And so the event

$$ \xi_2^{(p)} := \xi_2^{(p)}(\delta) = \left\{ \begin{array}{l} \forall (l, r) \in J_n, \forall S \subset [p], \left| \frac{r}{2} \left\| \bar{\epsilon}_{l,+r}^{(S)} - \bar{\epsilon}_{l,-r}^{(S)} \right\|^2 - \sigma^2 s \right| \\ \le 4 \bar{c}_{\text{conc}} L^2 \left( \log \left( \frac{n}{r\delta} \right) + s \log \left( \frac{2ep}{s} \right) \right) \end{array} \right\} \quad (39) $$

has probability larger than $1 - 2\delta$.

Set now

$$ \xi^{(p)} := \xi_{1}^{(p)} \cap \xi_{2}^{(p)}. $$

Note that

$$ \mathbb{P}(\xi^{(p)}) \geq 1 - 3\delta. $$

**Step 2: Study in the ‘no change-point’ situation** Consider $(l, r) \in J_n$ such that $\{\tau_k, k \in [K]\} \cap [l - r, l + r) = \emptyset$, and $S \subset [p]$. Note that since $\{\tau_k, k \in [K]\} \cap [l - r, l + r) = \emptyset$, we have $\bar{\theta}_{l,-r}^{(S)} = \bar{\theta}_{l,+r}^{(S)}$ so that

$$ \frac{r}{2} \left\| \bar{\theta}_{l,-r}^{(S)} - \bar{\theta}_{l,+r}^{(S)} \right\|^2 = 0, $$

and

$$ r\langle\bar{\epsilon}_{l,+r}^{(S)} - \bar{\epsilon}_{l,-r}^{(S)}, \bar{\theta}_{l,+r}^{(S)} - \bar{\theta}_{l,-r}^{(S)}\rangle = 0. $$

Moreover we have on $\xi^{(p)}$ that – see Equation (39)

$$ \left| \frac{r}{2} \left\| \bar{\epsilon}_{l,+r}^{(S)} - \bar{\epsilon}_{l,-r}^{(S)} \right\|^2 - \sigma^2 s \right| \le 4 \bar{c}_{\text{conc}} L^2 \left( \log \left( \frac{n}{r\delta} \right) + s \log \left( \frac{2ep}{s} \right) \right) \le \sigma^2 x_r^{(p)}, $$

for $\bar{c}_{\text{thresh}} \ge 4\bar{c}_{\text{conc}}$ – note that $\bar{c}_{\text{conc}} > 0$ is a universal constant. And so

$$ \Psi_{l,r}^{(p)} \le x_r^{(p)}, $$

so that on $\xi^{(d)}$,

$$ T_{l,r}^{(p)} = 0. $$

This concludes the proof of the first part of the proposition.

**Step 3: Study in the ‘change-point’ situation** Consider $k \in [K]$ such that $\tau_k$ is a $\kappa$-sparse high-energy change-point, – see Equation (18). Since $S_k$ is the support of $\mu_k - \mu_{k-1}$ – and therefore of $\bar{\theta}_{\tau_k, -\bar{r}_k^{(s)}} - \bar{\theta}_{\tau_k, +\bar{r}_k^{(s)}}$ – we have

$$ \frac{\bar{r}_k^{(s)}}{2} \left\| \bar{\theta}_{\tau_k, -\bar{r}_k^{(s)}} - \bar{\theta}_{\tau_k, +\bar{r}_k^{(s)}} \right\|^2 \ge \frac{\kappa}{8} L^2 \left( s_k \log \left( \frac{2ep}{s_k} \right) + \log \left( \frac{n}{\bar{r}_k^{(s)} \delta} \right) \right). $$

So on $\xi^{(p)}$ this implies that – see Lemma 15

$$ \bar{r}_k^{(d)} \left| \langle \bar{\epsilon}_{\tau_k, +\bar{r}_k^{(s)}}^{(S_k)} - \bar{\epsilon}_{\tau_k, -\bar{r}_k^{(s)}}^{(S_k)}, \bar{\theta}_{\tau_k, +\bar{r}_k^{(s)}}^{(S_k)} - \bar{\theta}_{\tau_k, -\bar{r}_k^{(s)}}^{(S_k)} \rangle \right| \le \frac{\bar{r}_k^{(s)}}{4} \left\| \bar{\theta}_{\tau_k, +\bar{r}_k^{(s)}} - \bar{\theta}_{\tau_k, -\bar{r}_k^{(s)}} \right\|^2. $$
tics, 16(2):117-186, 1945.

[40] Daren Wang, Yi Yu, and Alessandro Rinaldo. Optimal covariance change point localization in high dimension. *arXiv preprint arXiv:1712.09912*, 2017.

[41] Daren Wang, Yi Yu, and Alessandro Rinaldo. Optimal change point detection and localization in sparse dynamic networks. *arXiv preprint arXiv:1809.09602*, 2018.

[42] Daren Wang, Yi Yu, and Alessandro Rinaldo. Univariate mean change point detection: Personalization, CUSUM and optimality. *Electron. J. Stat.*, 14(1):1917-1961, 2020.

[43] Daren Wang, Yi Yu, Alessandro Rinaldo, and Rebecca Willett. Localizing changes in high-dimensional vector autoregressive processes. *arXiv preprint arXiv:1909.06359*, 2019.

[44] Runmin Wang and Xiaofeng Shao. Dating the break in high-dimensional data. *arXiv preprint arXiv:2002.04115*, 2020.

[45] Runmin Wang, Stanislav Volgushev, and Xiaofeng Shao. Inference for change points in high dimensional data. *arXiv preprint arXiv:1905.08446*, 2019.

[46] Tengyao Wang and Richard J. Samworth. High dimensional change point estimation via sparse projection. *J. R. Stat. Soc. Ser. B. Stat. Methodol.*, 80(1):57-83, 2018.

[47] Yihong Wu. Lecture notes for ece598yw: Information-theoretic methods for high-dimensional statistics, 2017.

[48] Mengjia Yu and Xiaohui Chen. Finite sample change point inference and identification for high-dimensional mean vectors. *Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 83(2):247-270, 2021.
Moreover we have on $\xi^{(p)}$ that – see Equation (39)

$$ \left| \frac{\bar{r}_k^{(s)}}{2} \left\| \bar{\bar{\epsilon}}^{(S_k)}_{\tau_k, +\bar{r}_k^{(s)}} - \bar{\bar{\epsilon}}^{(S_k)}_{\tau_k, -\bar{r}_k^{(s)}} \right\|^2 - \sigma^2 s \right| \le 4\bar{c}_{\text{conc}}L^2 \left( \log \left( \frac{n}{\bar{r}_k^{(s)}\delta} \right) + 2s_k \log \left( \frac{2ep}{s_k} \right) \right) \\ \le \sigma^2 x_{\bar{r}_k^{(s)}}^{(p)}, $$

for $\bar{c}_{\text{thresh}} \ge 4\bar{c}_{\text{conc}}$ – note that $\bar{c}_{\text{conc}} > 0$ is a universal constant. And so on $\xi^{(p)}$, combining the three previous displayed equations implies

$$ \Psi_{\tau_k, \bar{r}_k^{(s)}}^{(p)} \ge \frac{\bar{r}_k^{(d)}}{4\sigma^2} \left\| \bar{\bar{\theta}}_{\tau_k, +\bar{r}_k^{(s)}} - \bar{\bar{\theta}}_{\tau_k, -\bar{r}_k^{(s)}} \right\|^2 - x_{\bar{r}_k^{(s)}}^{(p)} \\ \ge \left( \frac{\kappa}{16} - \bar{c}_{\text{thresh}} \right) \frac{L^2}{\sigma^2} \left( \log \left( \frac{n}{\bar{r}_k^{(s)}\delta} \right) + s_k \log \left( \frac{2ep}{s_k} \right) \right) > x_{\bar{r}_k^{(s)}}^{(p)}, $$

since $\kappa > 32\bar{c}_{\text{thresh}}$. And so on $\xi^{(p)}$

$$ T_{\tau_k, \bar{r}_k^{(s)}}^{(p)} = 1. $$

This concludes the proof of the second part of the proposition.

### B.3.3. Proof of Corollary 5

Let $\xi^{(d)}$ and $\xi^{(s)}$ be two events such that Proposition 3 and Proposition 4 both hold with probability $1-3\delta$, and write $\xi = \xi^{(d)} \cap \xi^{(p)}$. From now on, we work on the event $\xi$, which holds with probability $1-6\delta$. Define here simply $\bar{\tau}_k = \tau_k$. Note that by definition of $\bar{r}_k$ in the sub-Gaussian regime:

$$ \bar{r}_k = \begin{cases} \bar{r}_k^{(d)} & \text{if } s_k \log \left( \frac{ep}{s_k} \right) > \sqrt{p \log \left( \frac{n}{r_k \delta} \right)} \\ \bar{r}_k^{(s)} & \text{if } s_k \log \left( \frac{ep}{s_k} \right) \le \sqrt{p \log \left( \frac{n}{r_k \delta} \right)} \end{cases} $$

According to Theorem 1, it is sufficient to prove that $\mathcal{A}(\Theta, T, \mathcal{K}^*, (\bar{\tau}_k, \bar{r}_k)_{k \in \mathcal{K}^*})$ holds.

1. (No false positive): $T_{l,r} = T_{l,r}^{(p)} \lor T_{l,r}^{(d)} = 0$ for any $(l,r) \in \mathcal{G}_F \cap \mathcal{H}_0$. by Proposition 3 and Proposition 4.

2. (Significant change-point detection): for every $k \in \mathcal{K}^*$ (see (20)), we have by definition of $\bar{r}_k$:

$$ 4(\bar{r}_k - 1) \le r_k. $$

Now if $s_k \log \left( \frac{ep}{s_k} \right) \ge \sqrt{p \log \left( \frac{n}{r_k \delta} \right)}$, we have $T_{\bar{\tau}_k, \bar{r}_k}^{(d)} = 1$ by Proposition 3, by definition of $c_0$, and for $\bar{c}_{\text{thresh}}^{(d)}$ as in Proposition 3.
If $s_k \log \left( \frac{ep}{s_k} \right) \le \sqrt{p \log \left( \frac{n}{r_k \delta} \right)}$, we have $T_{\hat{\tau}_k, \hat{r}_k}^{(p)} = 1$ by Proposition 4, by definition of $c_0$, and for $\tilde{c}_{\text{thresh}}^{(p)}$ as in Proposition 4.

Theorem 1 ensures that for all $k \in \mathcal{K}^*$, there exists $k' \in [\hat{K}]$ such that

$$|\hat{\tau}_{k'} - \tau_k| \le \bar{r}_k - 1.$$

This concludes the proof since $4(\bar{r}_k - 1) \le r_k$ for $k \in \mathcal{K}^*$.

### B.4. Proof of Theorem 2

Let us fix $(r, s) \in [1, n/4] \times [1, p]$. Let $\Delta$ be such that

$$r\Delta^2 = \frac{1}{2}\sigma^2 \left[ s \log \left( 1 + u \frac{\sqrt{p}}{s} \sqrt{\log \left( \frac{n}{r} \right)} \right) + u \log \left( \frac{n}{r} \right) \right],$$

for some $u \le \frac{1}{8}$.

In what follows, we consider any change-point detection method that outputs an estimator $\hat{\tau}$ of the change-points, associated to a number $\hat{K}$ of detected change-points, i.e. the length of $\hat{\tau}$. We also write $\mathbb{P}_{\Theta}$ for the distribution of the data when the mean parameter or the time series is fixed to a $n \times p$ matrix $\Theta$, i.e. of $\Theta + \varepsilon$ where the noise entries $(\varepsilon_t)_j$ are i.i.d. and follow $\mathcal{N}(0, \sigma^2)$ as in Section 3. Also abusing slightly notations, we write $\mathbb{P}_0$ for the distribution of the data when the parameter is constant and equal to 0.

Consider also any prior $\pi$ over the set of $n \times p$ matrices $\Theta$ such that the number of true change-points over the support of the prior is larger than 1 – i.e. the prior puts mass only on problems where more than one change-point occurs. Let $\mathbb{P}_{\pi}$ be the corresponding distribution of the data, namely the distribution of the matrix of data when the mean parameter of the time series is the random matrix $\hat{\Theta} \sim \pi$. Otherwise said, $\mathbb{P}_{\pi}$ is the distribution of $\hat{\Theta} + \varepsilon$ where $\hat{\Theta} \sim \pi$.

We remind that in our setting $K$ is the number of true change-points in a given problem – which would be either 0 under $\mathbb{P}_0$, or more than 1 under $\mathbb{P}_{\pi}$. If the support of $\pi_1$ is included in $\mathcal{P}(r, s)$, then

$$ \begin{aligned} \sup_{\Theta \in \mathcal{P}(r,s)} \mathbb{P}_{\Theta}(\hat{K} \neq K) &\ge \frac{1}{2} (\mathbb{P}_{\pi}(\hat{K} = 0) + \mathbb{P}_{0}(\hat{K} \neq 0)) \\ &\ge \frac{1}{2} (1 - d_{TV}(\bar{\mathbb{P}}_{\pi}, \mathbb{P}_{0})), \end{aligned} \quad (40) $$

where $d_{TV}$ is the total variation distance. From the Cauchy-Schwarz inequality, we have

$$d_{TV}(\bar{\mathbb{P}}_{\pi}, \mathbb{P}_{0}) \le \frac{1}{2} \sqrt{\chi^2(\bar{\mathbb{P}}_{\pi}, \mathbb{P}_{0})}, \quad (41)$$

where $\chi^2$ is the divergence between probability distributions:

$$\chi^2(\bar{\mathbb{P}}_{\pi}, \mathbb{P}_0) = \mathbb{E}_{\mathbb{P}_0} \left[ \left( \frac{d\bar{\mathbb{P}}_{\pi}}{d\mathbb{P}_0} - 1 \right)^2 \right].$$
By a simple computation that can be found for example in [47]

$$ \chi^2(\bar{\mathbb{P}}_{\pi}, \mathbb{P}_0) = \mathbb{E}_{\tilde{\Theta}, \tilde{\Theta}'} \left[ e^{\frac{1}{\sigma^2} \langle \tilde{\Theta}, \tilde{\Theta}' \rangle} \right] - 1, \quad (42) $$

where $\tilde{\Theta}$ and $\tilde{\Theta}'$ are i.i.d. and distributed according to $\pi$, $\langle \tilde{\Theta}, \tilde{\Theta}' \rangle = \text{Tr}(\tilde{\Theta}' \tilde{\Theta}^T)$ is the standard scalar product, and $\mathbb{E}_{\tilde{\Theta}, \tilde{\Theta}'}$ is the expectation according to $\tilde{\Theta}$ and $\tilde{\Theta}'$.

Let us consider the three following cases for the couple $(r, s)$:

**Case 1:** $u \log\left(\frac{n}{r}\right) \le s \log\left(1 + u \frac{\sqrt{p}}{s} \sqrt{\log\left(\frac{n}{r}\right)}\right)$ and $s \le u \sqrt{p \log\left(\frac{n}{r}\right)}$,

**Case 2:** $u \log\left(\frac{n}{r}\right) \le s \log\left(1 + u \frac{\sqrt{p}}{s} \sqrt{\log\left(\frac{n}{r}\right)}\right)$ and $s > u \sqrt{p \log\left(\frac{n}{r}\right)}$,

**Case 3:** $u \log\left(\frac{n}{r}\right) > s \log\left(1 + u \frac{\sqrt{p}}{s} \sqrt{\log\left(\frac{n}{r}\right)}\right)$.

Each case corresponds to the regime of detection of one of the three statistics. The first one corresponds to the Berk-Jones statistic, the second one to the dense statistic and the last one to the partial norm statistic.

**Case 1:** In that case, $r\Delta^2 \le \sigma^2 s \log(4u \frac{p}{s^2} \log(\frac{n}{r}))$. Let us define a probability distribution on the parameter $\Theta \in \mathcal{P}(r, s)$. For $\zeta = \lfloor \frac{n}{r} \rfloor - 1$ and $l \in \tilde{D}_r = \{1, r+1, 2r+1, \dots, \zeta r+1\}$, define the column vector $v_l = \sum_{j=l}^{l+r-1} e_j$, where $e_j$ is the $j^{th}$ element of the canonical basis of $\mathbb{R}^n$. Let $a$ be a random variable uniformly distributed in $\{x \in \{0, 1\}^p, |x|_0 = s\}$ and $\nu$ be a random variable independent from $a$ and uniformly distributed on $\{v_l : l \in \tilde{D}_r\}$. Let

$$ \tilde{\Theta}_{(1)} = \frac{\Delta}{\sqrt{s}} a \nu^T \in \mathbb{R}^{p \times n}, $$

and $\pi_1$ be the distribution of the random variable $\tilde{\Theta}_{(1)}$, and $\bar{\mathbb{P}}_{\pi_1}$ be the corresponding distribution of the data.

Consider two independent copies $\tilde{\Theta}_{(1)}$ and $\tilde{\Theta}'_{(1)}$ that are distributed like $\pi_1$. The probability that $\tilde{\Theta}_{(1)}$ and $\tilde{\Theta}'_{(1)}$ have the same support is exactly $\frac{1}{\zeta+1}$. Hence, from Equation (42)

$$ \chi^2(\bar{\mathbb{P}}_{\pi_1}, \mathbb{P}_0) = \frac{1}{\zeta+1} \left( \mathbb{E}_{a,a'} \left[ e^{\frac{r\Delta^2}{s\sigma^2} \langle a,a' \rangle} - 1 \right] \right), \quad (43) $$

where $a'$ is an independent copy of $a$, and $\mathbb{E}_{a,a'}$ is the expectation according to $a, a'$. Remark by symmetry that $\langle a, a' \rangle$ has the same law as $\sum_{i=1}^s a_i$. Hence

$$ \mathbb{E}_{a,a'} \left[ e^{\frac{r\Delta^2}{s\sigma^2} \langle a,a' \rangle} \right] = \mathbb{E}_a \left[ e^{\frac{r\Delta^2}{s\sigma^2} \sum_{i=1}^s a_i} \right], $$
where $\mathbb{E}_a$ is the expectation according to a.

Remark that $(a_1, \dots, a_p)$ has the same distribution as a random sampling without replacement of the list of length $p$ containing $(1, \dots, 1, 0, \dots, 0)$ – the list containing exactly $s$ times the quantity 1 and otherwise only 0. The following lemma allows us to replace the variables $a_i$ by independent Bernoulli random variables $Z_i \sim \mathcal{B}(s/p)$.

**Lemma 17.** Let $c = (c_1, \dots, c_p) \in \mathbb{R}^p$. We associate to the list $c$ two random sampling processes: (i) the sampling process without replacement $(X_i)_{i=1\dots s}$ of $s$ elements uniformly on the list $c$ and (ii) the sampling process with replacement $(Z_i)_{i=1\dots s}$ of $s$ elements uniformly in the list. Then for any convex function $f$,

$$ \mathbb{E} \left[ f \left( \sum_{i=1}^{s} X_i \right) \right] \le \mathbb{E} \left[ f \left( \sum_{i=1}^{s} Z_i \right) \right]. $$

The proof of this lemma can be found in [19]. Thus, if $(Z_i)_{i=1\dots s}$ is an i.i.d sequence of Bernoulli variables with parameter $\frac{s}{p}$ as described above, we obtain

$$ \chi^2(\mathbb{P}_{\pi_1}, \mathbb{P}_0) \le \frac{1}{\zeta+1} \left( \mathbb{E}_Z \left[ e^{\frac{r\Delta^2}{s\sigma^2} \sum_{i=1}^s Z_i} \right] - 1 \right) \quad (44) $$

$$ = \frac{1}{\zeta+1} \left[ \left( \frac{s}{p} e^{\frac{r\Delta^2}{s\sigma^2}} + 1 - \frac{s}{p} \right)^s - 1 \right] \le \frac{1}{\zeta+1} \left[ e^{\frac{s^2}{p} \left( e^{\frac{r\Delta^2}{s\sigma^2}} - 1 \right)} - 1 \right] $$

$$ \le 2 \frac{r}{n} e^{\frac{s^2}{p} \left( e^{\log(4u^2 \frac{p}{s^2} \log(\frac{n}{r}))} \right)} \le 2 \left( \frac{r}{n} \right)^{1-4u^2} \le 1, \quad (45) $$

where $\mathbb{E}_Z$ is the expectation according to the $(Z_i)_i$ and where in the last inequality we used $u \le 1/3$ and $n \ge 4r$.

**Case 2:** In that case, $r\Delta^2 \le \sigma^2 u \sqrt{p \log(\frac{n}{r})}$. Let $s_0 = \left\lfloor u \sqrt{p \log(\frac{n}{r})} \right\rfloor$ and $b$ be a random variable uniformly distributed in $\{x \in \{0, 1\}^p, |x|_0 = s_0\}$ and $\nu$ be defined as in **Case 1**. Let

$$ \tilde{\Theta}_{(2)} = \frac{\Delta}{\sqrt{p}} b \nu^T, $$

let $\pi_2$ be the distribution of $\tilde{\Theta}_{(2)}$ and $\mathbb{P}_{\pi_2}$ be the associated probability distribution of the data. Doing the same reasoning and similar computations as for **Case 1**, see in particular the steps of Equations (43) and (44) – replacing $s$ by $s_0$ and $a$ by $b$ – we have

$$ \begin{aligned} \chi^2(\mathbb{P}_{\pi_2}, \mathbb{P}_0) &= \mathbb{E}_{\tilde{\Theta}_{(2)}, \tilde{\Theta}'_{(2)}} \left[ e^{\frac{1}{s\sigma^2} \langle \tilde{\Theta}_{(2)}, \tilde{\Theta}'_{(2)} \rangle} \right] - 1 \\ &= \frac{1}{\zeta+1} \mathbb{E}_{b,b'} \left[ e^{\frac{r\Delta^2}{p\sigma^2} \langle b,b' \rangle} - 1 \right] \le \frac{1}{\zeta+1} \left[ e^{\frac{s_0^2}{p} \left( e^{\frac{r\Delta^2}{s_0\sigma^2}} - 1 \right)} - 1 \right] \\ &\le \frac{1}{\zeta+1} e^{2 \frac{s_0 r \Delta^2}{p \sigma^2}} \le 2 \frac{r}{n} e^{4u \log \frac{n}{r}} = 2 \left( \frac{r}{n} \right)^{1-4u} \le 1, \end{aligned} $$
where $\mathbb{E}_{\tilde{\Theta}_{(2)}, \tilde{\Theta}'_{(2)}}$ is the expectation according to $\tilde{\Theta}_{(2)}, \tilde{\Theta}'_{(2)}$ (where $\tilde{\Theta}'_{(2)}$ is an independent copy of $\tilde{\Theta}_{(2)}$) and where $\mathbb{E}_{b, b'}$ is the expectation according to $b, b'$ (where $b'$ is an independent copy of $b$), and where in the last step we used $u \le 1/8$ and $n \ge 4r$.

**Case 3:** In that case, $r\Delta^2 \le u \log(\frac{n}{r})$. Let $c = (1, 0, 0, \dots, 0)$ be the vector with 0 entries except the first one. Let $\nu$ be the random vector defined as in **Case 1**. Let

$$ \tilde{\Theta}_{(3)} = \Delta c \nu^T, $$

and $\pi_3$ be the distribution of the random variable $\tilde{\Theta}_{(3)}$ – and $\bar{\mathbb{P}}_{\pi_3}$ be the associated probability distribution of the data. Doing the same reasoning as in **Case 1** – see in particular the step of Equation (43) – replacing $a$ by $c$ and $s$ by $1$ – for the prior $\pi_3$, we obtain

$$ \begin{aligned} \chi^2(\bar{\mathbb{P}}_{\pi_3}, \mathbb{P}_0) &= \mathbb{E}_{\tilde{\Theta}_{(3)}, \tilde{\Theta}'_{(3)}} \left[ e^{\frac{1}{\sigma^2} \langle \tilde{\Theta}_{(3)}, \tilde{\Theta}'_{(3)} \rangle} \right] - 1 \\ &= \frac{1}{\zeta + 1} e^{\frac{r\Delta^2}{\sigma^2}} \\ &\le 2 \frac{r}{n} e^{u \log(\frac{n}{r})} \\ &\le 2 \left(\frac{r}{n}\right)^{1-u} \le 1, \end{aligned} \quad (46) $$

where $\mathbb{E}_{\tilde{\Theta}_{(3)}, \tilde{\Theta}'_{(3)}}$ is the expectation according to $\tilde{\Theta}_{(3)}, \tilde{\Theta}'_{(3)}$ (where $\tilde{\Theta}'_{(3)}$ is an independent copy of $\tilde{\Theta}_{(3)}$) and where in the last step we used $n \ge 4r$ and $u \le 1/2$.

Thus, in all cases – combining Equations (40) and (41) with Equations (45), (46) and (46) – we obtain in all three cases

$$ \sup_{\Theta \in \mathcal{P}(r,s)} \mathbb{P}_{\Theta}(\hat{K} \neq K) \ge \frac{1}{4}. $$

and this concludes the proof.

### B.5. Proofs for covariance and nonparametric change-point detection

*Proof of Proposition 5.* Consider an $r$-sample $(z_1, \dots, z_r)$ with mean zero and covariance matrix $\Sigma$ and Orlicz norm $B$. Koltchinskii and Lounici [23] have proved that, for any $x > 0$, the empirical covariance matrix $\hat{\Sigma} = r^{-1}(\sum_{i=1}^r z_i z_i^T)$ satisfies

$$ \|\hat{\Sigma} - \Sigma\|_{op} \le c'B^2 \left[ \sqrt{\frac{p}{r}} + \frac{p}{r} + \sqrt{\frac{x}{r}} + \frac{x}{r} \right], $$

with probability higher than $1 - \exp(-x)$. Here $c'$ is a suitable positive constant. Considering a union bound over all $(l, r) \in \mathcal{G}_D$ such that $\Sigma_t$ is constant over
$[l-r, l+r)$, we have, with probability higher than $1 - \delta/2$, that simultaneously on all such $r \in \mathcal{R}$ and $l \in \mathcal{D}_r$,

$$ \begin{aligned} \|\hat{\Sigma}_{l,r} - \hat{\Sigma}_{l,-r}\|_{op} &\le \|\hat{\Sigma}_{l,r} - \Sigma_l\|_{op} + \|\hat{\Sigma}_{l,-r} - \Sigma_l\|_{op} \\ &\le 8c'B^2 \left[ \sqrt{\frac{p}{r}} + \frac{p}{r} + \sqrt{\frac{\log(2n/(r\delta))}{r}} + \frac{\log(2n/(r\delta))}{r} \right], \end{aligned} $$

where the constant 8 comes from the union bound on all elements of the grid. As a consequence, the FWER of the multiple testing collection is at most $\delta/2$ provided that we choose $c_0 \le 8c'$.

Conversely, consider any high-energy change-point $\tau_k$. Let $\bar{r}_k$ be the smallest radius $r \in \mathcal{R}$ such that

$$ r \|\Sigma_{\tau_k} - \Sigma_{\tau_{k-1}}\|_{op}^2 \ge 0.25c_1 B^4 \left[ \left( p + \log \left( \frac{2n}{r\delta} \right) \right) \wedge r \right]. \quad (47) $$

and consider the closest location $l \in \mathcal{D}_r$ of $\tau_k$ so that $|l - \tau_k| \le r/2$. To ease the notation, we still write $r$ for $\bar{r}_k$. Without loss of generality, we assume that $l \ge \tau_k$. Let us decompose the statistic $\hat{\Sigma}_{l,-r} = \frac{r-l+\tau_k}{r} \hat{\Sigma}_{\tau_k,-(r-l+\tau_k)} + \frac{l-\tau_k}{r} \hat{\Sigma}_{l,-(l-\tau_k)}$. Since $r \le r_k/2$, $\Sigma_l$ is constant over $[l-r, \tau_k)$ and over $[\tau_k, l+r)$. Then, we apply three times the deviation inequality of Koltchinskii and Lounici [23] to get

$$ \begin{aligned} \|\hat{\Sigma}_{l,r} - \hat{\Sigma}_{l,-r}\|_{op} &\ge \frac{r-l+\tau_k}{r} \|\Sigma_{\tau_k} - \Sigma_{\tau_{k-1}}\|_{op} - \|\hat{\Sigma}_{l,r} - \Sigma_{\tau_k}\|_{op} \\ &\quad - \frac{l-\tau_k}{r} \|\hat{\Sigma}_{l,-(l-\tau_k)} - \Sigma_{\tau_k}\|_{op} - \frac{r-l+\tau_k}{r} \|\hat{\Sigma}_{\tau_k,-(r-l+\tau_k)} - \Sigma_{\tau_{k-1}}\|_{op} \\ &\ge \frac{1}{2} \|\Sigma_{\tau_k} - \Sigma_{\tau_{k-1}}\|_{op} - c'' B^2 \left[ \sqrt{\frac{p}{r}} + \frac{p}{r} + \sqrt{\frac{\log(2n/(r\delta))}{r}} + \frac{\log(2n/(r\delta))}{r} \right], \end{aligned} $$

with probability higher than $1 - 0.5\delta[r/(2n)]^2$. As a consequence, we have $T_{l,r} = 1$ provided that

$$ \|\Sigma_{\tau_k} - \Sigma_{\tau_{k-1}}\|_{op} \ge 2(c'' + c_0)B^2 \left[ \sqrt{\frac{p}{r}} + \frac{p}{r} + \sqrt{\frac{\log(2n/(r\delta))}{r}} + \frac{\log(2n/(r\delta))}{r} \right]. $$

Since $\|\Sigma_{\tau_k} - \Sigma_{\tau_{k-1}}\|_{op} \le 2B^2$ and if we choose $c_1 \ge 17 \vee 32(c'' + c_0)$, the bound (47) is achievable only if $r \ge p + \log(2n/(r\delta))$ and we deduce from (47) that $T_{l,r} = 1$.

Taking a union bound over all high-energy change-points, we deduce from Theorem 1 that, with probability higher than $1 - \delta$, $\hat{\tau}$ achieves (NoSp) and detects all high-energy change-points. Besides, the localization error (25) is a consequence of the definition (47) together with Theorem 1. $\square$

*Proof of Proposition 6.* As in the proof of Theorem 2, we only consider a specific setting where one aims at testing $K = 0$ with $\Sigma_1 = I_p$ versus $K = 2$ with $\tau_1 \in$
(n/4; 3n/4), $\tau_2 = \tau_1 + r$, $\Sigma_1 = \Sigma_{\tau_2} = I_p$ and $\Sigma_{\tau_1} = I_p + \zeta uu^T$ for some unknown unit vector $u$ in $\mathbb{R}^p$. Obviously, we have $r_1 = r_2 = r$ and $\|\Sigma_{\tau_1} - \Sigma_{\tau_0}\|_{op} = \|\Sigma_{\tau_2} - \Sigma_{\tau_1}\|_{op} = \zeta$ so that it suffices to prove that the sum of the type I and type II error probabilities of any test of these hypotheses is bounded away from zero. We consider two subcases:

**Case 1:** $\zeta \le c'\sqrt{p/r} \wedge \frac{1}{\sqrt{2}}$. Then, we focus on the specific alternative hypothesis where $\tau_1 = \lfloor n/2 \rfloor$ and $\tau_2 = \tau_1 + r$, so that the problem reduces exactly to testing whether the covariance matrix $\Sigma$ of a $r$-sample satisfies $\Sigma = I_p$ or whether $\Sigma = I_p + \zeta uu^T$. This hypothesis testing problem for covariance matrices is well understood. In particular, one can deduce from Theorem 5.1 in [3] that, as soon as $\zeta \le c'[\sqrt{p/r} \wedge 1]$, for some $c'$ sufficiently small, one has

$$ \inf_{\hat{\tau}} \sup_{\Theta \in \mathcal{P}(r, \zeta)} \mathbb{P}_{\Theta}(\hat{K} \neq K) \geq \frac{1}{4}. $$

**Case 2:** $\zeta \le c'\sqrt{\log(n/r)/r} \wedge 1/\sqrt{2}$. Here, we consider another specific class of alternative hypotheses where we fix $u = (1, 0, \dots, 0)$ but $\tau_1$ can take different values, i.e. $\tau_1 \in \{\lfloor n/4 \rfloor, \lfloor n/4 \rfloor + r, \dots, \lfloor n/4 \rfloor + r \lfloor n/2r \rfloor \}$. It turns out that this is equivalent to a univariate variance testing problem where one observes $q = \lfloor n/(2r) \rfloor$ samples of size $r$ with distributions $\mathcal{N}(0, \sigma_1^2)$, $\dots$, $\mathcal{N}(0, \sigma_q^2)$. Under the null, we have $\sigma_1 = \sigma_2 = \dots = \sigma_q = 1$. Under the alternative, for some $j \in [q]$, we have $\sigma_j = \sqrt{1+\zeta}$ and $\sigma_l = 1$ for $l \neq j$. For $j = 1, \dots, q$, write $\mathbb{P}_j$ for the distributions of the $j$-th sample of size $r$ when $\sigma_j^2 = 1 + \zeta$ and $\sigma_l = 1$ for $l \neq j$. Besides, we write $L_j$ for the corresponding likelihood ratio with the null distribution $\mathbb{P}_0$. Then, the mixture distribution is defined as $\bar{\mathbb{P}} = \frac{1}{q} \sum_{j=1}^q \mathbb{P}_j$ whereas $\bar{L}$ stands for the mean likelihood ratio. Following the classical path of Le Cam's method we obtain that, for any test $T$,

$$ \mathbb{P}_0[T=1] + \sup_{j=1,\dots,q} \mathbb{P}_j[T=0] \ge \mathbb{P}_0[T=1] + \bar{\mathbb{P}}[T=0] \ge 1 - \|\mathbb{P}_0 - \bar{\mathbb{P}}\|_{TV}, $$

where $\|\cdot\|_{TV}$ is the total variation norm. Using Cauchy-Schwarz inequality, we bound this total variation distance between the covariates

$$ \begin{aligned} \| \mathbb{P}_0 - \bar{\mathbb{P}} \|_{TV} &\le \mathbb{E}_0 [\bar{L}^2] - 1 \\ &= \frac{1}{q} (\mathbb{E}_0 [L_i^2] - 1) \\ &= \frac{1}{q} [(1 - \zeta^2)^{-r/2} - 1] \le \frac{1}{q} [e^{r\zeta^2} - 1], \end{aligned} $$

since $\zeta \in (0, 1/2)$. As a consequence, we derive that $\|\mathbb{P}_0 - \bar{\mathbb{P}}\|_{TV} \le 1/4$ as long as $r\zeta^2 \le c' \log(q) \wedge 1$. The result follows. $\square$

*Proof of Proposition 7.* The proof is based on an application of
Dvoretzky-Kiefer-Wolfowitz (DKW) inequality [4] together with a union bound. For a $q$ sample of a univariate distribution with empirical distribution function $\hat{F}$ and true distribution function $F$, DKW inequality ensures that

$$ \mathbb{P}\left[\|\hat{F} - F\|_{\infty} \ge \sqrt{\frac{x}{2q}}\right] \le 2e^{-x}. $$

Applying two-times DKW inequality to each statistic $T_{l,r}$ such that no-change-point occurs on $(l-r, l+r)$, we deduce that, setting $c_1$ sufficiently larger, the FWER of $(T_{l,r})$ is at most $\delta/2$ by summing the probabilities over all scales $r \in \mathcal{R}$ and by a union bound on all $l \in \mathcal{D}_r$.

Turning to the high-energy change points, we consider $\tau_k$ satisfying (26). Let $\bar{r}_k$ be the smallest radius $r \in \mathcal{R}$ such that

$$ r \|F_{\tau_k} - F_{\tau_{k-1}}\|_{\infty}^{2} \geq 0.25c_1 \frac{\log\left(\frac{n}{r\delta}\right)}{m}, \quad (48) $$

and consider the closest location $l \in \mathcal{D}_r$ of $\tau_k$ so that $|l - \tau_k| \le r/2$ and $2r \le r_k$. To ease the notation, we still write $r$ for $\bar{r}_k$. As in the proof of Proposition 5, we decompose the statistic

$$ \sum_{t=l}^{l+r-1} \hat{F}_t - \sum_{t=l-r}^{l-1} \hat{F}_t = \sum_{t=l}^{l+r-1} \hat{F}_t - \sum_{t=l-r}^{\tau_k-1} \hat{F}_t - \sum_{t=\tau_k}^{l-1} \hat{F}_t, $$

and apply DKW inequality to each of three sums. Taking the union bound over all possible $T_{l,r}$ we deduce that, with probability higher than $1 - \delta/2$

$$ r^{-1} \left\| \sum_{t=l}^{l+r-1} \hat{F}_t - \sum_{t=l-r}^{l-1} \hat{F}_t \right\|_{\infty} \geq \frac{1}{2} \left\| F_{\tau_k} - F_{\tau_{k-1}} \right\|_{\infty} - c'' \sqrt{\frac{\log(4n/r\delta)}{mr}}, $$

so that in view of Condition (48) implies that $T_{l,r} = 1$. Applying Theorem 1 allows us to conclude. $\square$

**Proof of Proposition 8.** As in the proof of Proposition 6, we focus on a simpler testing problem. Write $U$ for the cumulative distribution function of the uniform distribution on $[0, 1]$, i.e. $U(x) = x$ for any $x \in [0, 1]$. Given $\zeta \in (0, 1/4)$, we define the cumulative distribution function $U_\zeta$ by $U_\zeta(x) = (1 + 2\zeta)x$ for $x \in [0, 1/2]$ and $U_\zeta(x) = (1/2 + \zeta) + (1 - 2\zeta)(x - 1/2)$ for $x \in [1/2, 1]$. Note that $\|U_\zeta - U\|_\infty = \zeta$.

We focus on a testing problem where, under the null, $F_t = U$ for all $t = 1, \dots, n$, whereas under the alternative there exists

$\tau_1 \in \{\lfloor n/4 \rfloor, \lfloor n/4 \rfloor + r, \dots, \lfloor n/4 \rfloor + (r-1)\lfloor n/(2r) \rfloor\}$ such that $F_t = U_\zeta$ for $t = \tau_1, \dots, \tau_1 + r - 1$ and $F_t = U$ otherwise. Defining $q = \lfloor n/(2r) \rfloor$, we observe that this amounts to testing whether $q$ samples of size $rm$ are distributed according to the null distribution or whether exactly one of them is distributed according to $U_\zeta$. Arguing again in the proof of Proposition 6, we only need to bound the total variation distance between the distribution $\mathbb{P}_0$ under the
null and the mixture distribution $q^{-1} \sum_{j=1}^{q} \mathbb{P}_j$ of the $q$ possible alternatives – here $\mathbb{P}_0 = \otimes_{k=1}^{q} U^{\otimes(rm)}$ is the distribution of the samples when $F_t = U$ and $\mathcal{P}_j = [\otimes_{k=1}^{j-1} U^{\otimes(rm)}] \otimes U_{\zeta}^{\otimes(rm)} \otimes [\otimes_{k=j+1}^{q} U^{\otimes(rm)}]$, is for $j \ge 1$ the distribution of the samples when $F_t = U$ except for $t \in [jr, (j+1)r)$, in which case $F_t = U_{\zeta}$.

Let $z$ be a uniform random variable over $[0, 1]$ and $w$ be an independent Bernoulli random variable with parameter $1/2$. Then, one easily checks that $z/2 + w/2$ is uniformly distributed on $[0, 1]$. If $w$ is a Bernoulli random variable with parameter $1/2 - 2\zeta$, then one easily checks that the cumulative distribution function of $z/2 + w/2$ is $F_{\zeta}$. As a consequence, by a standard data-processing inequality [47], one derives that

$$ \|\mathbb{P}_0 - q^{-1} \sum_{j=1}^{q} \mathbb{P}_j\|_{TV} \le \|\tilde{\mathbb{P}}_0 - q^{-1} \sum_{j=1}^{q} \tilde{\mathbb{P}}_j\|_{TV}, $$

where under $\tilde{\mathbb{P}}_0$ one observes $q$ independent Binomial random variables with parameter $(mr, 1/2)$, whereas under $\mathbb{P}_j$, the $j$-th observation follows a Binomial distribution with parameter $(mr, 1/2 - 2\zeta)$. Using Cauchy-Schwarz inequality, we upper bound the square of the total variation distance by the $\chi^2$ distance and then compute it explicitly. This leads us to

$$ \|\tilde{\mathbb{P}}_0 - q^{-1} \sum_{j=1}^{q} \tilde{\mathbb{P}}_j\|_{TV}^2 \le \frac{1}{q} [(1 + 16\zeta^2)^{rm} - 1], $$

which is smaller than $1/4$ provided that $16rm\zeta^2 \le \log(q/4 + 1)$. If we choose $c'$ small enough in the statement of the proposition, this last condition holds and the result follows. $\square$

## Acknowledgments

The authors are grateful to two anonymous referees for their helpful comments that improved the presentation of the manuscript.

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