette (2019) discuss statistical inference for the detection and the localization of gradual changes in the jump characteristic of a discretely observed Itô semimartingale.

Quessy (2019) proposed a general class of consistent test statistics for the detection of gradual changes in copulas and developed their large-sample properties.

Now, let us turn to our problem. We shall study the least squares estimator \({\widehat{t}}_0\) for \(t_0\), which is obtained by minimizing

$$\begin{aligned} S(b_0,b_1,t_*)=\sum _{t=1}^n[X_t-(b_0+b_1 g(t,t_*)) X_{t-1}]^2 \end{aligned}$$

with respect to (w.r.t.) \(b_0,b_1\in \mathbb {R}\), \(t_*=0,1,\ldots ,\lfloor n(1-\delta )\rfloor \), \(\delta >0\) arbitrarily small, i.e.,

$$\begin{aligned} S({\widehat{b}}_0,{\widehat{b}}_1,{\widehat{t}}_0)=\min _{b_0,b_1,t_*} S(b_0,b_1,t_*) =\min _{t_*} \min _{b_0,b_1} S(b_0,b_1,t_*). \end{aligned}$$
(1.7)

Remark 2

The technical condition \(t_*\le \lfloor n(1-\delta )\rfloor \), with \(\delta >0\) fixed, could be weakened to allow for \(\delta =\delta _n\rightarrow 0\) \((n\rightarrow \infty )\) at a certain rate, which, however, would depend on the parameter \(\beta _1 =\beta _{1,n}\) from (1.2) and the function \(g=g_n\) from (1.3) as well. Since \(\beta _{1,n}\) is unknown, one should choose \(\delta >0\) fixed, but small, for practical use.

Via partial derivatives, it is not difficult to show that, for fixed \(t_*\),

$$\begin{aligned}&{\widehat{b}}_0(t_*) =\frac{\sum _{t=1}^nX_t X_{t-1}}{\sum _{t=1}^nX_{t-1}^2}- {\widehat{b}}_1(t_*) \frac{\sum _{t=1}^ng(t,t_*)X_{t-1}^2}{\sum _{t=1}^nX_{t-1}^2}\qquad \text {and} \end{aligned}$$
(1.8)
$$\begin{aligned}&{\widehat{b}}_1(t_*) = \frac{\sum _{t=1}^nX_t X_{t-1}g(t,t_*)-\frac{\sum _{j=1}^nX_j X_{j-1}}{\sum _{j=1}^nX_{j-1}^2}\sum _{t=1}^nX_{t-1}^2g(t,t_*)}{\sum _{t=1}^nX_{t-1}^2g^2(t,t_*)-\frac{\big (\sum _{t=1}^ng(t,t_*)X_{t-1}^2\big )^2}{\sum _{t=1}^nX_{t-1}^2}}. \end{aligned}$$
(1.9)

On plugging this into (1.7), we obtain

$$\begin{aligned}&S({\widehat{b}}_0,{\widehat{b}}_1,{\widehat{t}}_0)= \min _{t_*}\Bigg [\sum _{t=1}^n\Big (X_t- \frac{\sum _{j=1}^nX_j X_{j-1}}{\sum _{j=1}^nX_{j-1}^2}X_{t-1}\Big )^2\nonumber \\&\qquad -{\widehat{b}}_1^2(t_*)\sum _{t=1}^nX_{t-1}^2\Big (g(t,t_*)-\frac{\sum _{j=1}^nX_{j-1}^2 g(j,t_*)}{\sum _{j=1}^nX_{j-1}^2}\Big )^2 \Bigg ].\nonumber \\ \end{aligned}$$
(1.10)

Since the first term in (1.10) does not depend on \(t_*\), a combination of (1.7)–(1.10) eventually results in

$$\begin{aligned} {\widehat{t}}_0=\mathop {\mathrm{arg\,max}}\limits _{t_*} \frac{\Big [\sum _{t=1}^nX_t X_{t-1}g(t,t_*)-\frac{\sum _{j=1}^nX_j X_{j-1}}{\sum _{j=1}^nX_{j-1}^2}\sum _{t=1}^nX_{t-1}^2g(t,t_*)\Big ]^2}{\sum _{t=1}^nX_{t-1}^2g^2(t,t_*)-\frac{\big (\sum _{t=1}^ng(t,t_*)X_{t-1}^2\big )^2}{\sum _{t=1}^nX_{t-1}^2}}.\nonumber \\ \end{aligned}$$
(1.11)

Remark 3

Note that

$$\begin{aligned} \max _{t_*}\frac{\Big |\sum _{t=1}^nX_t X_{t-1}g(t,t_*)-\frac{\sum _{j=1}^nX_j X_{j-1}}{\sum _{j=1}^nX_{j-1}^2}\sum _{t=1}^nX_{t-1}^2g(t,t_*)\Big |}{\Big (\sum _{t=1}^nX_{t-1}^2g^2(t,t_*)-\frac{\big (\sum _{t=1}^ng(t,t_*)X_{t-1}^2\big )^2}{\sum _{t=1}^nX_{t-1}^2}\Big )^{1/2}} \end{aligned}$$
(1.12)

can be used as a test statistic for testing “no change” versus “there is a change”, even if the true function g is unknown, just some integral has to be nonzero (see, e.g., Hušková and Steinebach (2002)). In practice, before starting to estimate \(t_0=\lfloor n\tau _0\rfloor \), one should first carry out such a test for the existence of a change-point \(\tau _0\), with \(0<\tau _0<1\).

For our theoretical studies of \({\widehat{t}}_0\) below, it will be convenient to make use of the model equation (1.1) and rewrite (1.11), after a multiplication with 1/n, as

$$\begin{aligned} {\widehat{t}}_0 = \mathop {\mathrm{arg\,max}}\limits _{t_*}&\frac{\Big [\beta _1\Big (\frac{1}{n}\sum _{t=1}^ng(t,t_0)g(t,t_*)X_{t-1}^2- \frac{\frac{1}{n}\sum _{j=1}^ng(j,t_0)X_{j-1}^2\frac{1}{n}\sum _{j=1}^ng(j,t_*)X_{j-1}^2}{\frac{1}{n}\sum _{j=1}^nX_{j-1}^2}\Big )}{\frac{1}{n}\sum _{t=1}^ng^2(t,t_*)X_{t-1}^2-\frac{\big (\frac{1}{n}\sum _{j=1}^ng(j,t_*)X_{j-1}^2\big )^2}{\frac{1}{n}\sum _{j=1}^nX_{j-1}^2}} \nonumber \\&\frac{+\frac{1}{n}\sum _{t=1}^ne_t X_{t-1}g(t,t_*)- \frac{\frac{1}{n}\sum _{j=1}^ne_j X_{j-1} \frac{1}{n}\sum _{j=1}^ng(j,t_*)X_{j-1}^2}{\frac{1}{n}\sum _{j=1}^nX_{j-1}^2} \Big ]^2}{\frac{1}{n}\sum _{t=1}^ng^2(t,t_*)X_{t-1}^2-\frac{\big (\frac{1}{n}\sum _{j=1}^ng(j,t_*)X_{j-1}^2\big )^2}{\frac{1}{n}\sum _{j=1}^nX_{j-1}^2}}. \end{aligned}$$
(1.13)

For later asymptotics it may also be convenient to express \(\widehat{t}_0\) as

$$\begin{aligned} {\widehat{t}}_0 = \mathop {\mathrm{arg\,max}}\limits _{t_*}&\frac{\Big [\beta _1 \frac{1}{n}\sum _{t=1}^n{\widetilde{g}}_{n}(t, t_0) {\widetilde{g}}_n(t, t_*)X_{t-1}^2 +\frac{1}{n}\sum _{t=1}^ne_t X_{t-1}{\widetilde{g}}_n(t, t_*)\Big ]^2}{\frac{1}{n}\sum _{t=1}^n{\widetilde{g}}^2_n(t, t_*) X_{t-1}^2}, \end{aligned}$$
(1.14)

where

$$\begin{aligned} {\widetilde{g}}_n(t,t_*)= g(t,t_*) -\frac{\sum _{j=1}^ng(j,t_*)X_{j-1}^2}{\sum _{j=1}^nX_{j-1}^2}. \end{aligned}$$
(1.15)

The paper is organized as follows. Based on the required assumptions, which are collected first, Sect. 2 presents the main results of our work. As a first statement it can be shown in Theorem 1 that \(\widehat{t}_0/n\) is a consistent estimator for \(\tau _0\), where \(t_0 =\lfloor n\tau _0\rfloor \), with \(0<\tau _0<1\), i.e., \(\widehat{t}_0/n \,{\mathop {\rightarrow }\limits ^{P}}\,\tau _0\) \((n\rightarrow \infty )\). Based on the rates obtained in the proof of Theorem 1, a rough convergence rate estimate can immediately be given (see Theorem 2). Under somewhat stronger assumptions, a precise rate can then be derived in Theorem 3 by showing that our estimator has an asymptotically normal limit distribution. In Sect. 3, some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator. Section 4 collects some auxiliary results which are used in the proofs of the main theorems. The latter are finally given in Sect. 5.

2 Assumptions and main results

For our asymptotic results we assume the gradual change function \(g(\cdot ,t_*)\) to satisfy the following assumptions:

  1. (A.1)

    For every \(t_*=0,1,\ldots ,n-1\), the function \(g(\cdot ,t_*)\) is of the form

    $$\begin{aligned} g(t,t_*)= g_0\Big (\frac{t-t_*}{n}\Big ),\qquad t=0,1,\ldots ,n, \end{aligned}$$

    where \(g_0:(-\infty ,1]\rightarrow \mathbb {R}\) is a real function satisfying:

  2. (A.2)

    It holds that

    $$\begin{aligned} g_0(x)= 0\qquad (x\le 0)\qquad \text {and}\qquad g_0(x)> 0\qquad (0<x\le 1). \end{aligned}$$
  3. (A.3)

    The function \(g_0:(-\infty ,1]\rightarrow \mathbb {R}\) is bounded and Lipschitz continuous, i.e.

    $$\begin{aligned} |g_0(x)|\le D_1\qquad \text {and}\qquad |g_0(x)-g_0(y)|\le D_2|x-y|,\qquad x,y\le 1, \end{aligned}$$

    with some positive constants \(D_1\) and \(D_2\).

  4. (A.4)

    It holds that

    $$\begin{aligned}&|g_0(x)-g_0(y)-(x-y) g_0'(y)| \le D_3|x-y|^{1+\Delta },\quad 0<x,y<1,\\&|g_0(x)-x g'_{0+}(0)| \le D_3|x|^{1+\Delta },\quad 0\le x<1 ,\\&\Big (\int _0^{1}{\widetilde{g}}_0(x-\tau _0) \,{\widetilde{g}}_0'(x-\tau _0) dx\Big )^2 < \int _0^{1}{\widetilde{g}}_0^2(x-\tau _0) dx \int _0^{1}{\widetilde{g}}_0^{\prime 2}(x-\tau _0) dx,\\&\text {with}\quad {\widetilde{g}}_0 (x) = g_0 (x)- \int _0^{1}g_0(y-\tau _0)dy, \end{aligned}$$

    where \(g'_{0+}(0) \) denotes the right derivative at 0, \( g_0'(\cdot )\) denotes the derivative, assumed to be bounded and Riemann integrable, \(1/2<\Delta \le 1\), and \(D_3\) is a positive constant.

Remark 4

  1. (a)

    The case of a negative change, i.e., \(g_0(x)<0\) for \(0<x\le 1\), can be reduced to the positive one by just reparametrizing \({\bar{\beta }}_1:=-\beta _1\) and \({\bar{g}}_0(\cdot )=-g_0(\cdot )\).

  2. (b)

    The function \(g_0\), for example, could be such that \(g_0(x)=\pm \,x_+^\kappa \) \((x\le 1)\), where \(x_+\) denotes the positive part of x and \(\kappa \ge 1\) is a fixed exponent.

In our first result it will be shown that \({\widehat{t}}_0/n\) is a consistent estimator for \(\tau _0\), where \(t_0=\lfloor n\tau _0\rfloor \), with \(0<\tau _0<1\), i.e.,  \({\widehat{t}}_0/n\,{\mathop {\rightarrow }\limits ^{P}}\,\tau _0\) \((n\rightarrow \infty )\).

Theorem 1

Let Assumptions (A.1)–(A.3) be satisfied. Then, under the model (1.1) and the corresponding conditions formulated above, the estimator \({\widehat{t}}_0\) from (1.11) is consistent, i.e.,

$$\begin{aligned} \frac{{\widehat{t}}_0}{n}\,{\mathop {\rightarrow }\limits ^{P}}\,\tau _0\qquad (n\rightarrow \infty ). \end{aligned}$$
(2.1)

Remark 5

In case of an unknown change function g, it will be obvious from the proof of Theorem 1 that (2.1) still holds, if g in (1.11) is replaced by an estimator \(\widehat{g}_n\) at a rate \(o_P(\beta _1)\), more precisely, if there is an estimating function \(\widehat{g}_0=\widehat{g}_{0,n}\) such that, as \(n\rightarrow \infty \),

$$\begin{aligned} \max _{x\in [0,1]}\big |\widehat{g}_{0,n}(x)-g_0(x)\big |=o_P(\beta _1). \end{aligned}$$
(2.2)

In this case, \(\widehat{g}_n(t,t_*)=\widehat{g}_0((t-t_*)/n)\) can be used in (1.11) resp. (1.13) instead of \(g(t,t_*)=g_0((t-t_*)/n)\) and, in view of the rate \(o_P(\beta _1)\), the convergence in (2.1) will be retained.

If, for example, \(g_0(x)=x_+^\kappa \), with some \(\kappa \ge 1\), it would be sufficient to have an estimator \(\widehat{\kappa }_n\) such that \(|\widehat{\kappa }_n-\kappa |=o_P(\beta _1)\), e.g., \(|\widehat{\kappa }_n-\kappa |=O_P(1/\sqrt{n})\) as \(n\rightarrow \infty \). Such estimates have been obtained in other settings (cf., e.g., Döring and Jensen (2015) for a regression model). In our time series setting, it is an open question and has to be left for future research.

Another possible model to deal with would be the case \(\beta _1 g_0(x)=\beta _1 x_+ +\beta _2 x_+^2\), with unknown parameters \(\beta _1,\beta _2\). Here, least squares estimation means to minimize

$$\begin{aligned} \widetilde{S}(b_0,b_1,b_2,t_*)=\sum _{t=1}^n\Big [X_t-\Big (b_0+b_1 \Big (\frac{t-t_*}{n}\Big )_+ +b_2 \Big (\frac{t-t_*}{n}\Big )_+^2\Big )\, X_{t-1}\Big ]^2 \end{aligned}$$

w.r.t. \(b_0,b_1,b_2\in \mathbb {R}\), \(t_*=0,1,\ldots ,\lfloor n(1-\delta )\rfloor \), \(\delta >0\), and then to modify the corresponding steps in the proofs. For the sake of conciseness, this may also be left for further investigations.

The proofs of Theorem 1 and Remark 5 are postponed to Sect. 5.

Remark 6

It would also be quite straightforward to get a consistent estimator of \(\beta _1\), i.e., \({\widehat{b}}_1({\widehat{t}}_0)\), together with some limiting properties. For the sake of conciseness, we want to omit details here.

Also, if the function \(g(\cdot )\) is only known up to a multiplicative constant, then the resulting estimator is still consistent, but the limit distribution below, however, would depend on this multiplicative constant, which is unknown.

On checking the proof of Theorem 1 more carefully, a rough rate of consistency for our estimator \(\widehat{t}_0\) can be obtained as follows.

Theorem 2

Under the conditions of Theorem 1, assume that the limit function \(f(\tau _*)\) in (5.4) is twice continuously differentiable in a small neighbourhood of \(\tau _0\), with \(f''(\tau _*) >D\) for some \(D>0\). Then, with \(\widehat{t}_0=\lfloor n\widehat{\tau }_0\rfloor \), for every sequence \(\{\varepsilon _n\}\) with \(\varepsilon _n \rightarrow 0\),

$$\begin{aligned} |\widehat{\tau }_0 -\tau _0| = O_P\big (|\beta _1|^{1/2}\big )+o_P\Bigg (\frac{1}{|\beta _1|^{1/2} \,\varepsilon _n\, n^{1/4}}\Bigg ) \qquad (n\rightarrow \infty ). \end{aligned}$$
(2.3)

Remark 7

If (2.2) in Remark 5 is replaced by

$$\begin{aligned} \max _{x\in [0,1]}\big |\widehat{g}_{0,n}(x)-g_0(x)\big |=O_P\Big (\frac{1}{\sqrt{n}}\Big ), \end{aligned}$$
(2.4)

the approximation rate \(O_P\big (|\beta _1|^{1/2}\big )+o_P\big (1/(|\beta _1|^{1/2}\,\varepsilon _n\,n^{1/4})\big )\) in Theorem 2 can be retained.

The proofs of Theorem 2 and Remark 7 are also postponed to Sect. 5.

Remark 8

If, for example, \(\beta _1=n^{-\alpha }\), with \(0<\alpha < 1/2\), then \(\varepsilon _n\) could be chosen as \((\log n)^{-p}\), with \(p>0\), so that one would have the polynomial consistency rate

$$\begin{aligned} |\widehat{\tau }_0 -\tau _0|= {\left\{ \begin{array}{ll} O_P\big (\frac{1}{n^{\alpha /2}}\big ), &{} \text {if}\quad 0< \alpha< 1/4, \\ o_P\big (\frac{\log ^p n}{n^{1/4 -\alpha /2}}\big ), &{} \text {if}\quad 1/4 \le \alpha < 1/2, \end{array}\right. } \qquad (n\rightarrow \infty ). \end{aligned}$$
(2.5)

Next it will be shown that the estimator \({\widehat{t}}_0\) of \(t_0\) (or equivalently \({\widehat{\tau }}_0\) of \(\tau _0\)) has an asymptotically normal limit distribution.

Theorem 3

Let Assumptions (A.1)–(A.4) be satisfied. Then, as \(n\rightarrow \infty \),

$$\begin{aligned} \beta _1 \sqrt{n} \,{\widetilde{H}} \frac{\sigma ^2}{1-\beta _0^2}\frac{{\widehat{t}}_0 - t_0}{n}{\mathop {\rightarrow }\limits ^{d}} {\mathcal {N}}\Big (0,\frac{\sigma ^4}{1-\beta _0^2} {\widetilde{H}}\Big ), \end{aligned}$$
(2.6)

or, in a standardized form,

$$\begin{aligned} \frac{\beta _1}{ \sqrt{1-\beta _0^2}}\sqrt{{\widetilde{H}}}\sqrt{n}\, \frac{{\widehat{t}}_0 -t_0}{n}{\mathop {\rightarrow }\limits ^{d}} {\mathcal {N}}(0,1), \end{aligned}$$
(2.7)

equivalently

$$\begin{aligned} \frac{\beta _1}{ \sqrt{1-\beta _0^2}}\sqrt{{\widetilde{H}}}\sqrt{n} \,({\widehat{\tau }}_0 -\tau _0){\mathop {\rightarrow }\limits ^{d}} {\mathcal {N}}(0,1), \end{aligned}$$
(2.8)

where

$$\begin{aligned} {\widetilde{H}} = \int _0^{1} {\widetilde{g}}_0'^2(x-\tau _0)dx- \frac{\big (\int _0^{1}{\widetilde{g}}_0(x-\tau _0) {\widetilde{g}}_0'(x-\tau _0) dx\big )^2}{ \int _0^{1} {\widetilde{g}}_0^2 (x-\tau _0) dx}. \end{aligned}$$
(2.9)

Remark 9

Note that the limit distribution in Theorem 3 does not depend on \(\sigma ^2.\)

For the proof of Theorem 3 see also Sect. 5.

3 Some simulations

In this section, before we turn to the proofs of Theorems 13, we first present some results from a small simulation study. We simulated observations of the time series (1.1) with the function \(g_0(x) = x_+\), \(x\le 1\), for various combinations of \(\beta _0\) and \(\beta _1\) and for various change-points \(t_0\). The errors \(e_t\) were considered to be i.i.d. with a standard normal distribution. The first 50 simulated values were deleted to start computations with stationary observations \(X_t\) for \(t=1, \ldots , t_0.\) We simulated either \(n=500\), 1000 or 5000 observations of (1.1). For each realization of \(\{X_t, t=1,\ldots , n\}\), we estimated the change-point \({\widehat{t}}_0 \) according to (1.11) with the given function \(g_0\) and for \(t_*\) running from 0 to \(\lfloor n(1-\delta )\rfloor \), where \(\delta > 0\) denotes the proportion of observations, which were excluded. For each combination we used 10 000 simulation runs.

The change-point was chosen to be either \(t_0 =n/4\), n/2 or 3/4n, \(\tau _0 = t_0/n\), and \(\delta = 0.05\). The parameters \(\beta _0\) and \(\beta _1\) should satisfy the asymptotic relation (1.2), i.e., \(|\beta _0| <1, \beta _1 \rightarrow 0\), and \(|\beta _1|\sqrt{n} \rightarrow \infty \) as \(n\rightarrow \infty \). For \(\beta _1\) we used small multiples of \(1/\sqrt{\log \log n}\) such that the asymptotic condition (1.2) and also the condition \(|\beta _0 + \beta _1 g_0((t-t_0)/n)| <1\) for \(t=1,\ldots , \lfloor n(1-\delta )\rfloor \) were satisfied. Though in fact \(\beta _1 \) depends on n, we used the same value always for all considered sample sizes n, to make the presentation of the results more transparent.

In Tables 1234, the estimator \({\widehat{\tau }}_0\) computed as \({\widehat{\tau }}_0 = {\widehat{t}}_0/n\), the 90% confidence interval for \(\tau _0\) computed from the Monte Carlo percentiles of \(\sqrt{n}({\widehat{\tau }}_0 -\tau _0)\), and estimators of \(\beta _0, \beta _1\) computed from (1.8) and (1.9) with \(t_{*} = {\widehat{t}}_0\) are presented. The empirical standard deviations of the corresponding point estimators are given in parentheses.

It can be seen that the point estimators of \(\tau _0\) (or \(t_0\), respectively) and of \(\beta _0\) are systematically slightly underestimated, but otherwise behave quite well for all considered variants. The estimators of \(\beta _1\) are more volatile, but they converge to the true value with growing number of observations. To study the behaviour of the estimator of the change-point in finite samples, we also displayed histograms and Q-Q plots of the standardized statistic in (2.8). Some results are presented in Figs. 123456. The simulations show that the behaviour of the estimator \({\widehat{t}}_0\) depends both on the size of the sample and on the parameters \(\beta _0\) and \(\beta _1.\) The assumption \(|\beta _0 + \beta _1 g_0((t-t_0)/n)| <1\) guarantees the stability of the time series (1.1) even after the local change, expressed by the time varying part of the autoregressive coefficient. Due to the local character of the change the convergence to the normal distribution is slow and it can only be expected for sufficiently large samples. In Figs. 123456 convergence to normality is demonstrated for respective sample sizes \(n=500\) (left panels) and \(n=5000\) (right panels). The role of the parameter \(\beta _1\), which represents the rate of the change of the autoregressive coefficient, can be seen on comparing Figs. 3 and 4. In Fig. 3, the value of \(\beta _0 + \beta _1 g_0((t-t_0)/n)\) changes from 0 to 0.9, while in Fig. 4 it varies from 0.5 to 0.9 for the same values of \(t=t_0 + 1,\ldots , n\). The gradual change in the first case is much faster than in the second one, where the change is slow and the increments to the autoregressive coefficient are very small. In this case, the graph of the function on the right-hand side of () can be very flat and its global maximum can be incorrectly detected or it is detected either very soon or very late. This explains the large values of the side column in the histogram and the large skewness of the test statistic, especially in the left part of Fig. 4. Also the position of \(t_0\), and thus the length of the stationary part of the time series under consideration, plays a role. In general, we observe better results, when the change occurs in the middle of the observed time intervals. For smaller sample sizes the kurtosis of the standardized statistics is larger and the finite sample distribution has heavier tails than the normal distribution, but it improves with growing sample size.

Table 1 Estimators of \(\tau _0\), \(\beta _0\), \(\beta _1\); \(t_0=n/2\)
Table 2 Estimators of \(\tau _0\), \(\beta _0\), \(\beta _1\); \(t_0=n/4\)
Table 3 Estimators of \(\tau _0\), \(\beta _0\), \(\beta _1\); \(t_0=3n/4\)
Table 4 Estimators of \(\tau _0\), \(\beta _0\), \(\beta _1\); \(t_0=n/2\) (negative values of \(\beta _0\))
Fig. 1
figure 1

Histograms and Q-Q plots of \(\sqrt{n}(\widehat{\tau }_0 -\tau _0)\), left: \(n=500,\) right: \(n=5\,000\), \(t_0=n/2\), \(\beta _0 = 0\), \(\beta _1 = 1.8\)

Fig. 2
figure 2

Histograms and Q-Q plots of \(\sqrt{n}(\widehat{\tau }_0 -\tau _0)\), left: \(n=500,\) right: \(n=5\,000\), \(t_0=n/2\), \(\beta _0 = 0.3\), \(\beta _1 = 1.2\)

Fig. 3
figure 3

Histograms and Q-Q plots of \(\sqrt{n}(\widehat{\tau }_0 -\tau _0)\), left: \(n=500,\) right: \(n=5\,000\), \(t_0=3n/4\), \(\beta _0 = 0\), \(\beta _1 = 3.6\)

Fig. 4
figure 4

Histograms and Q-Q plots of \(\sqrt{n}(\widehat{\tau }_0 -\tau _0),\) left: \(n=500,\) right: \(n=5\,000\), \(t_0=3n/4\), \(\beta _0 = 0.5\), \(\beta _1 = 1.6\)

Fig. 5
figure 5

Histograms and Q-Q plots of \(\sqrt{n}(\widehat{\tau }_0 -\tau _0),\) left: \(n=500,\) right: \(n=5\,000\), \(t_0= n/2\), \(\beta _0 = -0.8\), \(\beta _1 = 3.4\)

Fig. 6
figure 6

Histograms and Q-Q plots of \(\sqrt{n}(\widehat{\tau }_0 -\tau _0),\) left: \(n=500,\) right: \(n=5\,000\), \(t_0=n/2\), \(\beta _0 = -0.5\), \(\beta _1 = 2.5\)

4 Some auxiliary results

In this section, we collect a series of auxiliary results, which are used in the proofs of our Theorems 13. In the sequel, C denotes a generic positive constant, independent of \(t,t_*\) and n, which may vary from case to case.

Lemma 1

Under the assumptions of Theorem 1, as \(n\rightarrow \infty \),

$$\begin{aligned}&\max _{t=1,\ldots ,n} EX_{t-1}^2=O(1)\quad \text {and}\quad \max _{t=1,\ldots ,n} EX_{t-1}^4=O(1), \end{aligned}$$
(4.1)
$$\begin{aligned}&\quad \Big |\frac{1}{n} \sum _{t=1}^nEX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\Big |=O\big (|\beta _1|\big ), \end{aligned}$$
(4.2)
$$\begin{aligned}&\quad \Big |\frac{1}{n} \sum _{t=1}^ng(t,t_0)EX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\int _0^1 g_0(x-\tau _0)dx\Big |=O\big (|\beta _1|\big ). \end{aligned}$$
(4.3)

Proof

  1. (a)

    In view of (1.5) and the independence and moment assumptions on \(\{e_t\}\),

    $$\begin{aligned} EX_{t-1}^2=\sigma ^2 \sum _{j=0}^{t-1} c_j^2\quad \text {with}\quad c_0=1,~ c_j=\prod _{i=0}^{j-1}\big (\beta _0+\beta _1g(t-1-i,t_0)\big ),~j=1,\ldots ,t-1. \end{aligned}$$

    Now, according to (1.2), for sufficiently large n,

    $$\begin{aligned}&\big (\beta _0+\beta _1g(t-1-i,t_0)\big )^2\\&\quad =\beta _0^2+\beta _1\big (2\beta _0 g(t-1-i,t_0)+\beta _1 g^2(t-1-i,t_0)\big ) \le \beta _0^2+|\beta _1|\,C=: q_1, \end{aligned}$$

    where \(0\le q_1<1\). So,  \(c_j^2\le q_1^j\)  and

    $$\begin{aligned} EX_{t-1}^2\le \sigma ^2 \frac{1-q_1^{t}}{1-q_1}\le \sigma ^2 \frac{1}{1-q_1}=O(1). \end{aligned}$$

    Again due to the moment and independence assumptions on \(\{e_t\}\), a similar estimation yields

    $$\begin{aligned} EX_{t-1}^4 =E e_0^4\, \sum _{j=0}^{t-1} c_j^4+3\sigma ^4\,\sum _{j\ne k} c_j^2 c_k^2 \le E e_0^4\, \sum _{j=0}^{t-1} c_j^4+3\sigma ^4\,\Big (\sum _{j=0}^{t-1} c_j^2\Big )^2=O(1), \end{aligned}$$

    which proves (4.1).

  2. (b)

    Moreover, since \(\beta _1\rightarrow 0\) as \(n\rightarrow \infty \),

    $$\begin{aligned}&EX_{t-1}^2-\frac{\sigma ^2}{1-\beta _0^2}\le \sigma ^2\Big (\frac{1}{1-\beta _0^2-|\beta _1|\,C}-\frac{1}{1-\beta _0^2}\Big )\\&\quad =\sigma ^2\frac{|\beta _1|\,C}{(1-\beta _0^2-|\beta _1|\,C)(1-\beta _0^2)}\le C|\beta _1|. \end{aligned}$$

    Analogously,

    $$\begin{aligned} EX_{t-1}^2\ge \sigma ^2 \frac{1-q_2^{t}}{1-q_2}, \end{aligned}$$

    with \(0\le q_2:=\beta _0^2-|\beta _1|\,C<1\), and, as \(n\rightarrow \infty \),

    $$\begin{aligned} EX_{t-1}^2-\frac{\sigma ^2}{1-\beta _0^2}\ge \sigma ^2\Big (\frac{1}{1-\beta _0^2+|\beta _1|\,C}-\frac{1}{1-\beta _0^2}\Big ) -\sigma ^2\frac{q_2^{t}}{1-q_2}\ge -C\big (|\beta _1|+q_2^{t}\big ). \end{aligned}$$

    Hence,

    $$\begin{aligned} \Big |EX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\Big |\le C\big (|\beta _1|+q_2^{t}\big ), \end{aligned}$$
    (4.4)

    which suffices to prove (4.2), since \(\sum _{t=1}^nq_2^{t}\le C\) and \(\beta _1\gg 1/n\) as \(n\rightarrow \infty \).

  3. (c)

    In view of (4.4) and the assumptions on \(\beta _1\) and g, as \(n\rightarrow \infty \),

    $$\begin{aligned}&\Big |\frac{1}{n} \sum _{t=1}^ng(t,t_0)EX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\int _0^1 g_0(x-\tau _0)dx\Big |\\&\le ~ \frac{1}{n} \sum _{t=1}^n\Big |g(t,t_0)\Big | \Big |EX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\Big | +\frac{\sigma ^2}{1-\beta _0^2}\Big |\frac{1}{n} \sum _{t=1}^ng(t,t_0)- \int _0^1 g_0(x-\tau _0)dx\Big |\\&\le ~ C\Big (|\beta _1| +\frac{1}{n}\sum _{t=1}^nq_2^t +\sum _{t=1}^n\int _{\frac{t-1}{n}}^{\frac{t}{n}}\Big |g_0\Big (\frac{t-t_0}{n}\Big ) -g_0(x-\tau _0)\Big |dx\Big )\\&\le ~ C\Big (|\beta _1| +\frac{1}{n} +\sum _{t=1}^n\int _{\frac{t-1}{n}}^{\frac{t}{n}}\Big |\frac{t}{n} -x-\frac{t_0}{n}+\tau _0\Big |dx\Big ) \le C\Big (|\beta _1| +\frac{1}{n}\Big )=O\big (|\beta _1|\big ), \end{aligned}$$

    which completes the proof. \(\square \)

Lemma 2

Under the assumptions of Theorem 1, as \(n\rightarrow \infty \), with \(t_*=\lfloor n\tau _*\rfloor \),

$$\begin{aligned} \max _{t_*}\Big |\frac{1}{n} \sum _{t=1}^ng(t,t_0)g(t,t_*)EX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\int _0^1 g_0(x-\tau _0)g_0(x-\tau _*)dx\Big |=O\big (|\beta _1|\big ), \end{aligned}$$
(4.5)
$$\begin{aligned} \max _{t_*}\Big |\frac{1}{n} \sum _{t=1}^ng(t,t_*)EX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\int _0^1 g_0(x-\tau _*)dx\Big |=O\big (|\beta _1|\big ),\nonumber \\ \end{aligned}$$
(4.6)

and

$$\begin{aligned} \max _{t_*}\Big |\frac{1}{n} \sum _{t=1}^ng^2(t,t_*)EX_{t-1}^2- \frac{\sigma ^2}{1-\beta _0^2}\int _0^1 g_0^2(x-\tau _*)dx\Big |=O\big (|\beta _1|\big ). \end{aligned}$$
(4.7)

Proof

The proof of (4.5) – (4.7) is similar to that of (4.3), so that details can be omitted. Note that the functions \(g_0(\cdot -\tau _0)g_0(\cdot -\tau _*)\), \(g_0(\cdot -\tau _*)\) and \(g_0^2(\cdot -\tau _*)\) are also bounded and Lipschitz continuous, uniformly in \(0\le \tau _*\le 1\).\(\square \)

Remark 10

It is obvious from the proofs of Lemmas 1 and 2 that, if the function \(g_0\) is just continuous (instead of Lipschitz continuous), assertions (4.3)–(4.7) still hold true with \(t_*=\lfloor n\tau _*\rfloor \), \(\tau _*\in [0,1]\) fixed (instead of \(\max _{t_*}\)), but with \(O\big (|\beta _1|\big )\) being replaced by o(1).

Lemma 3

Under the assumptions of Theorem 1, as \(n\rightarrow \infty \), with \(t_*=\lfloor n\tau _*\rfloor \),

$$\begin{aligned}&\Big |\sum _{j=1}^ne_j X_{j-1}\Big |=O_P\big (\sqrt{n}\big ), \end{aligned}$$
(4.8)
$$\begin{aligned} \nonumber \\&\Big |\sum _{t=1}^ne_t X_{t-1}g(t,t_*)\Big |=O_P\big (\sqrt{n}\big ),\quad \text {for every fixed}~~\tau _*\in [0,1], \end{aligned}$$
(4.9)

and

$$\begin{aligned} \frac{\varepsilon _n}{\sqrt{n}} ~\max _{t_*}\Big |\sum _{t=1}^ne_t X_{t-1}g(t,t_*)\Big |=o_P(1), \quad \text {for every sequence }\{\varepsilon _n\}_{n=1,2,\ldots }\text { with }\varepsilon _n\rightarrow 0.\nonumber \\ \end{aligned}$$
(4.10)

Proof

  1. (a)

    Let \({{\mathcal {F}}}_{j-1}\) denote the \(\sigma \)-field generated by \(e_0,\ldots ,e_{j-1}\). Then, in view of our assumptions on \(\{e_t\}\) and (4.2), as \(n\rightarrow \infty \),

    $$\begin{aligned}&E\Bigg (\sum _{j=1}^ne_j X_{j-1}\Bigg )^2 = \sum _{j=1}^nE\big (e_j^2 X_{j-1}^2\big ) +2\sum _{i<j} E\big (e_i X_{i-1} e_j X_{j-1}\big )\\&\qquad = \sum _{j=1}^nE\big (E\big [e_j^2 X_{j-1}^2\big |{{\mathcal {F}}}_{j-1}\big ]\big ) +2\sum _{i<j} E\big (E\big [e_i X_{i-1} e_j X_{j-1}|{{\mathcal {F}}}_{j-1}\big ]\big ) = \sigma ^2\sum _{j=1}^nEX_{j-1}^2 =O(n), \end{aligned}$$

    which suffices to prove (4.8).

  2. (b)

    Via similar arguments, along the lines of proof of (4.3), as \(n\rightarrow \infty \),

    $$\begin{aligned} E\Big (\sum _{t=1}^ne_t X_{t-1}g(t,t_*)\Big )^2=\sigma ^2 \sum _{t=1}^ng^2(t,t_*)EX_{t-1}^2=O(n), \end{aligned}$$

    which proves (4.9).

  3. (c)

    Consider, with \(t_*=\lfloor n\tau _*\rfloor \), the sequence of stochastic processes \(\{X_n(\cdot )\}_{n=1,2,\ldots }\) in D[0, 1], where

    $$\begin{aligned} X_n(\tau _*)=\frac{\varepsilon _n}{\sqrt{n}}~\sum _{t=1}^ne_t X_{t-1}g(t,t_*),\quad 0\le \tau _*\le 1. \end{aligned}$$

    In view of (4.9), the finite-dimensional distributions of \(X_n\) tend to 0. Moreover, for \(0\le \tau _1\le \tau _*\le \tau _2\le 1\),

    $$\begin{aligned} E|X_n(\tau _*)-X_n(\tau _1)||X_n(\tau _2)-X_n(\tau _*)| \le \big (E|X_n(\tau _*)-X_n(\tau _1)|^2\big )^{1/2} \big (E|X_n(\tau _2)-X_n(\tau _*)|^2\big )^{1/2}, \end{aligned}$$

    and, according to our assumptions on g, with \(t_1=\lfloor n\tau _1\rfloor \),

    $$\begin{aligned} E|X_n(\tau _*)-X_n(\tau _1)|^2&= \,\sigma ^2\,\frac{\varepsilon _n^2}{n}\,\sum _{t=1}^n|g(t,t_*)-g(t,t_1)|^2 EX_{t-1}^2\\&\le \,C\,\varepsilon _n^2\,\Big |\frac{t_*-t_1}{n}\Big |^2\le \,C\,\varepsilon _n^2\,|\tau _2-\tau _1|^2. \end{aligned}$$

    Analogously, with \(t_2=\lfloor n\tau _2\rfloor \),

    $$\begin{aligned} E|X_n(\tau _2)-X_n(\tau _*)|^2 \le \,C\,\varepsilon _n^2\,\Big |\frac{t_2-t_*}{n}\Big |^2\le \,C\,\varepsilon _n^2\,|\tau _2-\tau _1|^2, \end{aligned}$$

    so that

    $$\begin{aligned} E|X_n(\tau _*)-X_n(\tau _1)||X_n(\tau _2)-X_n(\tau _*)| \le \,C\,\varepsilon _n^2\,|\tau _2-\tau _1|^2. \end{aligned}$$

    In view of Billingsley (1968), Theorem 15.6, this proves that

    $$\begin{aligned} X_n \,{\mathop {\longrightarrow }\limits ^{\mathcal {D}[0,1]}}\,0\qquad \text {as}\qquad n\rightarrow \infty , \end{aligned}$$

    which suffices for the proof of (4.10).

\(\square \)

Lemma 4

Under the assumptions of Theorem 1, as \(n\rightarrow \infty \), with \(t_*=\lfloor n\tau _*\rfloor \),

$$\begin{aligned}&\Big |\sum _{t=1}^n\big (X_{t-1}^2-EX_{t-1}^2\big )\Big |=O_P\big (\sqrt{n}\big ), \end{aligned}$$
(4.11)
$$\begin{aligned}&\quad \Big |\sum _{t=1}^ng(t,t_0)g(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big )\Big |=O_P\big (\sqrt{n}\big ), \quad \text {for every fixed}~~\tau _*\in [0,1], \end{aligned}$$
(4.12)
$$\begin{aligned}&\quad \Big |\sum _{t=1}^ng(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big )\Big |=O_P\big (\sqrt{n}\big ), \quad \text {for every fixed}~~\tau _*\in [0,1], \end{aligned}$$
(4.13)
$$\begin{aligned}&\quad \Big |\sum _{t=1}^ng^2(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big )\Big |=O_P\big (\sqrt{n}\big ), \quad \text {for every fixed}~~\tau _*\in [0,1], \end{aligned}$$
(4.14)

and, for every sequence \(\{\varepsilon _n\}_{n=1,2,\ldots }\) with \(\varepsilon _n\rightarrow 0\),

$$\begin{aligned}&\frac{\varepsilon _n}{\sqrt{n}} ~\max _{t_*}\Big |\sum _{t=1}^ng(t,t_0)g(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big )\Big |=o_P(1), \end{aligned}$$
(4.15)
$$\begin{aligned}&\quad \frac{\varepsilon _n}{\sqrt{n}} ~\max _{t_*}\Big |\sum _{t=1}^ng(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big )\Big |=o_P(1), \end{aligned}$$
(4.16)
$$\begin{aligned}&\quad \frac{\varepsilon _n}{\sqrt{n}} ~\max _{t_*}\Big |\sum _{t=1}^ng^2(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big )\Big |=o_P(1). \end{aligned}$$
(4.17)

Proof

We only give a proof of (4.13) and (4.16) here. The other assertions can be shown in a similar manner.

  1. (a)

    In view of (1.5),

    $$\begin{aligned}&\sum _{t=1}^ng(t,t_*)X_{t-1}^2= \sum _{t=1}^ng(t,t_*)\Big (e_{t-1} +\sum _{j=1}^{t-1}e_{t-1-j}\prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big )\Big )^2 \nonumber \\&\quad = \sum _{t=1}^ng(t,t_*)e_{t-1}^2 +\sum _{t=1}^ng(t,t_*)\sum _{j=1}^{t-1}e_{t-1-j}^2\prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big )^2 \nonumber \\&\qquad + 2 \sum _{t=1}^ng(t,t_*)e_{t-1} \sum _{j=1}^{t-1}e_{t-1-j}\prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big ) \nonumber \\&\qquad + 2 \sum _{t=1}^ng(t,t_*)\sum _{j_1<j_2}e_{t-1-j_1}e_{t-1-j_2}\prod _{i_1=1}^{j_1-1}\big (\beta _0+\beta _1 g(t-1-i_1,t_0)\big )\nonumber \\&\qquad \times \prod _{i_2=1}^{j_2-1}\big (\beta _0+\beta _1 g(t-1-i_2,t_0)\big ) \nonumber \\&\quad =: S_1 +S_2 +2S_3 +2 S_4. \end{aligned}$$
    (4.18)

    Since g is bounded and \(\{e_t\}\) is an i.i.d. sequence,

    $$\begin{aligned} E(S_1-ES_1)^2 =\sum _{t=1}^ng^2(t,t_*)E\big (e_t^2-Ee_t^2\big )^2 \le C\,n. \end{aligned}$$
    (4.19)

    Next, with \(v^2:=Var(e_0^2)\) and b as in (1.4), due to the independence of \(e_{t_1-1-j}^2\) and \(e_{t_2-1-k}^2\), if \(t_1-1-j\ne t_2-1-k\), i.e., if \(k\ne j+t_2-t_1\), for sufficiently large n,

    $$\begin{aligned}&E(S_2-ES_2)^2 =E\Big (\sum _{t=1}^ng(t,t_*)\sum _{j=1}^{t-1}\big (e_{t-1-j}^2-Ee_{t-1-j}^2\big ) \prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big )^2\Big )^2 \nonumber \\&= v^2 \sum _{t=1}^ng^2(t,t_*)\sum _{j=1}^{t-1}\prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big )^4 \nonumber \\&\quad +2 v^2 \sum _{t_1<t_2}g(t_1,t_*)g(t_2,t_*)\sum _{j=1}^{t_1-1} \prod _{i_1=1}^{j-1} \big (\beta _0+\beta _1 g(t_1-1-i_1,t_0)\big )^2 \prod _{i_2=1}^{j-1+t_2-t_1}\big (\beta _0+\beta _1 g(t_2-1-i_2,t_0)\big )^2 \nonumber \\&\le C\,\Big (\sum _{t=1}^ng^2(t,t_*)\sum _{j=1}^{t-1}b^{4(j-1)}+ \sum _{t_2=2}^n g(t_2,t_*) \sum _{t_1=1}^{t_2-1} g(t_1,t_*) \sum _{j=1}^{t_1-1} b^{4(j-1)+2(t_2-t_1)}\Big )\le C\,n. \end{aligned}$$
    (4.20)

    Similarly, since \(ES_3=0\),

    $$\begin{aligned}&E(S_3-ES_3)^2 =E\Big (\sum _{t=1}^ng(t,t_*)e_{t-1} \sum _{j=1}^{t-1}e_{t-1-j} \prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big )^2\Big )^2 \nonumber \\ \qquad&= \sigma ^2 \sum _{t=1}^ng^2(t,t_*)E\Big (\sum _{j=1}^{t-1}e_{t-1-j} \prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big )^2\Big )^2 \nonumber \\ \qquad&= \sigma ^4 \sum _{t=1}^ng^2(t,t_*)\sum _{j=1}^{t-1}\prod _{i=1}^{j-1}\big (\beta _0+\beta _1 g(t-1-i,t_0)\big )^4 \,\le C\,\sum _{t=1}^ng^2(t,t_*)\sum _{j=1}^{t-1}b^{4(j-1)} \le C\,n. \end{aligned}$$
    (4.21)

    Finally, via a corresponding estimation,

    $$\begin{aligned}&E(S_4-ES_4)^2 \nonumber \\&=\sum _{t=1}^ng^2(t,t_*)E\Big (\sum _{j_1<j_2}e_{t-1-j_1}e_{t-1-j_2}\prod _{i_1=1}^{j_1-1}\big (\beta _0+\beta _1 g(t-1-i_1,t_0)\big ) \prod _{i_2=1}^{j_2-1}\big (\beta _0+\beta _1 g(t-1-i_2,t_0)\big )\Big )^2 \nonumber \\&\le C\sum _{t=1}^ng^2(t,t_*)\le C\,n. \end{aligned}$$
    (4.22)

    On combining (4.18)–(4.22), we see that

    $$\begin{aligned} E\Bigg (\sum _{t=1}^ng(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big )\Bigg )^2 \le C\,n, \end{aligned}$$

    which suffices to prove (4.13).

  2. b)

    The proof of (4.16) can be given analogously to that of (4.10) in Lemma 3. Consider, with \(t_*=\lfloor n\tau _*\rfloor \), the sequence of stochastic processes \(\{\widetilde{X}_n(\cdot )\}_{n=1,2,\ldots }\) in D[0, 1], where

    $$\begin{aligned} \widetilde{X}_n(\tau _*)=\frac{\varepsilon _n}{\sqrt{n}}~\sum _{t=1}^ng(t,t_*)\big (X_{t-1}^2-EX_{t-1}^2\big ),\quad 0\le \tau _*\le 1. \end{aligned}$$

    In view of (4.13), the finite-dimensional distributions of \(\widetilde{X}_n\) tend to 0. Moreover, for \(0\le \tau _1\le \tau _*\le \tau _1\le 1\), by just replacing \(g(t,t_*)\) in the estimations (4.18)-(4.22) by \(g(t,t_*)-g(t,t_1)\) and taking the Lipschitz continuity of \(g_0\) into account,

    $$\begin{aligned} E\big (\widetilde{X}_n(\tau _*)-\widetilde{X}_n(\tau _1)\big )^2 \le \,C\,\frac{\varepsilon _n^2}{n}\,\sum _{t=1}^n|g(t,t_*)-g(t,t_1)|^2 \le \,C\,\varepsilon _n^2\,|\tau _2-\tau _1|^2. \end{aligned}$$

    Analogously,

    $$\begin{aligned} E\big (\widetilde{X}_n(\tau _2)-\widetilde{X}_n(\tau _*)\big )^2 \le \,C\,\varepsilon _n^2\,|\tau _2-\tau _1|^2, \end{aligned}$$

    and, via the Cauchy-Schwarz inequality,

    $$\begin{aligned} E\big |\widetilde{X}_n(\tau _*)-\widetilde{X}_n(\tau _1)\big |\big |\widetilde{X}_n(\tau _2)-\widetilde{X}_n(\tau _*)\big | \le \,C\,\varepsilon _n^2\,|\tau _2-\tau _1|^2. \end{aligned}$$

    In view of Billingsley (1968), Theorem 15.6, this proves that

    $$\begin{aligned} \widetilde{X}_n \,{\mathop {\longrightarrow }\limits ^{\mathcal {D}[0,1]}}\,0\qquad \text {as}\qquad n\rightarrow \infty , \end{aligned}$$

    which suffices for the proof of (4.16).

\(\square \)

The following two lemmas from real analysis will also be used in the proof of Theorem 1.

Lemma 5

Let \(\{b_n\}_{n=1,2,\ldots }\) be a real sequence such that \(b_n\rightarrow \infty \) as \(n\rightarrow \infty \). Then there is a sequence \(\{\varepsilon _n\}_{n=1,2,\ldots }\) of positive reals, with \(\varepsilon _n\rightarrow 0\), such that still \(\varepsilon _n b_n\rightarrow \infty \) as \(n\rightarrow \infty \).

Proof

For n sufficiently large, there exists an integer \(k_n\) such that \(2^{k_n}\le b_n <2^{k_n+1}\). Obviously, \(k_n\rightarrow \infty \) as \(n\rightarrow \infty \), so that a choice of \(\varepsilon _n =1/k_n\) completes the proof.

\(\square \)

Lemma 6

Let f be a continuous real function on a compact set K and \(x_0\) be a unique maximizer of  f, i.e., \(x_0 = \mathrm{arg\,max}_x f(x)\). Furthermore assume that \(\lim _{n\rightarrow \infty }\max _{x\in K} |f_n(x)-f(x)|=0\) and let \(\widehat{x}_n =\mathrm{arg\,max}_x f_n(x)\) be a maximizer of \(f_n\) (not necessarily unique). Then,

$$\begin{aligned} \widehat{x}_n \rightarrow x_0\qquad \text {as}\qquad n\rightarrow \infty . \end{aligned}$$

Proof

 Suppose \(\widehat{x}_n \nrightarrow x_0\) as \(n\rightarrow \infty \). Since K is compact, there exists a subsequence \(\{\widehat{x}_{k_n}\}\) and an \(x_1\ne x_0\) such that \(\widehat{x}_{k_n}\rightarrow x_1\), hence

$$\begin{aligned} \big |f_{k_n}(\widehat{x}_{k_n})-f(x_1)\big |\le \max _x \big |f_{k_n}(x)-f(x)\big | + \big |f(\widehat{x}_{k_n})-f(x_1)\big | \rightarrow 0\quad \text {as}\quad n\rightarrow \infty , \end{aligned}$$

because of the continuity of f. Since \(x_0\) is the unique maximizer of f, it holds \(f(x_1) < f(x_0)\) implying that

$$\begin{aligned} \limsup _{n\rightarrow \infty } \big (f_{k_n}(\widehat{x}_{k_n})-f(x_0)\big ) < 0. \end{aligned}$$

This, however, contradicts our assumptions, since

$$\begin{aligned} \big |f_n(\widehat{x}_n)-f(x_0)\big | = \big |\max _x f_n(x)-\max _x f(x)\big | \le \max _x \big |f_n(x)-f(x)\big | \rightarrow 0 \quad \text {as}\quad n\rightarrow \infty , \end{aligned}$$

so that the proof is complete.\(\square \)

Next, we need some extensions of Lemmas 14. Particularly, we study properties of the following quantities for \(|t_0-t_*|\le b_n\), with \(b_n\rightarrow \infty ,~b_n/n \rightarrow 0\):

$$\begin{aligned} L_0(t_0, t_*)&= \frac{1}{\sqrt{n}}\sum _{t=1}^ne_t X_{t-1} \frac{g(t,t_*)- g(t,t_0)}{{ (t_0-t_*)}/n} \\ L_1(t_0,t_*)&= \frac{1}{\sqrt{n}}\sum _{t=1}^n\,e_t X_{t-1}{\widetilde{h}}_n(t,t_*,t_0), ~~\text {with} ~~{\widetilde{h}}_n(t,t_*,t_0) ~\text {as in (4.33) below,} \\ L_2(t_0, t_*)&= \frac{1}{\sqrt{n}}\sum _{t=1}^ne_t X_{t-1} \frac{{\widetilde{g}}_n(t,t_*)-{\widetilde{g}}_{n}(t,t_0)}{(t_*-t_0)/n}, \\ L_3(t_0,t_*)&= \frac{1}{n} \sum _{t=1}^nX_{t-1}^2 \,{\widetilde{g}}_n(t,t_0) \frac{{\widetilde{g}}_n(t, t_0)-{\widetilde{g}}_n(t,t_*)}{(t_*-t_0)/n}, \\ L_{4}(t_0,t_*)&=\frac{1}{n}\sum _{t=1}^nX_{t-1}^2 \frac{({\widetilde{g}}_{n}(t,t_0)- {\widetilde{g}}_n(t,t_*))^2}{((t_*-t_0)/n)^{2}}. \end{aligned}$$

We start with an extension of Lemma 3.

Lemma 7

Let the assumptions of Theorem 3 be satisfied and let \(t_*\) be such that \(|t_*-t_0|\le r_n\,\sqrt{n} \,|\beta _1|^{-1}\), with

$$\begin{aligned} r_n\rightarrow \infty ,\quad \frac{|\beta _1|\sqrt{n}}{r_n}\rightarrow \infty . \end{aligned}$$
(4.23)

Then, for \(j=0,1,2\), \(L_j(t_0,t_*)\) has an asymptotically normal limit distribution, with zero mean and variance \( \sigma _j^2\), where

$$\begin{aligned} \sigma _0^2&=\frac{\sigma ^4}{1-\beta _0^2}\int _0^{1}g_0^{\prime 2}(x-\tau _0) dx, \\ \sigma _1^2&=\frac{\sigma ^4}{1-\beta _0^2}\Big (\int _0^{1}{\widetilde{g}}_0'^2(x-\tau _0) dx- \frac{\big (\int _0^{1}{\widetilde{g}}_0(x-\tau _0) {\widetilde{g}}_0'(x-\tau _0) dx\big )^2}{\int _0^{1}{\widetilde{g}}_0^2(x-\tau _0) dx}\Big ), \\ \sigma _2^2&=\frac{\sigma ^2}{1-\beta _0^2}\int _0^{1}{\widetilde{g}}_0^{\prime 2} (x-\tau _0) dx. \end{aligned}$$

Moreover, as \(n\rightarrow \infty \),

$$\begin{aligned} \max _{|t_0-t_{**}|\le b_n }|L_j(t_0,t_*)- L_j(t_0,t_{**})|=o_P(c_n),\quad j=0,1,2, \end{aligned}$$
(4.24)

for some \(c_n\rightarrow 0\) and any \(t_{*} \) such that \(|t_0-t_{*}|\le b_n,\, b_n\rightarrow \infty ,\, b_n/n\rightarrow 0\).

Proof

We focus on \(L_0(t_0,t_*)\). The desired results for \(L_1(t_0, t_*)\) and \(L_2(t_0, t_*)\) can be derived in the same way, the obtained expressions are just somewhat more complex, but can be omitted.

Note that, for fixed \(t_*\) and \(t_0\), \(L_j (t_0, t_*),\, j=0,1,2\), are sums of martingale difference arrays. Hence to obtain their limit properties, we can apply Theorem 24.3 in Davidson (1994), p. 383, which means for \(L_0(t_0,t_*)\) to verify validity of the conditions

$$\begin{aligned}&\frac{\frac{1}{ n}\sum _{t=1}^n\big [ e_t X_{t-1} \frac{g(t,t_*)- g(t,t_0)}{(t_0-t_*)/n}\big ]^2}{ \frac{1}{ n} \sum _{j=1}^n{ E\big [e_j X_{j-1} \frac{g(j,t_*)- g(j,t_0)}{(t_0-t_*)/n}\big ]^2}}{\mathop {\rightarrow }\limits ^{P}}1, \end{aligned}$$
(4.25)
$$\begin{aligned}&\max _t\frac{\big [ e_t X_{t-1} \frac{g(t,t_*)-g(t,t_0)}{(t_0-t_*)/n}\big ]^2}{\sum _{j=1}^nE\big [ e_j X_{j-1} \frac{g(j,t_*)- g(j,t_0)}{(t_0-t_*)/n}\big ]^2}{\mathop {\rightarrow }\limits ^{P}}0. \end{aligned}$$
(4.26)

We make repeated use of assumption (A.4), particularly

$$\begin{aligned} \Big |\frac{g(j,t_*)- g(j,t_0)}{(t_0-t_*)/n}-g_0' ((j-t_0)/n)\Big |\le D |((t_*-t_0)/n)|^{\Delta }. \end{aligned}$$

Consider first the denominator in (4.25). Direct calculations give

$$\begin{aligned}&\frac{1}{ n}\sum _{j=1}^nE\Big [ e_j X_{j-1} \frac{g(j,t_*)-g(j,t_0)}{(t_0-t_*)/n}\Big ]^2\\&= \frac{\sigma ^4}{1-\beta ^2_0}\frac{1}{n}\sum _{j=1}^n\Big [\frac{g(j,t_*)-g(j,t_0)}{(t_0-t_*)/n}\Big ]^2 +\frac{\sigma ^2}{ n}\sum _{j=1}^n\Big [ E X^2_{j-1}-\frac{\sigma ^2}{1-\beta _0^2}\Big ] \Big [\frac{g(j,t_*)-g(j,t_0)}{(t_0-t_*)/n}\Big ]^2. \end{aligned}$$

By Assumption (A.4) and using the same arguments as in the proofs of Lemmas 2 and 4 , we get

$$\begin{aligned}&\frac{1}{n}\sum _{j=1}^n\Big [ \frac{g(j,t_*)- g(j,t_0)}{{ (t_0-t_*)}/n} - g'_0 ((j-t_0)/n))\Big ]^2 \le C \,| (t_*-t_0)/n |^{2\Delta } \le \,C \,\Big (\frac{ r_n}{|\beta _1| \sqrt{n}}\Big )^{2\Delta } , \\&\Big |\frac{1}{n}\sum _{j=1}^n\Big [ E X^2_{t-1}-\frac{\sigma ^2}{1-\beta _0^2}\Big ] \Big [\frac{g(j,t_*)- g(j,t_0)}{{ (t_0-t_*)}/n} - g'_0((j-t_0)/n)\big ]^2 \Big |\, \le C | (t_*-t_0)/n |^{2\Delta }\\&\le C \Big (\frac{ r_n}{|\beta _1| \sqrt{n}}\Big )^{2\Delta }, \\&\frac{1}{n} \sum _{t=1}^ng_0^{\prime 2}((t-t_0)/n) =\int _0^{1} g_0^{\prime 2}(x-\tau _0) dx +o(1), \\&\frac{1}{n} \sum _{t=1}^n\Big ( E X^2_{t-1}-\frac{\sigma ^2}{1-\beta _0^2}\Big ) g_0^{\prime 2}((t-t_0)/n)\big )^2 = O(|\beta _1|), \end{aligned}$$

as \(n\rightarrow \infty \), and combining all these results we have

$$\begin{aligned} \frac{1}{ n}\sum _{t=1}^nE\Big [ e_t X_{t-1} \frac{g(t,t_*)-g(t,t_0)}{(t_0-t_*)/n}\Big ]^2 = \frac{\sigma ^4}{1-\beta ^2_0}\int _0^{1} g_0^{\prime 2}(x-\tau _0) dx +o(1).\nonumber \\ \end{aligned}$$
(4.27)

For the numerator in (4.25) we have

$$\begin{aligned}&\frac{1}{ n}\sum _{t=1}^n\Big [ e_t X_{t-1} \frac{g(t,t_*)-g(t,t_0)}{(t_0-t_*)/n}\Big ]^2 \\&= \frac{1}{ n}\sum _{t=1}^n\big [e_t^2- \sigma ^2\big ] X_{t-1}^2 \Big [\frac{g(t,t_*)- g(t,t_0)}{(t_0-t_*)/n}\Big ]^2 + \frac{\sigma ^2}{ n}\sum _{t=1}^nX^2_{t-1}\Big [\frac{g(t,t_*)- g(t,t_0)}{(t_0-t_*)/n}\Big ]^2. \end{aligned}$$

The first term on the r.h.s. is the sum of martingale difference arrays, with zero mean and variance

$$\begin{aligned} E\Big [ \frac{1}{ n}\sum _{t=1}^n\Big ( \big (e_t^2- \sigma ^2\big ) X^2_{t-1} \frac{g(t,t_*)- g(t,t_0)}{(t_0-t_*)/n}\Big )^2\Big ]^2 =O\big (n^{-1}\big ), \end{aligned}$$

which follows from the finiteness of \(E e_t^4,\) Assumption (A.3) and the uniform boundedness of \(EX_{t-1}^4\) (cf. (4.1)). Thus,

$$\begin{aligned} \frac{1}{ n}\sum _{t=1}^n\big [e_t^2- \sigma ^2\big ] X_{t-1}^2 \Big [ \frac{g(t,t_*)- g(t,t_0)}{(t_0-t_*)/n}\Big ]^2 = O_P\big (n^{-1/2}\big ). \end{aligned}$$
(4.28)

Next, proceeding in the same way as in the proofs of Lemmas 1 and 4, we get

$$\begin{aligned} \frac{\sigma ^2}{ n}\sum _{t=1}^nX^2_{t-1} \Big [ \frac{g(t,t_*)- g(t,t_0)}{(t_0-t_*)/n}\Big ]^2 =\frac{\sigma ^4}{1-\beta ^2_0} \int _0^{1} g_0^{\prime 2}(x-\tau _0) dx +o_P(1),\nonumber \\ \end{aligned}$$
(4.29)

and combining (4.28), (4.29) and (4.27) we obtain (4.25).

To verify (4.26) note that, due to (4.27), it suffices to prove

$$\begin{aligned} \max _{t}\frac{1}{n}\Big [ e_t X_{t-1} \frac{ g(t,t_*)-g(t,t_0)}{(t_0-t_*)/n}\Big ]^2{\mathop {\rightarrow }\limits ^{P}}0. \end{aligned}$$

For this we have, using similar arguments as above,

$$\begin{aligned}&P\Big (\max _{t} \Big [ e_t X_{t-1} \frac{g(t,t_*)-g(t,t_0)}{(t_0-t_*)/n}\Big ]^2 \ge n\epsilon \Big ) \le \,\sum _{t=1}^nP\Big (\Big [ e_t X_{t-1} \frac{g(t,t_*)-g(t,t_0)}{(t_0-t_*)/n}\Big ]^2 \ge n \epsilon \Big ) \\&\le \frac{1}{n^2 \epsilon ^2}\sum _{t=1}^nE e_t^4 EX_{t-1}^4 \Big [ \frac{g(t,t_*)- g(t,t_0)}{(t_0-t_*)/n}\Big ]^4 = O(n^{-1}), \end{aligned}$$

and we can conclude that the asymptotic normality of \(L_0 (t_0, t_*)\) holds for fixed \(t_*\).

Next we show (4.24) for \(j=0\). We proceed as in the proof of Lemma 3, part c). Toward this we study for \(t_0<t_*<t_{**}\) and \(|t_0-t_*|+ |t_*-t_{**}|\le b_n\), satisfying \(b_n\rightarrow \infty ,\, b_n/n\rightarrow 0\), the quantity

$$\begin{aligned} E\big [{ L_0(t_0,t_*)- L_0(t_0,t_{**})}\big ]^2&=\frac{1}{n}\sum _{t=1}^n\sigma ^2 E (X_{t-1}^2)\Big (\frac{ g(t,t_*) -g (t,t_0)}{ (t_0-t_*)/n} -\frac{ g(t,t_{**}) -g (t,t_0)}{ (t_0-t_{**})/n}\Big )^2 \\&\le C \frac{1}{n}\sum _{t=1}^n\Big (\frac{ g(t,t_*) -g (t,t_0)}{ (t_0-t_*)/n} -\frac{ g(t,t_{**}) -g (t,t_0)}{ (t_0-t_{**})/n}\Big )^2, \end{aligned}$$

for some \(C>0\). By Assumption (A.4)

$$\begin{aligned} \big | g(t,t_{**})- g(t,t_{*})- g'_0((t-t_{*}) /n)((t_{*}-t_{**})/n)\big | \le C \big |(t_{**}-t_{*})/n\big |^{1+\Delta } \end{aligned}$$

which gives

$$\begin{aligned}&\frac{1}{n}\sum _{t=1}^n\Big (\frac{ g(t,t_*) -g (t,t_0)}{ (t_0-t_*)/n} -\frac{ g(t,t_{**}) -g(t,t_0)}{(t_0-t_{**})/n}\Big )^2 \\&=\frac{1}{n}\sum _{t=1}^n\Big [\frac{g(t,t_*) -g (t,t_0)}{(t_0-t_*)/n} \Big (1 -\frac{(t_0-t_*)/n}{(t_0-t_{**})/n}\Big ) -\frac{ g(t,t_{**}) -g(t,t_*)}{ (t_0-t_{**})/n}\Big ]^2 \\&= \frac{1}{n}\sum _{t=1}^n\Big [\Big (\frac{ g(t,t_*) -g (t,t_0)}{ (t_0-t_*)/n} - g_0'((t-t_{*})/n) \Big )\frac{(t_*-t_{**})/n}{(t_0-t_{**})/n} +O\Big (\frac{|(t_*-t_{**})/n|^{1+\Delta }}{|(t_0-t_{**})/n|}\Big )\Big ]^2 \\&\le C \Big [\big |(t_0-t_{*})/n\big |^{\Delta }\frac{(t_{*}-t_{**})/n}{ (t_0-t_{**})/n} +\frac{|(t_*-t_{**})/n|^{1+\Delta }}{|(t_0-t_{**})/n|}\Big ]^2 \\&\le C \Big [\frac{|(t_*-t_{**})/n|}{|(t_0-t_{**})/n|^{1-\Delta }} +\frac{|(t_*-t_{**})/n|^{1+\Delta }}{|(t_0-t_{**})/n|}\Big ]^2 \le C \big |(t_{*}-t_{**})/n\big |^{2\Delta }, \end{aligned}$$

since \(|t_0-t_*|\vee |t_*-t_{**}|\le |t_0-t_{**}|\) and \(\Delta \le 1\).

In view of \(2\Delta >1\) by Assumption (A.4), assertion (4.24) can now be finished by again making use of Billingsley (1968), Theorem 15.6, as in the proof of Lemma 3, part c). \(\square \)

The next lemma is an extension of Lemma 4.

Lemma 8

Under the assumptions of Theorem 3 we have

$$\begin{aligned} L_{4}(t_0,t_*)&=\frac{\sigma ^2}{1-\beta _0^2}\Big [ \frac{1}{n}\sum _{t=1}^n\Big ( g_0'((t-t_0)/n)- \frac{1}{n}\sum _{j=1}^ng_0'((j-t_0)/n)\Big )^2\Big ] +O_P\Big (\big |(t_0-t_*)/n)\big |^{\Delta }\Big ) +o_P(1) \nonumber \\&= \frac{\sigma ^2}{1-\beta _0^2}\int _0^{1} {\widetilde{g}}_0^{\prime 2} (x-\tau _0) dx +O_P\Big (\big |(t_0-t_*)/n)\big |^{\Delta }\Big ) +o_P(1),\quad | t_*-t_0|\le b_n, \end{aligned}$$
(4.30)
$$\begin{aligned} L_3(t_0,t_*)&=\frac{\sigma ^2}{1-\beta _0^2}\int _0^{1} {\widetilde{g}}_0 (x-\tau _0) \,{\widetilde{g}}_0' (x-\tau _0) dx + o_P(1),\quad | t_*-t_0|\le b_n. \end{aligned}$$
(4.31)

Moreover, as \(n\rightarrow \infty \),

$$\begin{aligned} \max _{|t_0-t_{**}|\le b_n }\big |L_j(t_0,t_*)- L_j(t_0,t_{**})\big |=o_P(c_n),\quad j=3,4, \end{aligned}$$
(4.32)

for some \(c_n\rightarrow 0\) and any \(|t_{*}-t_0| \le b_n \), with \(b_n\rightarrow \infty ,~b_n/n\rightarrow 0\).

Proof

Note that

$$\begin{aligned} L_{4}(t_0,t_*)&=\frac{1}{n}\sum _{t=1}^nX_{t-1}^2 \frac{\big ({\widetilde{g}}_{n}(t,t_0) -{\widetilde{g}}_n(t,t_*)\big )^2}{((t_*-t_0)/n)^2} \\&=\frac{1}{n}\sum _{t=1}^nX_{t-1}^2 \frac{\big (g(t,t_0)-g(t,t_*)\big )^2}{((t_*-t_0)/n\big )^2} -\frac{\Big (\frac{1}{n}\sum _{j=1}^nX_{j-1}^2 \big (g(j,t_0)-g(j,t_*)\big )\Big )^2}{\frac{1}{n} \sum _{j=1}^nX_{j-1}^2 \big ((t_*-t_0)/n\big )^2} \\&=:L_{41}(t_0,t_*) -L_{42}(t_0,t_*). \end{aligned}$$

We have

$$\begin{aligned} L_{41}(t_0,t_*)&= \frac{1}{n} \sum _{j\ge \max (t_0, t_*)} X_{j-1}^2 \frac{\big (g(j,t_0)-g(j,t_*)\big )^2}{\big ((t_*-t_0)/n\big )^2}\\&\quad +\frac{1}{n}\sum _{\min (t_0,t_*)\le j\le \max (t_0, t_*)} X_{j-1}^2 \frac{\big (g(j,t_0)-g(j,t_*)\big )^2}{\big ((t_*-t_0)/n\big )^2}. \end{aligned}$$

The latter sum on the r.h.s has only \(|t_0-t_*|\le 2 b_n\) summands, with \(b_n\rightarrow \infty ,~b_n/n\rightarrow 0\), and the terms \( \big (g(j,t_0)-g(j,t_*)\big )\big / \big ((t_*-t_0)/n\big )\) are bounded in j, thus the respective terms are not influential. By (A.4),

$$\begin{aligned}&\frac{1}{n}\sum _{j\ge \max (t_0, t_*)} X_{j-1}^2 \Big ({ \frac{g(j,t_*)-g(j,t_0)}{(t_0-t_*)/n)}} - g_0'((j-t_0)/n)\Big )^2 \\&\le C \, \frac{1}{n}\sum _{j=1}^nX_{j-1}^2 \Big |\frac{t_*-t_0}{n}\Big |^{2\Delta } =\, O_P\Big (\Big |\frac{t_*-t_0}{n}\Big |^{2\Delta }\Big ), \end{aligned}$$

where the last relation follows from (4.1) and (4.11). Proceeding as in the proofs of Lemmas 1 and 2, we get

$$\begin{aligned} \frac{1}{n}\sum _{j=1}^nX_{j-1}^2 g_0^{\prime 2}((j-t_0)/n)=\frac{\sigma ^2}{1-\beta _0^2} \int _0^{1} g_0^{\prime 2}(x-\tau _0) dx +o_P(1) \end{aligned}$$

and hence

$$\begin{aligned} L_{41}(t_*,t_0)=\frac{\sigma ^2}{1-\beta _0^2} \int _0^{1} g_0^{\prime 2}(x-\tau _0) dx +o_P(1). \end{aligned}$$

The result for \( L_{42}(t_*,t_0)\) is obtained in the same way and thus the assertion on \(L_{4}(t_*,t_0)\) follows.

The proof for \( L_{3}(t_*,t_0)\) follows the same lines and can therefore be omitted.

To prove (4.32) we proceed as in the proof of (4.24). To avoid too many technicalities we study only

$$\begin{aligned} \big ( L_5(t_*,t_0)-L_5(t_{**},t_0)\big )^2 \end{aligned}$$

where

$$\begin{aligned} L_5(t_*,t_0)=\frac{1}{n}\sum _{t=1}^nX_{t-1}^2 \frac{(g(t,t_0)-g(t,t_*))^2}{((t_*-t_0)/n)^2}. \end{aligned}$$

Now

$$\begin{aligned} L_5(t_*,t_0)-L_5(t_{**},t_0)&=\frac{1}{n}\sum _{t=1}^n( X_{t-1}^2 -EX_{t-1}^2) \Bigg [\frac{(g(t,t_0)-g(t,t_*))^2}{((t_*-t_0)/n)^2}- \frac{(g(t,t_0)-g(t,t_{**}))^2}{((t_{**}-t_0)/n)^2}\Bigg ] \\&+ \frac{1}{n}\sum _{t=1}^nE X_{t-1}^2 \Bigg [\frac{(g(t,t_0)-g(t,t_*))^2}{((t_*-t_0)/n)^2}- \frac{(g(t,t_0)-g(t,t_{**}))^2}{((t_{**}-t_0)/n)^2}\Bigg ]. \end{aligned}$$

As in the proof of Lemma 4 and utilizing Assumption (A.4),

$$\begin{aligned}&E\Bigg [\frac{1}{n}\sum _{t=1}^n\big ( X_{t-1}^2 -EX_{t-1}^2\big )\Bigg (\frac{(g(t,t_0)-g(t,t_*))^2}{((t_*-t_0)/n)^2} -\frac{(g(t,t_0)-g(t,t_{**}))^2}{((t_{**}-t_0)/n)^2}\Bigg )\Bigg ]^2 \\&\le C \frac{1}{n^2}\sum _{t=1}^n\Bigg [\frac{(g(t,t_0)-g(t,t_*))^2}{((t_*-t_0)/n)^2} -\frac{(g(t,t_0)-g(t,t_{**}))^2}{((t_{**}-t_0)/n)^2}\Bigg ]^2 \\&\le { C} \frac{1}{n^2}\sum _{t=1}^n\Big |\frac{g(t,t_0)-g(t,t_*)}{(t_*-t_0)/n} -\frac{g(t,t_0)-g(t,t_{**})}{(t_{**}-t_0)/n}\Big |^2 \le C\,\frac{1}{n}\,\big |(t_*-t_{**})/n\big |^{2\Delta }, \end{aligned}$$

where we also used arguments from the proof of Lemma 7. Finally,

$$\begin{aligned}&\frac{1}{n}\Bigg |\sum _{t=1}^nE X^2_{t-1} \Bigg [\frac{(g(t,t_0)-g(t,t_*))^2}{((t_*-t_0)/n)^2} -\frac{(g(t,t_0)-g(t,t_{**}))^2}{((t_{**}-t_0)/n)^2}\Bigg ]\Bigg | \\&\le C \frac{1}{n}\sum _{t=1}^n\Big |\frac{g(t,t_0)-g(t,t_*)}{(t_*-t_0)/n} -\frac{g(t,t_0)-g(t,t_{**})}{(t_{**}-t_0)/n}\Big | \le C \,\big |(t_*-t_{**})/n\big |^{\Delta }. \end{aligned}$$

On combining all these estimates, we can conclude that (4.32) holds true.\(\square \)

By Theorem 1 and since \( Q(t_0)\) (defined below) does not depend on \(t_*\), the estimator \({\widehat{t}}_0\) has the same limit distribution as

$$\begin{aligned} {\widehat{t}}_0(b_n)= \mathop {\mathrm{arg\,max}}\limits _{ |t_*-t_0)|\le b_n} \big (Q(t_*)- Q(t_0)\big ), \end{aligned}$$

for some \( b_n\rightarrow \infty ,~b_n/n\rightarrow 0\), and

$$\begin{aligned} Q(t_*)&=\frac{1}{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_{n}^2(t,t_*) X_{t-1}^2} \Bigg (\beta _1 \frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_{n}(t,t_0) {\widetilde{g}}_n(t,t_*) X_{t-1}^2 +\frac{1}{n} \sum _{t=1}^ne_t X_{t-1} {\widetilde{g}}_n(t,t_*)\Bigg )^2, \\&= \beta _1^2 Q_1(t_*) +2\beta _1 Q_2(t_*) +Q_3(t_*), \\ Q_1(t_*)&=\frac{\Big (\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_{n}(t,t_0) {\widetilde{g}}_n(t,t_*) X_{t-1}^2\Big )^2}{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_n^2(t,t_*) X_{t-1}^2}, \\ Q_2(t_*)&= \frac{\Big (\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_{n}(t,t_0) {\widetilde{g}}_n(t,t_*) X_{t-1}^2\Big )\Big ( \frac{1}{n}\sum _{t=1}^ne_j X_{t-1} {\widetilde{g}}_n(t,t_*)\Big )}{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_n^2(t,t_*) X_{t-1}^2}, \\ Q_3(t_*)&=\frac{\Big ( \frac{1}{n}\sum _{t=1}^ne_j X_{t-1} \widetilde{g}_n(t,t_*)\Big )^2}{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_n^2(t,t_*) X_{t-1}^2}. \end{aligned}$$

We need to study the properties of \(Q_j(t_*)- Q_j(t_0),~j=1,2,3\), separately. This is formulated in the next three Propositions.

Proposition 1

Under the Assumptions (A.1)– (A.4) we get

$$\begin{aligned} \max _{1\le |t_*-t_0|\le b_n}\Big |\frac{Q_1(t_*)- Q_1(t_0)}{(\frac{t_* -t_0}{n})^2}+\frac{\sigma ^2}{1-\beta _0^2} {\widetilde{H}}\Big |= o_p( 1). \end{aligned}$$

Proof

Direct but long calculations give

$$\begin{aligned}&\Big ( Q_1(t_*)-Q_1(t_0)\Big )\Bigg (\frac{1}{n}\sum _{t=1}^nX_{t-1}^2{\widetilde{g}}^2_n(t,t_*)\Bigg )=\Bigg (\frac{1}{n} \sum _{t=1}^nX_{t-1}^2{\widetilde{g}}_{n}(t,t_0)\big ({\widetilde{g}}_{n}(t,t_0) -{\widetilde{g}}_n(t,t_*)\big )\Bigg )^2\\&- \Bigg (\frac{1}{n}\sum _{t=1}^nX_{t-1}^2 \big ({\widetilde{g}}_{n}(t,t_0) -{\widetilde{g}}_n(t,t_*)\big )^2\Bigg ) \Bigg (\frac{1}{n}\sum _{t=1}^nX_{t-1}^2{\widetilde{g}}_{n}^2(t,t_0)\Bigg ). \end{aligned}$$

Using the assertions in Lemma 1, 2, 4 and the calculations in Lemma 8, we get after some steps, uniformly in \(|t_*-t_0)|\le b_n\),

$$\begin{aligned}&Q_1(t_*)- Q_1(t_0) \\&=\big ((t_* -t_0)/n\big )^2\Bigg [ \frac{\big (\frac{1}{n}\sum _{t=1}^nX_{t-1}^2 {\widetilde{g}}_0((t-t_0)/n){\widetilde{g}}'_0((t-t_0)/n)\big )^2}{ \frac{1}{n}\sum _{t=1}^nX_{t-1}^2 {\widetilde{g}}_0^2((t-t_0)/n)} -\frac{1}{n}\sum _{t=1}^nX_{t-1}^2 {\widetilde{g}}_0^{\prime 2}((t-t_0)/n)\Bigg ] \\&\times \big (1+O_P(|\beta _1|+ (|t_*-t_0|/n)^{\Delta }\big ) \\&=\frac{\sigma ^2}{1-\beta _0^2}\Big (\frac{t_0-t_*}{n}\Big )^2 \,\Bigg [\frac{\big (\int _0^{1}{\widetilde{g}}_0(x-\tau _0) {\widetilde{g}}_0'(x-\tau _0) dx\big )^2}{ \int _0^{1} {\widetilde{g}}_0^2(x-\tau _0) dx} - \int _0^{1} {\widetilde{g}}_0^{\prime 2}(x-\tau _0) dx\Bigg ] \\&\times \big (1+O_P(|\beta _1|+ (|t_*-t_0|/n)^{\Delta }\big ), \end{aligned}$$

which suffices for the proof.\(\square \)

Proposition 2

Under Assumptions (A.1)–(A.4) we get

$$\begin{aligned}&\max _{ r_n \beta _1^{-1}\sqrt{n}<|t_*-t_0|\le b_n} \Big |\frac{Q_2(t_*)-Q_2(t_0)}{\beta _1(\frac{t_*-t_0}{n})^2}\Big | = o_p\big (n^{-1/2}+r_n^{-1}\big )\qquad \text {and} \\&\max _{1\le |t_*-t_0|\le r_n \beta _1^{-1}\sqrt{n}} \Big |\frac{Q_2(t_*)-Q_2(t_0)}{\frac{t_*-t_0}{n}}-\frac{1}{\sqrt{n}}\frac{1}{\sqrt{n}} \sum _{t=1}^ne_t X_{t-1} {\widetilde{h}}_n(t,t_*,t_0)\Big | \\&= O_P\Big (\max _{1\le |t_*-t_0|\le r_n \beta _1^{-1}\sqrt{n}}\frac{1}{\sqrt{n}}\frac{|t_*-t_0|}{n} \Big ) =O_P\Big (\frac{1}{\sqrt{n}}\frac{r_n}{|\beta _1|\sqrt{n}}\Big ), \end{aligned}$$

where \(r_n\) satisfies (4.23) and

$$\begin{aligned} {\widetilde{h}}_n(t, t_*,t_0)=\frac{1}{\big (\frac{t_*-t_0}{n}\big )} \Bigg ( \big ({\widetilde{g}}_n(t,t_*)-{\widetilde{g}}_{n}(t, t_0)\big )-{\widetilde{g}}_n(t,t_0) \frac{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_n(t,t_*) \big ({\widetilde{g}}(t,t_*) -{\widetilde{g}}_{n}(t, t_0)\big ) X_{t-1}^2}{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_n^2(t,t_*) X_{t-1}^2}\Bigg ).\nonumber \\ \end{aligned}$$
(4.33)

Moreover, the limit distribution of \(\frac{1}{\sqrt{n}}\sum _{t=1}^ne_t X_{t-1} {\widetilde{h}}_n(t,t_*,t_0)\) is normal   \(\mathcal N\big (0,\frac{\sigma ^4}{1-\beta _0^2} {\widetilde{H}}\big )\) for each \(|t_*-t_0|\le r_n|\beta _1|^{-1} \sqrt{n}\), which follows from Lemma 7.

Proof

Direct calculations give

$$\begin{aligned}&Q_2(t_*)- Q_2(t_0)= \frac{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_{n}(t,t_0) {\widetilde{g}}_n(t,t_*) X_{t-1}^2}{\frac{1}{n}\sum _{t=1}^n{\widetilde{g}}_n^2(t,t_*) X_{t-1}^2} \, \frac{1}{n}\sum _{t=1}^ne_t X_{t-1}{\widetilde{g}}_n(t,t_*) - \frac{1}{n}\sum _{t=1}^ne_j X_{t-1} {\widetilde{g}}_{n}(t,t_0)\\&= \frac{1}{n}\sum _{t=1}^ne_t X_{t-1} \Big [ \big ({\widetilde{g}}_n(t,t_*)-{\widetilde{g}}_{n}(t, t_0)\big ) -{\widetilde{g}}_n(t,t_0) \frac{\frac{1}{n} \sum _{j=1}^n{\widetilde{g}}_n(j,t_*)\big ({\widetilde{g}}_n(j,t_*)-{\widetilde{g}}_{n}(j, t_0)\big )X_{j-1}^2}{\frac{1}{n} \sum _{j=1}^n{\widetilde{g}}_n^2(j,t_*) X_{j-1}^2}\Big ]\\&\quad + \frac{1}{n}\sum _{t=1}^ne_t X_{t-1} \big ({\widetilde{g}}_n(t,t_*)-{\widetilde{g}}_n(t,t_0)\big ) \,\frac{\frac{1}{n} \sum _{j=1}^n{\widetilde{g}}_n(j,t_*) \big (-{\widetilde{g}}_n(j,t_*)+{\widetilde{g}}_{n}(j, t_0)\big ) X_{j-1}^2}{\frac{1}{n} \sum _{j=1}^n{\widetilde{g}}_n^2(j,t_*) X_{j-1}^2}. \end{aligned}$$

On applying Lemmas 13 and 7 to the above sums, we get that the first term on the r.h.s. is influential, while the latter one is negligible. So, both assertions follow from here. \(\square \)

Proposition 3

Under Assumptions (A.1)–(A.4) we get

$$\begin{aligned}&\max _{\sqrt{n}|\beta _1|^{-1} r_n<|t_*-t_0|\le b_n} \frac{|Q_3(t_*)-Q_3(t_0)|}{\beta _1^2 \big (\frac{t_* -t_0}{n}\big )^2} = O_P\Big (\frac{1}{\sqrt{n} \,|\beta _1| r_n}\Big )\qquad \text {and}\\&\max _{1\le |t_*-t_0|\le \sqrt{n}|\beta _1|^{-1} r_n} \frac{|Q_3(t_*)-Q_3(t_0)|}{|\beta _1| |\frac{t_* -t_0}{n}|} = O_P\Big ( \frac{1}{n\,|\beta _1|}\Big ), \end{aligned}$$

where \( r_n\) satisfies (4.23).

Proof

Direct calculations give

$$\begin{aligned}&Q_3(t_*)-Q_3(t_0) =\frac{\big ( \frac{1}{n}\sum _{t=1}^ne_t X_{t-1}{\widetilde{g}}_n(t,t_*)\big )^2}{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_n^2(t,t_*) X_{t-1}^2} -\frac{\big ( \frac{1}{n}\sum _{t=1}^ne_t X_{t-1} {\widetilde{g}}_{n}(t,t_0)\big )^2}{\frac{1}{n}\sum _{t=1}^nX^2_{t-1} {\widetilde{g}}_{n}^2(t,t_0)}\\&=\frac{\big [ \frac{1}{n}\sum _{t=1}^ne_t X_{t-1} (\widetilde{g}_n(t,t_*)-{\widetilde{g}}_{n}(t,t_0))\big ] \big [ \frac{1}{n}\sum _{t=1}^ne_t X_{t-1}({\widetilde{g}}_n(t,t_*)+{\widetilde{g}}_{n}(t,t_0))\big ]}{\frac{1}{n} \sum _{t=1}^n{\widetilde{g}}_n^2(t,t_*) X_{t-1}^2}\\&\quad + \Big ( \frac{1}{n}\sum _{t=1}^ne_t X_{t-1} {\widetilde{g}}_{n}(t,t_0)\Big )^2 \frac{\frac{1}{n}\sum _{t=1}^nX^2_{t-1} \big ( {\widetilde{g}}_{n}^2(t,t_0)- {\widetilde{g}}_n^2(t,t_*)\big )}{\big ( \frac{1}{n}\sum _{t=1}^nX^2_{t-1}{\widetilde{g}}_{n}^2(t,t_0)\big ) \big ( \frac{1}{n}\sum _{t=1}^nX^2_{t-1} {\widetilde{g}}_n^2(t,t_*)\big )}\\&= O_P\Big (\frac{1}{n}\frac{|t_*-t_0|}{n}\Big ), \end{aligned}$$

uniformly for \(|t_*-t_0|\le b_n\), where the last relation is implied by a combination of Lemmas 7 and 8. Then both assertions follow immediately.\(\square \)

5 Proofs of the main theorems

Now we are ready to turn to the proofs of Theorems 13. We first prove the consistency of our least squares estimator \(\widehat{t}_0\).

Proof of Theorem 1

In view of (1.13), we consider, for \(t_*=0,1,\ldots ,\lfloor n(1-\delta )\rfloor \),

$$\begin{aligned} h_n(t_*)=\frac{\Big [\beta _1\Big (\frac{1}{n}\sum _{t=1}^ng(t,t_0)g(t,t_*)X_{t-1}^2- \frac{\frac{1}{n}\sum _{j=1}^ng(j,t_0)X_{j-1}^2\frac{1}{n}\sum _{j=1}^ng(j,t_*)X_{j-1}^2}{\frac{1}{n}\sum _{j=1}^nX_{j-1}^2}\Big ) +R_n(t_*)\Big ]^2}{\frac{1}{n}\sum _{t=1}^ng^2(t,t_*)X_{t-1}^2-\frac{\big (\frac{1}{n}\sum _{j=1}^ng(j,t_*)X_{j-1}^2\big )^2}{\frac{1}{n}\sum _{j=1}^nX_{j-1}^2}},\nonumber \\ \end{aligned}$$
(5.1)

where

$$\begin{aligned} R_n(t_*)=\frac{1}{n}\sum _{t=1}^ne_t X_{t-1}g(t,t_*)- \frac{\frac{1}{n}\sum _{j=1}^ne_j X_{j-1} \frac{1}{n}\sum _{j=1}^ng(j,t_*)X_{j-1}^2}{\frac{1}{n}\sum _{j=1}^nX_{j-1}^2}. \end{aligned}$$
(5.2)

In view of Lemma 5 and the assumptions on \(\beta _1\), there exists a sequence \(\{\varepsilon _n\}\) of positive reals such that \(\varepsilon _n\rightarrow 0\), but \(|\beta _1|\,\varepsilon _n\sqrt{n}\rightarrow \infty \) as \(n\rightarrow \infty \). Then, on combining Lemmas 14, as \(n\rightarrow \infty \),

$$\begin{aligned} \max _{t_*}\big |R_n(t_*)\big |=o_P\Big (\frac{1}{\varepsilon _n\sqrt{n}}\Big )+O_P\Big (\frac{1}{\sqrt{n}}\Big ) =o_P\big (|\beta _1|\big ), \end{aligned}$$
(5.3)

and, with \(t_*=\lfloor n\tau _*\rfloor \), for every fixed \(\tau _*\in [0,1-\delta ]\),

$$\begin{aligned} f_n(\tau _*):= & {} \frac{1}{\beta _1^2}\,h_n(t_*)\,{\mathop {\rightarrow }\limits ^{P}}\,\frac{\sigma ^2}{1-\beta _0^2}\frac{\big [\int _0^1 g_0(x-\tau _0)g_0(x-\tau _*)dx- \int _0^1 g_0(x-\tau _0)dx \int _0^1 g_0(x-\tau _*)dx\big ]^2}{\int _0^1 g_0^2(x-\tau _*)dx-\big (\int _0^1 g_0(x-\tau _*)dx\big )^2}\nonumber \\&=:f(\tau _*), \end{aligned}$$
(5.4)

as a consequence of (5.1)–(5.3) in combination with the approximations obtained in Lemmas 14.

Note that the denominator in f is bounded away from 0 on \([0,1-\delta ]\), with \(0<\delta <1\), since, via Jensen’s inequality, first for \(\tau _* \in [\tilde{\delta },1-\delta ]\), with \(0<\tilde{\delta }<1-\delta \),

$$\begin{aligned}&\int _{\tau _*}^1 g_0^2(x-\tau _*)dx-(1-\tau _*)^2\Big (\int _{\tau _*}^1 g_0(x-\tau _*)\frac{dx}{1-\tau _*}\Big )^2 \nonumber \\&\quad \ge \int _{\tau _*}^1 g_0^2(x-\tau _*)dx-(1-\tau _*)^2 \int _{\tau _*}^1 g_0^2(x-\tau _*)\frac{dx}{1-\tau _*} = \tau _* \int _{\tau _*}^1 g_0^2(x-\tau _*)dx \nonumber \\&\quad \ge ~\tilde{\delta }\int _{\tau _*}^{1} g_0^2(x-\tau _*)dx = \tilde{\delta }\int _{0}^{1-\tau _*} g_0^2(x)dx \ge \tilde{\delta }\int _{0}^{\delta } g_0^2(x)dx >0. \end{aligned}$$
(5.5)

Secondly, since \(\int _0^1 g_0^2(x)dx-\big (\int _0^1 g_0(x)dx\big )^2 >0\) and the denominator in f is continuous in \(\tau _*\), also

$$\begin{aligned} \int _0^1 g_0^2(x-\tau _*)dx-\Big (\int _0^1 g_0(x-\tau _*)dx\Big )^2 >0, \end{aligned}$$
(5.6)

for \(\tau _* \in [0,\tilde{\delta })\), with some \(\tilde{\delta }>0\).

So, (5.5) and (5.6) prove the positivity on \([0,1-\delta ]\) and, in view of Lemmas 14, the latter positivity also implies that the convergence in (5.4) is uniform on \([0,1-\delta ]\), i.e.

$$\begin{aligned} \max _{\tau _*\in [0,1-\delta ]}\big |f_n(\tau _*)-f(\tau _*)\big |\,{\mathop {\rightarrow }\limits ^{P}}\,0\quad (n\rightarrow \infty ). \end{aligned}$$
(5.7)

We finally show that the limit function f has a unique maximum at \(\tau _*=\tau _0\). First, via the Cauchy-Schwarz inequality,

$$\begin{aligned}&\Bigg [\int _0^1 g_0(x-\tau _0)g_0(x-\tau _*)dx- \int _0^1 g_0(x-\tau _0)dx \int _0^1 g_0(x-\tau _*)dx\Bigg ]^2 \nonumber \\&\le ~\Bigg [\int _0^1 g_0^2(x-\tau _0)dx-\Bigg (\int _0^1 g_0(x-\tau _0)dx\Bigg )^2\Bigg ] \Bigg [\int _0^1 g_0^2(x-\tau _*)dx-\Bigg (\int _0^1 g_0(x-\tau _*)dx\Bigg )^2\Bigg ]. \end{aligned}$$
(5.8)

Hence

$$\begin{aligned} f(\tau _*)\le \frac{\sigma ^2}{1-\beta _0^2}\Bigg [\int _0^1 g_0^2(x-\tau _0)dx-\Bigg (\int _0^1 g_0^2(x-\tau _0)dx\Bigg )^2\Bigg ], \end{aligned}$$

and the bound is attained for \(\tau _* =\tau _0\).

It remains to prove that \(\tau _0\) is the unique maximizer of f. To do so, we show that strict inequality holds in (5.8), if \(\tau _*\ne \tau _0\). Assume equality. Then, there is a \(\lambda \ne 0\) such that, for almost every \(x\in [0,1]\),

$$\begin{aligned} \widetilde{g}_0(x-\tau _*):= & {} g_0(x-\tau _*)-\int _0^1 g_0(x-\tau _*)dx =\lambda \Bigg (g_0(x-\tau _0)-\int _0^1 g_0(x-\tau _0)dx\Bigg )\\=: & {} \lambda \widetilde{g}_0(x-\tau _0). \end{aligned}$$

This, however, is impossible, since, e.g., for \(\tau _* >\tau _0\),

$$\begin{aligned} \widetilde{g}_0(\cdot -\tau _*) ~~\text {is constant on} ~~[\tau _0,\tau _*], ~\text {but}\,\, \lambda \widetilde{g}_0(\cdot -\tau _0)~~\text {is not}. \end{aligned}$$

Similarly, for \(\tau _*<\tau _0\),

$$\begin{aligned} \lambda \widetilde{g}_0(\cdot -\tau _0) ~~\text {is constant on} ~~[\tau _*,\tau _0], ~\text {but} ~~\widetilde{g}_0(\cdot -\tau _*)~~\text {is not}, \end{aligned}$$

which proves that \(\tau _0\) is the unique maximizer of f.

Via the subsequence principle for convergence in probability, the proof of Theorem 1 can now be completed from (5.7) by applying Lemma 6. \(\square \)

Proof of Remark 5

In case of an unknown change function g, it is obvious from the proof of Theorem 1 that, under (2.2), \(g(t,t_*)=g_0((t-t_*)/n)\) in (1.11) resp. (1.13) can be replaced by \(\widehat{g}_n(t,t_*)=\widehat{g}_0((t-t_*)/n)\). The reason is that, in view of (5.1)–(5.3) and the rate \(o_P(\beta _1)\), the convergence in (5.4) still holds with estimated \(g(t,t_*)\)’s, so that the proof can be completed as before.

If \(g_0(x)=x_+^\kappa \), with some \(\kappa \ge 1\), and \(|\widehat{\kappa }_n-\kappa |=o_P(\beta _1)\) as \(n\rightarrow \infty \) for an estimator \(\widehat{\kappa }_n\), note that on \(\{\widehat{\kappa }_n>\kappa _1\}\), with \(0<\kappa _1<\kappa \), by the mean-value theorem, for some \({\bar{\kappa }}_n\) between \(\widehat{\kappa }_n\) and \(\kappa \),

$$\begin{aligned} \max _{x\in (0,1]}|x^{\widehat{\kappa }_n}-x^{\kappa }| = \max _{x\in (0,1]}|\log x||x^{{\bar{\kappa }}_n}||\widehat{\kappa }_n-\kappa | \le \max _{x\in (0,1]}|\log x||x^{\kappa _1}||\widehat{\kappa }_n-\kappa |. \end{aligned}$$

In view of \(|\log x||x^{\kappa _1}|\rightarrow 0\) as \(x\downarrow 0\), \(|\log x||x^{\kappa _1}|\) is bounded on (0, 1], which suffices to prove that

$$\begin{aligned} \max _{x\in (0,1]}|x^{\widehat{\kappa }_n}-x^{\kappa }|/\beta _1 = \max _{x\in [0,1]}|x^{\widehat{\kappa }_n}-x^{\kappa }|/\beta _1\,{\mathop {\rightarrow }\limits ^{P}}\,0, \end{aligned}$$

since the \(\max \) is attained for \(x>0\) and \(P(\widehat{\kappa }_n>\kappa _1)\rightarrow 1\) as \(n\rightarrow \infty \). \(\square \)

Proof of Theorem 2

In view of the consistency obtained in Theorem 1, it suffices to concentrate on a small neighbourhood \([\tau _1,\tau _2]\) of \(\tau _0\). With the notations in (5.4), we have

$$\begin{aligned}&f_n(\widehat{\tau }_0)-f(\tau _0)=\max _{\tau _*\in [\tau _1,\tau _2]} f_n(\tau _*) -\max _{\tau _*\in [\tau _1,\tau _2]} f(\tau _*) =\big ( f_n(\widehat{\tau }_0)-f(\widehat{\tau }_0)\big )+\big ( f(\widehat{\tau }_0)-f(\tau _0)\big ) \\&=\big ( f_n(\widehat{\tau }_0)-f(\widehat{\tau }_0)\big ) +\Big (f'(\tau _0)(\widehat{\tau }_0-\tau _0)+f''(\tau _n)\frac{(\widehat{\tau }_0-\tau _0)^2}{2}\Big ), \end{aligned}$$

where \(\tau _n\) is between \(\widehat{\tau }_0\) and \(\tau _0\).

Since \(f'(\tau _0)=0\) and \(|f''(\tau _n)|\ge C\), for some \(C>0\), this results in the estimate

$$\begin{aligned} |\widehat{\tau }_0 -\tau _0|^2&\le \frac{2}{C} \Big (\big |\max _{\tau _*\in [\tau _1,\tau _2]} f_n(\tau _*) -\max _{\tau _*\in [\tau _1,\tau _2]} f(\tau _*)\big | + \max _{\tau _*\in [\tau _1,\tau _2]} \big |f_n(\tau _*)-f(\tau _*)\big |\Big )\\&\le \frac{4}{C} \, \max _{\tau _*\in [\tau _1,\tau _2]} \big |f_n(\tau _*)-f(\tau _*)\big |. \end{aligned}$$

Now, on checking the steps in the proof of Theorem 1 more carefully, one can show that

$$\begin{aligned} \max _{\tau _*\in [\tau _1,\tau _2]} \big |f_n(\tau _*)-f(\tau _*)\big | = O_P\big (|\beta _1|\big )+o_P\Big (\frac{1}{|\beta _1|\,\varepsilon _n\sqrt{n}}\Big ). \end{aligned}$$
(5.9)

Note that, in view of Lemmas 24, the uniform rate of approximation for the denominator of \(f_n\) in (5.4) by that of f is \(O_P(|\beta _1|)+o_P(1/(\varepsilon _n\sqrt{n}))\), whereas the one for the numerator is \(O_P(|\beta _1|)+o_P(1/(|\beta _1|\,\varepsilon _n\sqrt{n}))\). This, together with the positivity of the denominator of f on \([\tau _1,\tau _2]\), results in (5.9) after some elementary calculation, which completes the proof. \(\square \)

Proof of Remark 7

Under (2.4), the approximation rate \(O_P\big (|\beta _1|\big )+o_P\big (1/(|\beta _1|\,\varepsilon _n\sqrt{n})\big )\) in (5.9) still holds, since only an additional rate \(O_P\big (1/\sqrt{n}\big )\) would have to be added in the denominator of \(f_n\) and \(O_P\big (1/(|\beta _1|\sqrt{n})\big )\) in the numerator, which are negligible compared to the other terms. \(\square \)

Proof of Theorem 3

The proof follows the usual lines of proofs of the limit behavior of estimators under a gradual change (see, e.g., Hušková (1998b) or Jarušková (1998a)). Therefore we will focus on the main steps only. \(\square \)

Gathering the assertions in Propositions 13 and Theorem 1, we can conclude that \({\widehat{t}}_0\) has the same asymptotic distribution as

$$\begin{aligned}&{\widehat{t}}_0(r_n\sqrt{n} |\beta _1|^{-1})=\mathop {\mathrm{arg\,max}}\limits _{|t_*-t_0|\le \sqrt{n}|\beta _1|^{-1} r_n}\\&\quad \Bigg \{-\beta _1^2\Big (\frac{|t_0-t_*|}{n}\Big )^2 \frac{\sigma ^2}{1-\beta _0^2}\,{\widetilde{H}} +\beta _1\Big (\frac{t_0-t_*}{n}\Big )\frac{2}{\sqrt{n}}\frac{1}{\sqrt{n}} \sum _{t=1}^ne_t X_{t-1} {\widetilde{h}}_n(t,t_*,t_0)\Bigg \}. \end{aligned}$$

By Lemma 7, particularly by (4.24), its limit behavior does not change if \({\widetilde{h}}_n(t,t_*,t_0)\) is replaced by \({\widetilde{h}}_n(t,t_{**},t_0)\), with any fixed \(t_{**}\) such that \(|t_{**}-t_0|\le r_n\sqrt{n} \,|\beta _1|^{-1}\). Also, by Lemma 7,

$$\begin{aligned} Z_n= \frac{1}{\sqrt{n}} \sum _{t=1}^ne_t X_{t-1} {\widetilde{h}}_n(t,t_{**},t_0) \end{aligned}$$

has an asymptotic normal distribution with zero mean and variance

$$\begin{aligned} E\Big ( \frac{1}{\sqrt{n}} \sum _{t=1}^ne_t X_{t-1} {\widetilde{h}}_n(t,t_{**},t_0)\Big )^2= \frac{\sigma ^4}{1-\beta _0^2}{\widetilde{H}} \big (1+o(1)\big ). \end{aligned}$$

Thus it suffices to study the estimator

$$\begin{aligned} {\widetilde{t}}_0\,\big (r_n\sqrt{n} \,|\beta _1|^{-1}\big ) =\mathop {\mathrm{arg\,max}}\limits _{|t_*-t_0|\le \sqrt{n}|\beta _1|^{-1} r_n} \Big \{-\beta _1^2\Big (\frac{|t_0-t_*|}{n}\Big )^2 \frac{\sigma ^2}{1-\beta _0^2}\,{\widetilde{H}} +\beta _1\,\sqrt{ {\widetilde{H}}}\,\,\frac{t_0-t_*}{n}\frac{2}{\sqrt{n}}\, \frac{Z_n}{\sqrt{ {\widetilde{H}}}}\Big \}. \end{aligned}$$

The boundary can only be a attained with probability tending to 0, since \(Z_n=O_P(1)\), and, on plugging the upper bound into the above expression, one gets

$$\begin{aligned} -\beta _1^2 \Big (\frac{\sqrt{n}\,|\beta _1|^{-1} r_n}{n}\Big )^2 \frac{\sigma ^2}{1-\beta _0^2} {\widetilde{H}} +\beta _1 \,\frac{\sqrt{n}\,|\beta _1|^{-1} r_n}{n}\frac{ 2}{\sqrt{n}}\,Z_n = -\frac{ r_n^2}{n} \frac{\sigma ^2}{1-\beta _0^2} {\widetilde{H}} +\frac{r_n}{n} \beta _1 |\beta _1|^{-1} \,2 Z_n. \end{aligned}$$

Using the assumption on \(r_n\) (see (4.23)) together with \(Z_n=O_P(1)\), it is now straightforward to check that this estimator satisfies

$$\begin{aligned} \sqrt{n} \,\beta _1\,\frac{\sigma ^2}{1-\beta _0^2}\,{\widetilde{H}}\, \frac{{\widetilde{t}}_0\,\big (r_n\sqrt{n} \,|\beta _1|^{-1}\big )-t_0}{n}= Z_n \,\big (1+o_P(1)\big ). \end{aligned}$$

From here and the asymptotic normality of \(Z_n\) it can be concluded that the assertion of Theorem 3 holds true. \(\square \)