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Спутники
РН
SpaceX
Роскосмос
NASA
СпутникиРНSpaceXРоскосмосNASAIn this paper, we propose a computationally efficient and theoretically justified group least absolute shrinkage and selection operator (Group LASSO; GLASSO) method for estimating multiple change-points in a piecewise stationary generalized integer-valued autoregressive process. The proposed method is particularly suitable for finite samples with many closely spaced change-points. We further develop an efficient implementation that combines least angle regression and optimal partitioning (OP). The overall computational complexity is $$O(Kn+K^2)$$ when OP is used and $$O(Kn+K^3)$$ when the backward elimination algorithm is used. In addition, we propose an iterative procedure for selecting a data-driven order $$\tilde{p}$$, which achieves satisfactory performance with relatively low computational cost. Simulation studies and a real data analysis demonstrate that the proposed method and iterative procedure perform well in practice and support the theoretical results.
Next, we give some lemmas used in the proof of Theorem 1. Proofs of all lemmas are provided at the end of this Appendix. In the following proofs, the constant C denotes a generic constant that may be different from line to line.
Lemma 1Under Assumptions H.1 and H.2, for any \(c_0>0\), there exists some constant \(C>0\) such that
$$\begin{aligned} \textrm{P}\left( \max _{1\le s\le n}\max _{1\le k\le \tilde{p}}\left| \sum _{i=s}^nX_i^k\eta _i\right| \ge c_0\sqrt{n\log n}\right) \le C(c_0^2\log n)^{-1-\varepsilon _X/2}, \end{aligned}$$
(6.1)
where \(\eta _i=\sigma _i\epsilon _i\), \(X_i^0=1\) and \(X_i^k=X_{i-k}\) for \(k=1,\cdots ,\tilde{p}\).
Lemma 2Under Assumptions H.1-3, for any positive constants \(a_n, x, C_1, C_2\), we have for any \(1\le j\le m_0+1\),
$$\begin{aligned}&\textrm{P}\left\{ \max \limits _{\begin{array}{c} ~~~|s-t|\ge a_n, \\ \tau _{j-1}^0+1\le s<t\le \tau _{j}^0 \end{array}}\left( \max _{0\le k\le \tilde{p}}(t-s)^{-1}\left| \sum _{i=s}^t X_i^k\eta _i\right| \ge 3x\right) \right\} \\ \le&4(\tilde{p}+1)n_j^2\left\{ \exp \left[ -\frac{x^2(t-s)}{4(b+\frac{C_1x a_n}{9\log n})}\right] +2n\beta ^{[C_2 \log n]}\right\} +c\left( 48\frac{C_2}{C_1}\right) ^{1+\frac{\varepsilon _X}{2}}\\&\quad (\tilde{p}+1)\frac{n_j(\log ^2n)^{1+\frac{\varepsilon _X}{2}}}{xa_n^{2+\frac{\varepsilon _X}{2}}} \end{aligned}$$
where \(0<\beta <1\), \(b=2(\max _{j}\textrm{E}X_{1,j}^2+1)\), \(n_j=\tau _{j}^0-\tau _{j-1}^0\) and \(c=\max _{1\le t\le n,0\le k\le \tilde{p}}\textrm{E}|X_t^k\eta _t|^{2+\varepsilon _X/2}\).
Lemma 3Let \(\hat{\varvec{\vartheta }}(n,\tilde{p})\) be defined by (2.5), and define \( \hat{e}_t = X_t- \sum _{i=1}^{t} \hat{\varvec{\vartheta }}_i^{\top }(\tilde{p}) {\varvec{X}}_{t-1}. \) Then, for every \(\hat{\tau }_j^{(1)}\in \widehat{\mathcal {J}}_n^{(1)}\),
$$\begin{aligned}&\sum _{t=\hat{\tau }_j^{(1)}+1}^{n} {\varvec{X}}_{t-1}\hat{e}_t - \frac{1}{2}n\lambda _n \frac{ \hat{\varvec{\vartheta }}_{\hat{\tau }_j^{(1)}+1}(\tilde{p}) }{ \left\| \hat{\varvec{\vartheta }}_{\hat{\tau }_j^{(1)}+1}(\tilde{p}) \right\| } = {\varvec{0}}, \end{aligned}$$
where \( \hat{\varvec{\vartheta }}_{\hat{\tau }_j^{(1)}+1}(\tilde{p}) \ne {\varvec{0}}. \) For every \(s=1,\ldots ,n\) such that \(\hat{\varvec{\vartheta }}_s(\tilde{p})={\varvec{0}}\),
$$\begin{aligned} \left\| \sum _{t=s}^{n} {\varvec{X}}_{t-1}\hat{e}_t \right\| \le \frac{1}{2}n\lambda _n. \end{aligned}$$
Furthermore, define the cumulative coefficient vector by \( \hat{\varvec{b}}_t(\tilde{p}) = \sum _{i=1}^{t} \hat{\varvec{\vartheta }}_i(\tilde{p}). \) Let
$$ 0=\hat{\tau }_0^{(1)}< \hat{\tau }_1^{(1)}< \cdots< \hat{\tau }_{\hat{m}^{(1)}}^{(1)} < \hat{\tau }_{\hat{m}^{(1)}+1}^{(1)}=n. $$
Then, for \(j=1,\ldots ,\hat{m}^{(1)}+1\),
$$ \hat{\varvec{b}}_t(\tilde{p}) = \hat{\varvec{b}}_{\hat{\tau }_{j-1}^{(1)}+1}(\tilde{p}) =:\hat{\varvec{\beta }}_j(\tilde{p}), \qquad \hat{\tau }_{j-1}^{(1)}+1 \le t\le \hat{\tau }_{j}^{(1)}. $$
As a direct consequence of the preceding KKT conditions, for any integers \(0\le r<s\le n\),
$$ \left\| \sum _{t=r+1}^{s} {\varvec{X}}_{t-1}\hat{e}_t \right\| \le n\lambda _n. $$
Under Assumptions H.1 - H.3, if \(m_0\) is known and \(\left| \widehat{\mathcal {J}}_n^{(1)}\right| =2m_0\), then
$$\begin{aligned} \textrm{P}\left( \max _{1\le j\le m_0}\left| \hat{\tau }_{2j-1}^{(1)}-\tau _j^0\right| \vee \left| \hat{\tau }_{2j}^{(1)}-(\tau _j^0+{p_{j+1}^0})\right| \le n\gamma _n\right) \rightarrow 1,~~\text {as}~~n\rightarrow \infty . \end{aligned}$$
We first prove \( \textrm{P}\left( \left| \widehat{\mathcal {J}}_n^{(1)}\right| \ge 2m_0 \right) \rightarrow 1. \) Suppose, on the contrary, that \(\left| \widehat{\mathcal {J}}_n^{(1)}\right| <2m_0\). Then there exist some \(i_0\in \{0,1,\ldots ,m_0-1\}\) and \(j_0\in \{0,1,\ldots ,\hat{m}^{(1)}\}\) such that \( \tau _{i_0+1}^0 - \left[ \left( \tau _{i_0}^0+p_{i_0+1}^0\right) \vee \hat{\tau }_{j_0}^{(1)} \right] \ge \frac{n\gamma _n}{3} \) and \( \left[ \tau _{i_0+2}^0 \wedge \hat{\tau }_{j_0+1}^{(1)} \right] - \left( \tau _{i_0+1}^0+p_{i_0+2}^0 \right) \ge \frac{n\gamma _n}{3}, \) where \( \hat{\tau }_0^{(1)}=0, \hat{\tau }_{\hat{m}^{(1)}+1}^{(1)}=n. \) Applying the KKT conditions in Lemma 3 to \( \left[ \left( \left( \tau _{i_0}^0+p_{i_0+1}^0\right) \vee \hat{\tau }_{j_0}^{(1)} \right) +1, \, \tau _{i_0+1}^0 \right] \) and \( \left[ \tau _{i_0+1}^0+p_{i_0+2}^0+1, \, \tau _{i_0+2}^0 \wedge \hat{\tau }_{j_0+1}^{(1)} \right] , \) and using the piecewise constancy of \(\hat{\varvec{b}}_t(\tilde{p})\), we obtain
$$\begin{aligned} & \left\| \sum _{t= \left( \left( \tau _{i_0}^0+p_{i_0+1}^0\right) \vee \hat{\tau }_{j_0}^{(1)} \right) +1 }^{\tau _{i_0+1}^0} {\varvec{X}}_{t-1} \left[ \varvec{\theta }_{i_0+1}^{0\top }(\tilde{p}) - \hat{\varvec{\beta }}_{j_0+1}^{\top }(\tilde{p}) \right] {\varvec{X}}_{t-1} \right\| \\ & \le n\lambda _n+ \left\| \sum _{t= \left( \left( \tau _{i_0}^0+p_{i_0+1}^0\right) \vee \hat{\tau }_{j_0}^{(1)} \right) +1 }^{\tau _{i_0+1}^0} {\varvec{X}}_{t-1}\eta _t \right\| \end{aligned}$$
and
$$\begin{aligned} \left\| \sum _{t=\tau _{i_0+1}^0+p_{i_0+2}^0+1}^{ \tau _{i_0+2}^0\wedge \hat{\tau }_{j_0+1}^{(1)} } {\varvec{X}}_{t-1} \left[ \varvec{\theta }_{i_0+2}^{0\top }(\tilde{p}) - \hat{\varvec{\beta }}_{j_0+1}^{\top }(\tilde{p}) \right] {\varvec{X}}_{t-1} \right\| \le n\lambda _n+ \left\| \sum _{t=\tau _{i_0+1}^0+p_{i_0+2}^0+1}^{ \tau _{i_0+2}^0\wedge \hat{\tau }_{j_0+1}^{(1)} } {\varvec{X}}_{t-1}\eta _t \right\| . \end{aligned}$$
Similar to the argument for (6.6), for any \(x>0\),
$$\begin{aligned} \left\| \textrm{E}\left( {\varvec{X}}_{\tau _{i_0+1}^0} {\varvec{X}}_{\tau _{i_0+1}^0}^{\top } \right) \left[ \varvec{\theta }_{i_0+1}^{0}(\tilde{p}) - \hat{\varvec{\beta }}_{j_0+1}(\tilde{p}) \right] \right\| \le \frac{2n\lambda _n}{ \tau _{i_0+1}^0- \left[ \left( \tau _{i_0}^0+p_{i_0+1}^0\right) \vee \hat{\tau }_{j_0}^{(1)} \right] } +2x \end{aligned}$$
and
$$\begin{aligned} \left\| \textrm{E}\left( {\varvec{X}}_{\tau _{i_0+2}^0} {\varvec{X}}_{\tau _{i_0+2}^0}^{\top } \right) \left[ \varvec{\theta }_{i_0+2}^{0}(\tilde{p}) - \hat{\varvec{\beta }}_{j_0+1}(\tilde{p}) \right] \right\| \le \frac{2n\lambda _n}{ \left[ \tau _{i_0+2}^0 \wedge \hat{\tau }_{j_0+1}^{(1)} \right] - \left( \tau _{i_0+1}^0+p_{i_0+2}^0 \right) } +2x \end{aligned}$$
with probability approaching one. Since
$$ \tau _{i_0+1}^0 - \left[ \left( \tau _{i_0}^0+p_{i_0+1}^0\right) \vee \hat{\tau }_{j_0}^{(1)} \right] \ge \frac{n\gamma _n}{3}, $$
$$ \left[ \tau _{i_0+2}^0 \wedge \hat{\tau }_{j_0+1}^{(1)} \right] - \left( \tau _{i_0+1}^0+p_{i_0+2}^0 \right) \ge \frac{n\gamma _n}{3}, $$
and \(\gamma _n/\lambda _n\rightarrow \infty \), the arbitrariness of x yields
$$ \left\| \varvec{\theta }_{i_0+1}^{0}(\tilde{p}) - \hat{\varvec{\beta }}_{j_0+1}(\tilde{p}) \right\| \xrightarrow {p}0 \quad \text {and}\quad \left\| \varvec{\theta }_{i_0+2}^{0}(\tilde{p}) - \hat{\varvec{\beta }}_{j_0+1}(\tilde{p}) \right\| \xrightarrow {p}0. $$
It follows that
$$ \left\| \varvec{\theta }_{i_0+1}^{0}(\tilde{p}) - \varvec{\theta }_{i_0+2}^{0}(\tilde{p}) \right\| \xrightarrow {p}0, $$
which contradicts Assumption H.1. Consequently,
$$ \textrm{P}\left( \left| \widehat{\mathcal {J}}_n^{(1)}\right| \ge 2m_0 \right) \rightarrow 1. $$
Next, we prove
$$ \textrm{P}\left( \max _{\tau _j^0\in \mathcal {J}_0} \min _{\hat{\tau }_j^{(1)} \in \widehat{\mathcal {J}}_n^{(1)}} \left| \hat{\tau }_j^{(1)}-\tau _j^0 \right| \le n\gamma _n \right) \rightarrow 1. $$
Based on the preceding result, it suffices to consider \(\left| \widehat{\mathcal {J}}_n^{(1)}\right| \ge 2m_0\). We first consider \(\left| \widehat{\mathcal {J}}_n^{(1)}\right| =2m_0\). According to Lemma 4,
$$\begin{aligned} \textrm{P}\left( \max _{1\le j\le m_0} \left\{ \left| \hat{\tau }_{2j-1}^{(1)}-\tau _j^0 \right| \vee \left| \hat{\tau }_{2j}^{(1)} - \left( \tau _j^0+p_{j+1}^0\right) \right| \right\} \le n\gamma _n \right) \rightarrow 1 \end{aligned}$$
as \(n\rightarrow \infty \). This implies that
$$ \textrm{P}\left( \max _{\tau _j^0\in \mathcal {J}_0} \min _{\hat{\tau }_j^{(1)} \in \widehat{\mathcal {J}}_n^{(1)}} \left| \hat{\tau }_j^{(1)}-\tau _j^0 \right| \le n\gamma _n \right) \rightarrow 1. $$
We next consider \(k=\left| \widehat{\mathcal {J}}_n^{(1)}\right| >2m_0\). There exists a subset of \(\widehat{\mathcal {J}}_n^{(1)}\), denoted by
$$ \widehat{\widetilde{\mathcal {J}}}_n^{(1)} = \left\{ \hat{\tau }_{s_1},\ldots ,\hat{\tau }_{s_{m_0}} \right\} \subseteq \widehat{\mathcal {J}}_n^{(1)}, $$
where
$$ \hat{\tau }_{s_j} \in \mathop \textrm{argmin}_{\hat{\tau }^{(1)} \in \widehat{\mathcal {J}}_n^{(1)}} \left| \hat{\tau }^{(1)}-\tau _j^0 \right| , \qquad 1\le j\le m_0. $$
Using Lemmas 2 and 3, together with an argument similar to that in the proof of Lemma 4, we obtain
$$ \max _{k>2m_0} \textrm{P}\left( \max _{1\le j\le m_0} \left| \hat{\tau }_{s_j}-\tau _j^0 \right| >n\gamma _n \right) \rightarrow 0. $$
A similar argument can be found in the proof of Proposition 4 in Harchaoui and Lévy-Leduc (2010) and Theorem 2.3 in Chan et al. (2014). The proof of Theorem 1 is completed. \(\square \)
Proof of Theorem 2Combining Theorem 2 in Sheng and Wang (2024a) and Theorem 2.5 in Chan et al. (2014), Theorem 2 can be proved. \(\square \)
Proof of Lemmas 1-3The proofs of these lemmas are similar to the proofs of Lemmas 1 - 3 in Chan et al. (2014), and we omit them. \(\square \)
Proof of Lemma 4Let \( A_{nj}^{(1)} = \left\{ \left| \hat{\tau }_{2j-1}^{(1)}-\tau _j^0 \right| >n\gamma _n \right\} \) and \( A_{nj}^{(2)} = \left\{ \left| \hat{\tau }_{2j}^{(1)} -\left( \tau _j^0+p_{j+1}^0\right) \right| \right. \left. >n\gamma _n \right\} \), \(j=1,\ldots ,m_0.\) Then,
$$\begin{aligned} & \textrm{P}\left( \max _{1\le j\le m_0} \left\{ \left| \hat{\tau }_{2j-1}^{(1)}-\tau _j^0 \right| \vee \left| \hat{\tau }_{2j}^{(1)} -\left( \tau _j^0+p_{j+1}^0\right) \right| \right\} >n\gamma _n \right) \\ & \le \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(1)}\right) + \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(2)}\right) . \end{aligned}$$
Put
$$ C_n^{(1)} = \left\{ \max _{1\le j\le m_0} \left| \hat{\tau }_{2j-1}^{(1)}-\tau _j^0 \right| \le \frac{1}{2} \min _{1\le j\le m_0+1} \left| \tau _j^0-\tau _{j-1}^0 \right| \right\} , $$
and
$$ C_n^{(2)} = \left\{ \max _{1\le j\le m_0} \left| \hat{\tau }_{2j}^{(1)} -\left( \tau _j^0+p_{j+1}^0\right) \right| \le \frac{1}{2} \min _{1\le j\le m_0+1} \left| \tau _j^0-\tau _{j-1}^0 \right| \right\} . $$
To prove Lemma 4, it suffices to show that
$$\begin{aligned}&\sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(1)}\cap C_n^{(1)}\right) \rightarrow 0, \qquad \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(1)}\cap C_n^{(1),c}\right) \rightarrow 0, \end{aligned}$$
(6.2)
$$\begin{aligned}&\sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(2)}\cap C_n^{(2)}\right) \rightarrow 0, \qquad \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(2)}\cap C_n^{(2),c}\right) \rightarrow 0, \end{aligned}$$
(6.3)
where \(C_n^{(1),c}\) and \(C_n^{(2),c}\) denote the complements of \(C_n^{(1)}\) and \(C_n^{(2)}\), respectively. The arguments for (6.2) and (6.3) are similar to those in the proof of Proposition 5 in Harchaoui and Lévy-Leduc (2010) and the proof of Theorem 2.2 in Chan et al. (2014). For readability, we give the proof of
$$ \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(1)}\cap C_n^{(1)}\right) \rightarrow 0. $$
We first consider the case where \(m_0\) is fixed. By the definition of \(C_n^{(1)}\),
$$ \tau _{j-1}^0< \hat{\tau }_{2j-1}^{(1)} < \tau _{j+1}^0, \qquad j=1,\ldots ,m_0. $$
We divide the event \(A_{nj}^{(1)}\) into the two cases
$$ \hat{\tau }_{2j-1}^{(1)}<\tau _j^0 \qquad \text {and}\qquad \hat{\tau }_{2j-1}^{(1)}>\tau _j^0. $$
Consider first the case \(\hat{\tau }_{2j-1}^{(1)}<\tau _j^0\). Applying the interval KKT bound in Lemma 3 to \([\hat{\tau }_{2j-1}^{(1)}+1,\tau _j^0]\) gives
$$\begin{aligned} \left\| \sum _{t=\hat{\tau }_{2j-1}^{(1)}+1}^{\tau _j^0} {\varvec{X}}_{t-1} \left[ X_t- \hat{\varvec{\beta }}_{2j}^{\top }(\tilde{p}) {\varvec{X}}_{t-1} \right] \right\| \le n\lambda _n. \end{aligned}$$
It follows that
$$\begin{aligned}&\Bigg \Vert \sum _{t=\hat{\tau }_{2j-1}^{(1)}+1}^{\tau _j^0} {\varvec{X}}_{t-1}\eta _t+ \sum _{t=\hat{\tau }_{2j-1}^{(1)}+1}^{\tau _j^0} {\varvec{X}}_{t-1} \left[ \varvec{\theta }_j^0(\tilde{p}) - \widetilde{\varvec{\theta }}_{j+1}^0(\tilde{p}) \right] ^{\top } {\varvec{X}}_{t-1}\\&\quad + \sum _{t=\hat{\tau }_{2j-1}^{(1)}+1}^{\tau _j^0} {\varvec{X}}_{t-1} \left[ \widetilde{\varvec{\theta }}_{j+1}^0(\tilde{p}) - \hat{\varvec{\beta }}_{2j}(\tilde{p}) \right] ^{\top } {\varvec{X}}_{t-1} \Bigg \Vert \\&=: \left\| {\varvec{\Delta }}_1+ {\varvec{\Delta }}_2+ {\varvec{\Delta }}_3 \right\| \le n\lambda _n. \end{aligned}$$
\(\square \)
Therefore,
$$\begin{aligned} \textrm{P}\left( A_{nj}^{(1)} \cap C_n^{(1)} \cap \left\{ \hat{\tau }_{2j-1}^{(1)}<\tau _j^0 \right\} \right)&\le \textrm{P}\left( \left\{ \frac{1}{3}\Vert {\varvec{\Delta }}_2\Vert \le n\lambda _n \right\} \cap A_{nj}^{(1)} \right) \\&\quad + \textrm{P}\left( \left\{ \Vert {\varvec{\Delta }}_1\Vert> \frac{1}{3}\Vert {\varvec{\Delta }}_2\Vert \right\} \cap A_{nj}^{(1)} \right) \\&\quad + \textrm{P}\left( \left\{ \Vert {\varvec{\Delta }}_3\Vert > \frac{1}{3}\Vert {\varvec{\Delta }}_2\Vert \right\} \cap A_{nj}^{(1)} \cap C_n^{(1)} \right) \\&=: \textrm{P}\left( A_{nj1}^{(1)}\right) + \textrm{P}\left( A_{nj2}^{(1)}\right) + \textrm{P}\left( A_{nj3}^{(1)}\right) . \end{aligned}$$
Analogously to the argument in the proof of Lemma 2 in Chan et al. (2014), for any \(x>0\), by taking \(C_1=x/8\) and choosing \(C_2\) such that \(n^4\beta ^{C_1\log n}\rightarrow 0\), we have
$$\begin{aligned} \textrm{P}\Bigg ( \sup _{\begin{array}{c} \tau _j^0-s\ge n\gamma _n\\ \tau _{j-1}^0+p_j^0\le s<\tau _j^0 \end{array}} \Bigg \Vert \frac{1}{\tau _j^0-s} \sum _{t=s+1}^{\tau _j^0} \Big [ {\varvec{X}}_{t-1}{\varvec{X}}_{t-1}^{\top } - \textrm{E}\big ( {\varvec{X}}_{t-1}{\varvec{X}}_{t-1}^{\top } \big ) \Big ] \Bigg \Vert >x \Bigg ) \rightarrow 0. \end{aligned}$$
(6.4)
It follows that, on the event \(\{|\hat{\tau }_{2j-1}^{(1)}-\tau _j^0|>n\gamma _n\}\),
$$\begin{aligned} \frac{1}{3}\Vert {\varvec{\Delta }}_2\Vert&\ge \frac{ |\tau _j^0-\hat{\tau }_{2j-1}^{(1)}| }{6} \Bigg \Vert \textrm{E}\left( {\varvec{X}}_{t-1}{\varvec{X}}_{t-1}^{\top } \right) \left[ \varvec{\theta }_j^0(\tilde{p}) - \widetilde{\varvec{\theta }}_{j+1}^0(\tilde{p}) \right] \Bigg \Vert \nonumber \\&\ge \frac{n\gamma _n}{6} \Bigg \Vert \textrm{E}\left( {\varvec{X}}_{t-1}{\varvec{X}}_{t-1}^{\top } \right) \left[ \varvec{\theta }_j^0(\tilde{p}) - \widetilde{\varvec{\theta }}_{j+1}^0(\tilde{p}) \right] \Bigg \Vert \nonumber \\&=:c_0n\gamma _n>0 \end{aligned}$$
(6.5)
with probability tending to one. Since \(\gamma _n/\lambda _n\rightarrow \infty \), this gives
$$ \textrm{P}\left( A_{nj1}^{(1)}\right) \rightarrow 0. $$
Similarly, by Lemma 2,
$$\begin{aligned}&\textrm{P}\left( \sup _{\begin{array}{c} \tau _j^0-s\ge n\gamma _n\\ \tau _{j-1}^0+p_j^0\le s<\tau _j^0 \end{array}} \max _{0\le k\le \tilde{p}} \left| \frac{1}{\tau _j^0-s} \sum _{i=s+1}^{\tau _j^0} X_i^k\eta _i \right| \ge c_0 \right) \\&\le C\left[ \frac{\tilde{p}+1}{n} + C_2^{1+\varepsilon _X/2} (\tilde{p}+1) \frac{ n_j(\log ^2n)^{1+\varepsilon _X/2} }{ c_0(n\gamma _n)^{2+\varepsilon _X/2} } \right] \rightarrow 0. \end{aligned}$$
Together with (6.5), this implies
$$ \textrm{P}\left( A_{nj2}^{(1)}\right) \rightarrow 0. $$
Next, we show that \(\textrm{P}(A_{nj3}^{(1)})\rightarrow 0\). Using the interval KKT conditions in Lemma 3, the piecewise constancy of the cumulative coefficient vector \(\hat{\varvec{b}}_t(\tilde{p})\), and Lemma 2, for any \(x>0\) we obtain
$$\begin{aligned} \left\| \widetilde{\varvec{\theta }}_{j+1}^0(\tilde{p}) - \hat{\varvec{\beta }}_{2j}(\tilde{p}) \right\| \le \frac{Cn\lambda _n}{ \tau _{j+1}^0-\tau _j^0 } +Cx \end{aligned}$$
(6.6)
with probability tending to one. Combining (6.6) and (6.4) gives
$$\begin{aligned} \Vert {\varvec{\Delta }}_3\Vert&\le C\left( \tau _j^0-\hat{\tau }_{2j-1}^{(1)} \right) \left[ \frac{n\lambda _n}{ \tau _{j+1}^0-\tau _j^0 } +x \right] \end{aligned}$$
(6.7)
with probability tending to one. On the other hand, (6.5) implies that
$$\begin{aligned} \frac{1}{3}\Vert {\varvec{\Delta }}_2\Vert \ge c_0 \left( \tau _j^0-\hat{\tau }_{2j-1}^{(1)} \right) \end{aligned}$$
(6.8)
with probability tending to one. By Assumption H.3,
$$ \frac{n\gamma _n}{ |\tau _{j+1}^0-\tau _j^0| } \rightarrow 0, \qquad \frac{\lambda _n}{\gamma _n}\rightarrow 0. $$
Taking \(x>0\) sufficiently small and combining (6.7) and (6.8), we obtain
$$ \textrm{P}\left( A_{nj3}^{(1)}\right) \rightarrow 0. $$
Consequently,
$$ \textrm{P}\left( A_{nj}^{(1)} \cap C_n^{(1)} \cap \left\{ \hat{\tau }_{2j-1}^{(1)}<\tau _j^0 \right\} \right) \rightarrow 0. $$
The case \(\hat{\tau }_{2j-1}^{(1)}>\tau _j^0\) can be treated analogously, and hence
$$ \textrm{P}\left( A_{nj}^{(1)} \cap C_n^{(1)} \cap \left\{ \hat{\tau }_{2j-1}^{(1)}>\tau _j^0 \right\} \right) \rightarrow 0. $$
Therefore,
$$ \textrm{P}\left( A_{nj}^{(1)} \cap C_n^{(1)} \right) \rightarrow 0, \qquad j=1,\ldots ,m_0. $$
The terms involving \(C_n^{(1),c}\) are controlled by an argument analogous to that in Proposition 5 of Harchaoui and Lévy-Leduc (2010) and Theorem 2.2 of Chan et al. (2014). Thus,
$$ \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(1)} \cap C_n^{(1),c} \right) \rightarrow 0. $$
For \(A_{nj}^{(2)}\), the same argument applies after replacing the first parameter transition \( \varvec{\theta }_j^0(\tilde{p}) - \widetilde{\varvec{\theta }}_{j+1}^0(\tilde{p}) \) by the second parameter transition \( \widetilde{\varvec{\theta }}_{j+1}^0(\tilde{p}) - \varvec{\theta }_{j+1}^0(\tilde{p}), \) and replacing the transition location \(\tau _j^0\) by \(\tau _j^0+p_{j+1}^0\). Consequently,
$$ \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(2)} \cap C_n^{(2)} \right) \rightarrow 0, \qquad \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(2)} \cap C_n^{(2),c} \right) \rightarrow 0. $$
When \(m_0\rightarrow \infty \), Lemma 2, together with suitable choices of \(C_1\) and \(C_2\), ensures that the preceding probability bounds are sufficiently uniform in j. Hence,
$$ \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(1)}\right) \rightarrow 0, \qquad \sum _{j=1}^{m_0} \textrm{P}\left( A_{nj}^{(2)}\right) \rightarrow 0. $$
It follows that
$$ \textrm{P}\left( \max _{1\le j\le m_0} \left\{ \left| \hat{\tau }_{2j-1}^{(1)}-\tau _j^0 \right| \vee \left| \hat{\tau }_{2j}^{(1)} -\left( \tau _j^0+p_{j+1}^0\right) \right| \right\} >n\gamma _n \right) \rightarrow 0. $$
The proof of Lemma 4 is completed.
Models (B1) - (B9) and (C1) - (C9)
Model (B):
$$\begin{aligned} X_t=\left\{ \begin{array}{lll} 0.5\circ X_{1,t-1}+Z_{1,t}& Z_{1,t}\overset{i.i.d}{\sim }\ \textrm{Poi} (0.5)& 0<t\le \tau _1^0,\\ 0.249\circ X_{2,t-\tau _1^0-1}+0.254\circ X_{2,t-\tau _1^0-2}+0.297\circ X_{2,t-\tau _1^0-3}+Z_{2,t}& Z_{2,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(1)& \tau _1^0<t\le \tau _2^0,\\ 0.4\circ X_{3,t-\tau _2^0-1}+Z_{3,t}& Z_{3,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(0.5)& \tau _2^0<t\le \tau _3^0,\\ 0.014\circ X_{4,t-\tau _3^0-1}+0.041\circ X_{4,t-\tau _3^0-2}+0.29\circ X_{4,t-\tau _3^0-3}& \\ \quad +0.454\circ X_{4,t-\tau _3^0-4}+Z_{4,t}& Z_{4,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(2)& \tau _3^0<t\le \tau _4^0,\\ 0.332\circ X_{5,t-\tau _4^0-1}+0.268\circ X_{5,t-\tau _4^0-2}+Z_{5,t}& Z_{5,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(0.5)& \tau _4^0<t\le \tau _5^0,\\ 0.2\circ X_{6,t-\tau _5^0-1}+Z_{6,t}& Z_{6,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(4)& \tau _5^0<t\le \tau _6^0,\\ 0.109\circ X_{7,t-\tau _6^0-1}+0.306\circ X_{7,t-\tau _6^0-2}+0.305\circ X_{7,t-\tau _6^0-3}+Z_{7,t}& Z_{7,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(3)& \tau _6^0<t\le \tau _7^0,\\ 0.3\circ X_{8,t-\tau _7^0-1}+Z_{8,t}& Z_{8,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(0.5)& \tau _7^0<t\le \tau _8^0,\\ 0.202\circ X_{9,t-\tau _8^0-1}+0.127\circ X_{9,t-\tau _8^0-2}+0.179\circ X_{9,t-\tau _8^0-3}& \\ \quad +0.392\circ X_{9,t-\tau _8^0-4}+Z_{9,t}& Z_{9,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(1)& \tau _8^0<t\le \tau _9^0,\\ 0.3\circ X_{10,t-\tau _9^0-1}+Z_{10,t}& Z_{10,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(2)& \tau _9^0<t\le n. \end{array}\right. \end{aligned}$$
Model (B1) consists of the first two segments of Model (B) and \(\tau _1^0=[0.5n]\). That is
$$\begin{aligned} X_t=\left\{ \begin{array}{lll} 0.5\circ X_{1,t-1}+Z_{1,t}& Z_{1,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(0.5)& 0<t\le [0.5n]\\ 0.249\circ X_{2,t-\tau _1^0-1}+0.254\circ X_{2,t-\tau _1^0-2}+0.297\circ X_{2,t-\tau _1^0-3}+Z_{2,t}& Z_{2,t}\overset{i.i.d}{\sim }\ \textrm{Poi}(1)& [0.5n]<t\le n \end{array}\right. \end{aligned}$$
Similarly, Models (B2)-(B9) are defined as follows.
Model (B2) consists of the first three segments of Model (B) and \((\tau _1^0,\tau _2^0)=([0.3n], [0.6n])\).
Model (B3) consists of the first four segments of Model (B) and \((\tau _1^0,\tau _2^0,\tau _3^0)=([0.2n], [0.5n], [0.8n])\).
Model (B4) consists of the first five segments of Model (B) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0) =([0.2n], [0.4n], [0.6n], [0.8n])\).
Model (B5) consists of the first six segments of Model (B) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0) = ([0.1n], [0.3n], [0.6n], [0.7n], [0.9n])\).
Model (B6) consists of the first seven segments of Model (B) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0) =([0.1n], [0.2n], [0.3n], [0.5n], [0.8n], [0.9n])\).
Model (B7) consists of the first eight segments of Model (B) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0,\tau _7^0)=([0.1n],[0.2n],[0.3n],[0.4n],[0.5n],[0.8n],[0.9n])\).
Model (B8) consists of the first nine segments of Model (B) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0,\tau _7^0,\tau _8^0)=([0.1n],[0.2n],[0.3n],[0.4n],[0.5n],[0.7n],[0.8n],[0.9n])\).
Model (B9) consists of the first ten segments of Model (B) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0,\tau _7^0,\tau _8^0,\tau _9^0)=([0.1n],[0.2n],[0.3n],[0.4n],[0.5n],[0.6n],[0.7n],[0.8n],[0.9n])\).
Similarly, Models (C1)-(C9) are constructed from Model (C).
Model (C):
$$\begin{aligned}{X_t=\left\{ \begin{array}{lll} 0.5*X_{1,t-1}+Z_{1,t}& Z_{1,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (1/3)& 0<t\le \tau _1^0,\\ 0.249*X_{2,t-\tau _1^0-1}+0.254*X_{2,t-\tau _1^0-2}+0.297*X_{2,t-\tau _1^0-3}+Z_{2,t}& Z_{2,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (1/2)& \tau _1^0<t\le \tau _2^0,\\ 0.4*X_{3,t-\tau _2^0-1}+Z_{3,t}& Z_{3,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (1/3)& \tau _2^0<t\le \tau _3^0,\\ 0.014*X_{4,t-\tau _3^0-1}+0.041*X_{4,t-\tau _3^0-2}& \\ \quad +0.29*X_{4,t-\tau _3^0-3}+0.454*X_{4,t-\tau _3^0-4}+Z_{4,t}& Z_{4,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (2/3)& \tau _3^0<t\le \tau _4^0,\\ 0.332*X_{5,t-\tau _4^0-1}+0.268*X_{5,t-\tau _4^0-2}+Z_{5,t}& Z_{5,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (1/3)& \tau _4^0<t\le \tau _5^0,\\ 0.2*X_{6,t-\tau _5^0-1}+Z_{6,t}& Z_{6,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (4/5)& \tau _5^0<t\le \tau _6^0,\\ 0.109*X_{7,t-\tau _6^0-1}+0.306*X_{7,t-\tau _6^0-2}+0.305*X_{7,t-\tau _6^0-3}+Z_{7,t}& Z_{7,t}\overset{i.i.d}{\sim }\textrm{Geo} (3/4)& \tau _6^0<t\le \tau _7^0,\\ 0.3*X_{8,t-\tau _7^0-1}+Z_{8,t}& Z_{8,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (1/3)& \tau _7^0<t\le \tau _8^0,\\ 0.202*X_{9,t-\tau _8^0-1}+0.127*X_{9,t-\tau _8^0-2}& \\ \quad +0.179*X_{9,t-\tau _8^0-3}+0.392*X_{9,t-\tau _8^0-4}+Z_{9,t}& Z_{9,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (1/2)& \tau _8^0<t\le \tau _9^0,\\ 0.3*X_{10,t-\tau _9^0-1}+Z_{10,t}& Z_{10,t}\overset{i.i.d}{\sim }\ \textrm{Geo} (2/3)& \tau _9^0<t\le n. \end{array}\right. } \end{aligned}$$
Model (C1) consists of the first two segments of Model (C) and \(\tau _1^0=[0.5n]\). That is
$$\begin{aligned}{X_t=\left\{ \begin{array}{lll} 0.5*X_{1,t-1}+Z_{1,t}& Z_{1,t}\overset{i.i.d}{\sim }\ \textrm{Geo}(1/3)& 0<t\le [0.5n]\\ 0.249*X_{2,t-\tau _1^0-1}+0.254*X_{2,t-\tau _1^0-2}+0.297*X_{2,t-\tau _1^0-3}+Z_{2,t}& Z_{2,t}\overset{i.i.d}{\sim }\ \textrm{Geo}(1/2)& [0.5n]<t\le n\\ \end{array}\right. } \end{aligned}$$
Similarly, Models (C2)-(C9) are defined as follows (see Tables 7 and 8).
Model (C2) consists of the first three segments of Model (C) and \((\tau _1^0,\tau _2^0)=([0.3n], [0.6n])\).
Model (C3) consists of the first four segments of Model (C) and \((\tau _1^0,\tau _2^0,\tau _3^0)=([0.2n], [0.5n], [0.8n])\).
Model (C4) consists of the first five segments of Model (C) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0) =([0.2n], [0.4n], [0.6n], [0.8n])\).
Model (C5) consists of the first six segments of Model (C) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0) = ([0.1n], [0.3n], [0.6n], [0.7n], [0.9n])\).
Model (C6) consists of the first seven segments of Model (C) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0) =([0.1n], [0.2n], [0.3n], [0.5n], [0.8n], [0.9n])\).
Model (C7) consists of the first eight segments of Model (C) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0,\tau _7^0)=([0.1n],[0.2n],[0.3n],[0.4n],[0.5n],[0.8n],[0.9n])\).
Model (C8) consists of the first nine segments of Model (C) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0,\tau _7^0,\tau _8^0)=([0.1n],[0.2n],[0.3n],[0.4n],[0.5n],[0.7n],[0.8n],[0.9n])\).
Model (C9) consists of the first ten segments of Model (C) and
\((\tau _1^0,\tau _2^0,\tau _3^0,\tau _4^0,\tau _5^0,\tau _6^0,\tau _7^0,\tau _8^0,\tau _9^0)=([0.1n],[0.2n],[0.3n],[0.4n],[0.5n],[0.6n],[0.7n],[0.8n],[0.9n])\) (See Tables 7 and 8).