SlowGuess/NeurIPS_2023_OCR huggingface dataset:
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A Additional simulation results
A.1 Robustness to Kernel Selection and Bandwidth Parameters
In this subsection, we provide additional simulation results in Table 2 to show that our proposed localization method, i.e. MNSBS, is robust against both the choice of Kernel functions and the choice of bandwidth parameters.
We consider the setting with  $T = 150$  and  $p = 3$  of Scenario 1. In addition to the Gaussian kernel used in the previous numeric experiments, we consider the Epanechnikov kernel and the Triweight kernel. We also let the bandwidth  $h = c_h \times (1 / T)^{1 / (2r + p)}$  with  $r = 2$  and  $c_{h} \in \{1,2,5,10,20\}$ . Out of the 500 iterations for each case, the table below reports the proportion of the number of change points  $K$  and the averaged localization errors.
Table 2: Additional localization results of Scenario 1.
Kernel	T=150,p=3 and r=2
ch=1	ch=2	ch=5	ch=10	ch=20
Propotion of times K≠K
Gaussian	0.138	0.070	0.066	0.070	0.068
Epanechnikov	0.748	0.184	0.070	0.070	0.064
Triweight	0.280	0.082	0.068	0.066	0.060
Average (standard deviation) of dh
Gaussian	0.038(0.050)	0.029(0.043)	0.025(0.039)	0.026(0.040)	0.026(0.038)
Epanechnikov	0.118(0.045)	0.037(0.049)	0.012(0.035)	0.011(0.035)	0.010(0.035)
Triweight	0.053(0.057)	0.017(0.038)	0.010(0.034)	0.010(0.035)	0.010(0.034)
A.2 Runtime and localization for Independent Data
Examination of Scenario 1 with independent data.
We examined Scenario 1 where  $p = \overline{3}$ ,  $n$  is in the set  $\{150, 300\}$ , and  $X_{t}$  is i.i.d. distributed as  $N(0_{p}, I_{p})$ . Our results indicate that MNSBS excels in change point localization. The refinement process further enhances its performance. Refer to Table 3 for specifics.
Runtime Comparison.
We benchmarked the runtime of our method against others. The tests were conducted on a machine powered by an Apple M2 chip with an 8-core CPU. The parameters were set at  $p = 3$  and  $n$  in the set  $\{150, 300\}$  for the independent setting. Our method performs comparably at  $n = 150$ . However, it is slower at  $n = 300$ , attributed to the computational demands of CUSUM. Detailed findings are presented in Table 3.
Table 3: Runtime and localization results of Scenario 1 with independent data.
METHOD	T=150p=3	T=300p=3
AVERAGE (STANDARD DEVIATION) OF RUNTIME (IN SECONDS)
MNSBS(INITIAL & REFINED)	1.130 (0.171)	13.134 (2.006)
NMP	0.812 (0.142)	10.899 (1.742)
ECP	0.117 (0.053)	0.646 (0.140)
SBS	0.619 (0.033)	0.656 (0.040)
DCBS	0.866 (0.092)	1.254 (0.134)
PROPORTION OF TIMES K≠K
MNSBS(INITIAL)	0.000	0.005
MNSBS(REFINED)	0.000	0.005
NMP	0.000	0.000
ECP	0.000	0.040
SBS	0.445	0.015
DCBS	0.075	0.065
AVERAGE (STANDARD DEVIATION) OF dh
MNSBS(INITIAL)	0.010 (0.013)	0.009 (0.012)
MNSBS(REFINED)	0.006 (0.011)	0.005 (0.011)
NMP	0.016 (0.019)	0.008 (0.010)
ECP	0.007 (0.011)	0.009 (0.028)
SBS	0.158 (0.157)	0.012 (0.041)
DCBS	0.021 (0.037)	0.011 (0.022)
B Real data application
We applied our proposed change point inference procedure to analyze stock price data $^2$ , which consisted of daily adjusted close price of the 3 major stock market indices (S&P 500, Dow Jones and NASDAQ) from Jan-01-2021 to Jan-19-2023. After removing missing values and standardizing the raw data, the sample size was  $n = 515$  and the dimension  $p = 3$ .
We localized 6 estimated change points and performed inference based on them; results are summarized in Table 4. We also implemented the NMP and ECP methods on the same dataset, the estimated change points being presented below. Except for the time point Aug-24-2022 estimated by ECP, all other estimated change points were located in the constructed  $99\%$  confidence intervals by our proposed method.
The transformed real data is illustrated in the figure below. These data correspond to the daily adjusted close price, from Jan-01-2021 to Jan-19-2023, of the 3 major stock market indices, S&P 500, Dow Jones and NASDAQ. Moreover, in Table 4, we present the estimated change point by our proposed method MNSBS on the data before mentioned, together with their respective inference.
Figure 2: Plot of the standardized daily close price, from Jan-01-2021 to Jan-19-2023, of the 3 major stock market indices.
Table 4: Confidence intervals constructed for change point locations in the Real data example.
\(\widehat{\eta}\)	\(\alpha=0.01\)	\(\alpha=0.05\)
LOWER BOUND	UPPER BOUND	LOWER BOUND	UPPER BOUND
APRIL-07-2021	APRIL-01-2021	APRIL-12-2021	APRIL-05-2021	APRIL-09-2021
JUNE-30-2021	JUNE-23-2021	JULY-09-2021	JUNE-25-2021	JULY-07-2021
OCT-19-2021	OCT-12-2021	OCT-26-2021	OCT-14-2021	OCT-22-2021
JAN-18-2022	JAN-12-2022	JAN-21-2022	JAN-13-2022	JAN-20-2022
APRIL-25-2022	APRIL-20-2022	APRIL-28-2022	APRIL-21-2022	APRIL-27-2022
OCT-27-2022	OCT-24-2022	NOV-01-2022	OCT-25-2022	OCT-31-2022
The result of the implementation of NMP and ECP methods on the same dataset are {April-01-2021, July-01-2021, Oct-19-2021, Jan-14-2022, April-21-2022, Oct-26-2022} and {April-08-2021, June-25-2021, Oct-18-2021, Jan-18-2022, April-28-2022, Aug-24-2022, Oct-27-2022} respectively.
2The stock price data are downloaded from https://fred.stlouisfed.org/series.
C  $\alpha$ -mixing coefficients
This paper focuses on multivariate time series that exhibit  $\alpha$ -mixing behavior with exponential decay coefficients. This condition is denoted as Assumption 1e. While the constant  $2c$  is present in the exponent of the exponential function, it plays a non-essential role in our theoretical framework. We include it solely for the sake of convenience during verification.
The  $\alpha$ -mixing condition with exponential decay as specified in Assumption 1e is a commonly held assumption in time series analysis. A broad spectrum of multivariate time series satisfies this condition, including linear/nonlinear VAR models [e.g. Liebscher (2005)], a comprehensive class of GARCH models [e.g. Boussama et al. (2011)], and various Markov processes [e.g. Chan & Tong (2001)]. To further elaborate, consider the  $p$  dimensional stationary VAR(1) model:
$$
X _ {t} = A X _ {t - 1} + \epsilon_ {t}
$$
where  $A$  is the  $p \times p$  transition matrix whose spectral norm satisfying  $\| A \| \in (0,1)$  and the innovations  $\epsilon_t$  are i.i.d. Gaussian vectors. Denote  $\Sigma = \operatorname{cov}(X_1)$ , and let  $\lambda_{\mathrm{max}}$  and  $\lambda_{\mathrm{min}}$  be the largest and smallest eigenvalues of  $\Sigma$ . Then by Theorem 3.1 in Han & Wu (2023), we have that for any  $k \geq 0$ , the  $\alpha$ -mixing coefficient of the time series  $X_t$  satisfying
$$
\alpha_ {k} \leq \sqrt {\frac {\lambda_ {\mathrm {m a x}}}{\lambda_ {\mathrm {m i n}}} | A | ^ {k}} \leq e ^ {- C \log (1 / | A |) k}
$$
where  $C > 0$  is some constant depending only on  $\sqrt{\lambda_{\mathrm{max}} / \lambda_{\mathrm{min}}}$ . In this example, the constant  $C\log (1 / |A|)$  corresponds to the constant  $2c$  in Assumption 1e. Essentially, Assumption 1e is useful to unlock several technical tools under temporal dependence, which include a Bernstein's inequality Merlevède et al. (2012), a moment inequality [see Proposition 2.5 in Fan & Yao (2003)], maximal inequalities (see Section G.1) and a central limit theorem (see Section G.2). For instance, we utilize the moment inequality to bound the autocovariances of a dependence process with all lags by  $\alpha$ -mixing coefficients, thereby demonstrating the existence of the long-run variance, which is the sum of all the autocovariances.
In this section, we present the proof of theorem Theorem 1.
Proof of Theorem 1. For any  $(s,e]\subseteq (0,T]$ , let
$$
\widetilde {f} _ {t} ^ {(s, e ]} (x) = \sqrt {\frac {e - t}{(e - s) (t - s)}} \sum_ {l = s + 1} ^ {t} f _ {l} (x) - \sqrt {\frac {t - s}{(e - s) (e - t)}} \sum_ {l = t + 1} ^ {e} f _ {l} (x), x \in \mathcal {X}.
$$
For any  $\widetilde{r} \in (\rho, T - \rho]$ , we consider
$$
\begin{array}{l} \mathcal{A}((s,e],\rho ,\lambda) = \Bigg\{\max_{t = s + \rho +1}\sup_{x\in \mathbb{R}^{p}}|\widetilde{F}_{t,h}^{s,e}(x) - \widetilde{f}_{t}^{s,e}(x)|\leq \lambda \Bigg\}; \\ \mathcal {B} (\widetilde {r}, \rho , \lambda) = \left\{\max _ {N = \rho} ^ {T - \widetilde {r}} \sup _ {x \in \mathbb {R} ^ {p}} \left| \frac {1}{\sqrt {N}} \sum_ {t = \widetilde {r} + 1} ^ {\widetilde {r} + N} F _ {t, h} (x) - \frac {1}{\sqrt {N}} \sum_ {t = \widetilde {r} + 1} ^ {\widetilde {r} + N} f _ {t} (x) \right| \leq \lambda \right\} \bigcup \\ \left\{\underset {N = \rho} {\widetilde {r}} \sup  _ {x \in \mathbb {R} ^ {p}} \left| \frac {1}{\sqrt {N}} \sum_ {t = \widetilde {r} - N + 1} ^ {\widetilde {r}} F _ {t, h} (x) - \frac {1}{\sqrt {N}} \sum_ {t = \widetilde {r} - N + 1} ^ {\widetilde {r}} f _ {t} (x) \right| \leq \lambda \right\}. \\ \end{array}
$$
From Algorithm 1, we have that
$$
\rho = \frac {\log (T)}{h ^ {p}}.
$$
Therefore, Proposition 2 imply that with
$$
\lambda = C _ {\lambda} \left(2 C \sqrt {\frac {\log T}{h ^ {p}}} + \frac {2 C _ {1} \sqrt {p}}{\sqrt {h ^ {p}}} + 2 C _ {2} \sqrt {T} h ^ {r}\right). \tag {9}
$$
for some diverging sequence  $C_{\lambda}$ , it holds that
$$
P \left\{\mathcal {A} ^ {c} ((s, e ], \rho , \lambda) \right\} \lesssim \frac {1}{T ^ {2}},
$$
$$
P \left\{\mathcal {B} ^ {c} (\widetilde {r}, \rho , \frac {\lambda}{2}) \right\} \lesssim \frac {1}{T ^ {2}}.
$$
Now, we notice that,
$$
\sum_ {k = 1} ^ {\Re} \widetilde {n} _ {k} = \sum_ {k = 1} ^ {\Re} (2 ^ {k} - 1) \leq \sum_ {k = 1} ^ {\Re} 2 ^ {k} \leq 2 (2 ^ {\left\lceil C _ {\widetilde {\Re}} \left(\frac {\log \left(\frac {T}{\Delta}\right)}{\log (2)}\right)\right\rceil} - 1) \leq 2 (2 ^ {\left\lceil C _ {\widetilde {\Re}} \left(\frac {\log (T)}{\log (2)}\right)\right\rceil} - 1) = O (T).
$$
since  $2^{-x} < 1$  for any  $x > 0$ . In addition, there are  $K = O(1)$  number of change points. In consequence, it follows that
$$
P \left\{\mathcal {A} (\mathcal {I}, \rho , \lambda) \text {f o r a l l} \mathcal {I} \in \mathcal {J} \right\} \geq 1 - \frac {1}{T}, \tag {10}
$$
$$
P \left\{\mathcal {B} (s, \rho , \lambda) \cup \mathcal {B} (e, \rho , \lambda) \text {f o r a l l} (s, e ] = \mathcal {I} \in \mathcal {J} \right\} \geq 1 - \frac {1}{T}, \tag {11}
$$
$$
P \left\{\mathcal {B} \left(\eta_ {k}, \rho , \lambda\right) \text {f o r a l l} 1 \leq k \leq K \right\} \geq 1 - \frac {1}{T}. \tag {12}
$$
The rest of the argument is made by assuming the events in equations (10), (11) and (12) hold. By Remark 1, we have that on these events, it is satisfied that
$$
\max_{t = s + \rho +1}^{e - \rho}||\widetilde{F}_{t,h}^{s,e}(x) - \widetilde{f}_{t}^{s,e}(x)||_{L_{2}}\leq \lambda .
$$
$$
\Upsilon_ {k} = C \log (T) \left(T ^ {\frac {p}{2 r + p}}\right) \kappa_ {k} ^ {- 2} \quad \text {a n d} \quad \Upsilon_ {\max } = C \log (T) \left(T ^ {\frac {p}{2 r + p}}\right) \kappa^ {- 2},
$$
where  $\kappa = \min \{\kappa_{1},\dots ,\kappa_{K}\}$ . Since  $\Upsilon_{k}$  is the desired localisation rate, by induction, it suffices to consider any generic interval  $(s,e]\subseteq (0,T]$  that satisfies the following three conditions:
$$
\eta_ {m - 1} \leq s \leq \eta_ {m} \leq \dots \leq \eta_ {m + q} \leq e \leq \eta_ {m + q + 1}, \quad q \geq - 1;
$$
either  $\eta_{m} - s\leq \Upsilon_{m}$  or  $s - \eta_{m - 1}\leq \Upsilon_{m - 1}$
either  $\eta_{m + q + 1} - e\leq \Upsilon_{m + q + 1}$  or  $e - \eta_{m + q}\leq \Upsilon_{m + q}$
Here  $q = -1$  indicates that there is no change point contained in  $(s, e]$ .
$$
\Delta_ {k} = \eta_ {k - 1} - \eta_ {k} \text {f o r} k = 1, \dots , K + 1 \quad \text {a n d} \quad \Delta = \min  \left\{\Delta_ {1}, \dots , \Delta_ {K + 1} \right\}.
$$
Observe that by assumption 3,
$$
\Upsilon_ {\mathrm {m a x}} = C \log (T) \left(T ^ {\frac {p}{2 r + p}}\right) \kappa^ {- 2} \leq \frac {\Delta}{4}
$$
Therefore, it has to be the case that for any true change point  $\eta_m \in (0,T]$ , either  $|\eta_m - s| \leq \Upsilon_m$  or  $|\eta_m - s| \geq \Delta - \Upsilon_{\max} \geq \frac{3}{4}\Delta$ . This means that  $\min \{| \eta_m - e|, |\eta_m - s| \} \leq \Upsilon_m$  indicates that  $\eta_m$  is a detected change point in the previous induction step, even if  $\eta_m \in (s,e]$ . We refer to  $\eta_m \in (s,e]$  as an undetected change point if  $\min \{\eta_m - s, \eta_m - e\} \geq \frac{3}{4}\Delta$ . To complete the induction step, it suffices to show that MNSBS((s,e], h,τ)
(i) will not detect any new change point in  $(s, e]$  if all the change points in that interval have been previously detected, and
(ii) will find a point  $D^{\mathcal{I}^*}$  in  $(s,e]$  such that  $|\eta_m - D^{\mathcal{I}^*}| \leq \Upsilon_m$  if there exists at least one undetected change point in  $(s,e]$ .
In order to accomplish this, we need the following series of steps.
Step 1. We first observe that if  $\eta_k \in \{\eta_k\}_{k=1}^K$  is any change point in the functional time series, by Lemma 5, there exists a seeded interval  $\mathcal{I}_k = (s_k, e_k]$  containing exactly one change point  $\eta_k$  such that
$$
\min  \left\{\eta_ {k} - s _ {k}, e _ {k} - \eta_ {k} \right\} \geq \frac {1}{1 6} \Delta , \quad \text {a n d} \quad \max  \left\{\eta_ {k} - s _ {k}, e _ {k} - \eta_ {k} \right\} \leq \frac {9}{1 0} \Delta
$$
Even more, we notice that if  $\eta_k \in (s, e]$  is any undetected change point in  $(s, e]$ . Then it must hold that
$$
s - \eta_ {k - 1} \leq \Upsilon_ {\max }
$$
Since  $\Upsilon_{\max} = O(\log (T)T^{\frac{p}{2r + p}})$  and by assumption 3, we have that  $\Upsilon_{\max} < \frac{1}{10}\Delta$ . Moreover,  $\eta_k - s_k \leq \frac{9}{10} (\eta_k - \eta_{k-1})$ , so that it holds that
$$
s _ {k} - \eta_ {k - 1} \geq \frac {1}{1 0} (\eta_ {k} - \eta_ {k - 1}) > \Upsilon_ {\max } \geq s - \eta_ {k - 1}
$$
and in consequence  $s_k \geq s$ . Similarly  $e_k \leq e$ . Therefore
$$
\mathcal {I} _ {k} = \left(s _ {k}, e _ {k} \right] \subseteq (s, e ].
$$
Step 2. Consider the collection of intervals  $\{\mathcal{I}_k = (s_k, e_k]\}_{k=1}^K$  in Step 1. In this step, it is shown that for each  $k \in \{1, \ldots, K\}$ , it holds that
$$
\max  _ {t = s _ {k} + \rho} ^ {t = e _ {k} - \rho} \left\| \widetilde {F} _ {t, h} ^ {\left(s _ {k}, e _ {k} \right]} \right\| _ {L _ {2}} \geq c _ {1} \sqrt {\Delta} \kappa_ {k}, \tag {13}
$$
for some sufficient small constant  $c_{1}$
Let  $k \in \{1, \dots, K\}$ . By Step 1,  $\mathcal{I}_k$  contains exactly one change point  $\eta_k$ . Since  $f_t$  is a one-dimensional population time series and there is only one change point in  $\mathcal{I}_k = (s_k, e_k]$ , it holds that
$$
f _ {s _ {k} + 1} = \dots = f _ {\eta_ {k}} \neq f _ {\eta_ {k} + 1} = \dots = f _ {e _ {k}}
$$
which implies, for  $s_k < t < \eta_k$
$$
\begin{array}{l} \widetilde {f} _ {t} ^ {\left(s _ {k}, e _ {k} \right]} = \sqrt {\frac {e _ {k} - t}{\left(e _ {k} - s _ {k}\right) \left(t - s _ {k}\right)}} \sum_ {l = s _ {k} + 1} ^ {t} f _ {\eta_ {k}} - \sqrt {\frac {t - s _ {k}}{\left(e _ {k} - s _ {k}\right) \left(e _ {k} - t\right)}} \sum_ {l = t + 1} ^ {\eta_ {k}} f _ {\eta_ {k}} \\ - \sqrt {\frac {t - s _ {k}}{(e _ {k} - s _ {k}) (e _ {k} - t)}} \sum_ {l = \eta_ {k} + 1} ^ {e _ {k}} f _ {\eta_ {k} + 1} \\ = (t - s _ {k}) \sqrt {\frac {e _ {k} - t}{(e _ {k} - s _ {k}) (t - s _ {k})}} f _ {\eta_ {k}} - (\eta_ {k} - t) \sqrt {\frac {t - s _ {k}}{(e _ {k} - s _ {k}) (e _ {k} - t)}} f _ {\eta_ {k}} \\ - \left(e _ {k} - \eta_ {k}\right) \sqrt {\frac {t - s _ {k}}{\left(e _ {k} - s _ {k}\right) \left(e _ {k} - t\right)}} f _ {\eta_ {k} + 1} \\ = \sqrt {\frac {(t - s _ {k}) (e _ {k} - t)}{(e _ {k} - s _ {k})}} f _ {\eta_ {k}} - (\eta_ {k} - t) \sqrt {\frac {t - s _ {k}}{(e _ {k} - s _ {k}) (e _ {k} - t)}} f _ {\eta_ {k}} \\ - \left(e _ {k} - \eta_ {k}\right) \sqrt {\frac {t - s _ {k}}{\left(e _ {k} - s _ {k}\right) \left(e _ {k} - t\right)}} f _ {\eta_ {k} + 1} \\ = (e _ {k} - t) \sqrt {\frac {t - s _ {k}}{(e _ {k} - t) (e _ {k} - s _ {k})}} f _ {\eta_ {k}} - (\eta_ {k} - t) \sqrt {\frac {t - s _ {k}}{(e _ {k} - s _ {k}) (e _ {k} - t)}} f _ {\eta_ {k}} \\ - \left(e _ {k} - \eta_ {k}\right) \sqrt {\frac {t - s _ {k}}{\left(e _ {k} - s _ {k}\right) \left(e _ {k} - t\right)}} f _ {\eta_ {k} + 1} \\ = (e _ {k} - \eta_ {k}) \sqrt {\frac {t - s _ {k}}{(e _ {k} - t) (e _ {k} - s _ {k})}} f _ {\eta_ {k}} - (e _ {k} - \eta_ {k}) \sqrt {\frac {t - s _ {k}}{(e _ {k} - s _ {k}) (e _ {k} - t)}} f _ {\eta_ {k} + 1} \\ = (e _ {k} - \eta_ {k}) \sqrt {\frac {t - s _ {k}}{(e _ {k} - t) (e _ {k} - s _ {k})}} (f _ {\eta_ {k}} - f _ {\eta_ {k} + 1}). \\ \end{array}
$$
Similarly, for  $\eta_k \leq t \leq e_k$
$$
f _ {t} ^ {(s _ {k}, e _ {k} ]} = \sqrt {\frac {e _ {k} - t}{(e _ {k} - s _ {k}) (t - s _ {k})}} (\eta_ {k} - s _ {k}) (f _ {\eta_ {k}} - f _ {\eta_ {k} + 1}).
$$
$$
\widetilde {f} _ {t} ^ {\left(s _ {k}, e _ {k} \right]} = \left\{ \begin{array}{l l} \sqrt {\frac {t - s _ {k}}{\left(e _ {k} - s _ {k}\right) \left(e _ {k} - t\right)}} \left(e _ {k} - \eta_ {k}\right) \left(f _ {\eta_ {k}} - f _ {\eta_ {k} + 1}\right), & s _ {k} <   t <   \eta_ {k}; \\ \sqrt {\frac {e _ {k} - t}{\left(e _ {k} - s _ {k}\right) \left(t - s _ {k}\right)}} \left(\eta_ {k} - s _ {k}\right) \left(f _ {\eta_ {k}} - f _ {\eta_ {k} + 1}\right), & \eta_ {k} \leq t \leq e _ {k}. \end{array} \right. \tag {14}
$$
Since  $\rho = O(\log (T)T^{\frac{p}{2r + p}})$ , by Assumption 3, we have that
$$
\min  \left\{\eta_ {k} - s _ {k}, e _ {k} - \eta_ {k} \right\} \geq \frac {1}{1 6} \Delta > \rho , \tag {15}
$$
so that  $\eta_{k}\in [s_{k} + \rho ,e_{k} - \rho ]$  . Then, from (14), (15) and the fact that  $|e_k - s_k| <   \Delta$  and  $|\eta_k - s_k| <   \Delta$
$$
\left| \left| \widetilde {f} _ {\eta_ {k}} ^ {\left(s _ {k}, e _ {k} \right]} \right| \right| _ {L _ {2}} = \sqrt {\frac {e _ {k} - \eta_ {k}}{\left(e _ {k} - s _ {k}\right) \left(\eta_ {k} - s _ {k}\right)}} \left(\eta_ {k} - s _ {k}\right) \left| \left| f _ {\eta_ {k}} - f _ {\eta_ {k} + 1} \right| \right| _ {L _ {2}} \geq c _ {2} \sqrt {\Delta} \frac {3}{4} \kappa_ {k}. \tag {16}
$$
Therefore, it holds that
$$
\begin{array}{l} \max_{t = s_{k} + \rho}\left|\left|\widetilde{F}_{t,h}^{(s_{k},e_{k}]}\right|\right|_{L_{2}}\geq \left|\left|\widetilde{F}_{\eta_{k},h}^{(s_{k},e_{k}]}\right|\right|_{L_{2}} \\ \geq \left| \left| \widetilde {f} _ {\eta_ {k}} ^ {\left(s _ {k}, e _ {k} \right]} \right| \right| _ {L _ {2}} - \lambda \\ \geq c _ {2} \frac {3}{4} \sqrt {\Delta} \kappa_ {k} - \lambda , \\ \end{array}
$$
where the first inequality follows from the fact that  $\eta_k \in [s_k + \rho, e_k - \rho]$ , the second inequality follows from the good event in (10) and Remark 2, and the last inequality follows from (16). Next, we observe that by Assumption 3
$$
\log^ {\frac {1}{2}} (T) \sqrt {\frac {1}{h ^ {p}}} = \sqrt {T ^ {\frac {p}{2 r + p}}} \sqrt {\log (T)} \leq \frac {c _ {2}}{4} \sqrt {\Delta} \kappa_ {k},
$$
$$
\sqrt {T} h ^ {r} = T ^ {\frac {1}{2} - \frac {r}{2 r + p}} = T ^ {\frac {p}{4 r + 2 p}}.
$$
In consequence, since  $\kappa_{k}$  is a positive constant, by the upper bound of  $\lambda$  on Equation (9), for sufficiently large  $T$ , it holds that
$$
\frac {c _ {2}}{4} \sqrt {\Delta} \kappa_ {k} \geq \lambda .
$$
$$
\max_{t = s_{k} + \rho}^{t = e_{k} - \rho}||\widetilde{F}_{t,h}^{(s_{k},e_{k}]}||_{L_{2}}\geq \frac{c_{2}}{2}\sqrt{\Delta}\kappa_{k}.
$$
Therefore Equation (13) holds with  $c_{1} = \frac{c_{2}}{2}$ .
Step 3. In this step, it is shown that  $\mathrm{SBS}((s,e],h,\tau)$  can consistently detect or reject the existence of undetected change points within  $(s,e]$ .
Suppose  $\eta_k \in (s, e]$  is any undetected change point. Then by the second half of Step 1,  $\mathcal{I}_k \subseteq (s, e]$ , and moreover
$$
a _ {\mathcal {I} ^ {*}} \geq \max _ {t = s _ {k} + \rho} ^ {t = e _ {k} - \rho} | | \widetilde {F} _ {t, h} ^ {(s _ {k}, e _ {k} ]} | | _ {L _ {2}} \geq c _ {1} \sqrt {\Delta} \kappa_ {k} > \tau ,
$$
where the second inequality follows from Equation (13), and the last inequality follows from Assumption 3 and the choice of  $\tau = C_{\tau}\left(\log^{\frac{1}{2}}(T)\sqrt{\frac{1}{h^p}}\right)$ . Therefore,  $\mathcal{M}^{s,e} \neq \emptyset$ , since  $\mathcal{I}_k \in \mathcal{M}^{s,e}$ .
Suppose there does not exist any undetected change point in  $(s, e]$ . Then for any  $\mathcal{I} = (\alpha, \beta] \subseteq (s, e]$ , one of the following situations must hold,
(a) There is no change point within  $(\alpha, \beta]$ ;
(b) there exists only one change point  $\eta_{k}$  within  $(\alpha ,\beta ]$  and  $\min \{\eta_k - \alpha ,\beta -\eta_k\} \leq \Upsilon_k$
(c) there exist two change points  $\eta_k, \eta_{k+1}$  within  $(\alpha, \beta]$  and
$$
\eta_ {k} - \alpha \leq \Upsilon_ {k} \quad \text {a n d} \quad \beta - \eta_ {k + 1} \leq \Upsilon_ {k + 1}.
$$
Observe that if (a) holds, then we have
$$
\max  _ {\alpha + \rho <   t <   \beta - \rho} | | \widetilde {F} _ {t, h} ^ {(\alpha , \beta ]} | | _ {L _ {2}} \leq \max  _ {\alpha + \rho <   t <   \beta - \rho} | | \widetilde {f} _ {t} ^ {(\alpha , \beta ]} | | _ {L _ {2}} + \lambda = \lambda .
$$
Cases (b) and (c) can be dealt with using similar arguments. We will only work on (c) here. It follows that, in the good event in Equation (10),