SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \max  _ {\alpha + \rho <   t <   \beta - \rho} | | \widetilde {F} _ {t, h} ^ {(\alpha , \beta ]} | | _ {L _ {2}} \leq \max  _ {\alpha <   t <   \beta} | | \widetilde {f} _ {t} ^ {(\alpha , \beta ]} | | _ {L _ {2}} + \lambda (17) \\ \leq \sqrt {e - \eta_ {k}} \kappa_ {k + 1} + \sqrt {\eta_ {k} - s} \kappa_ {k} + \lambda (18) \\ \leq 2 \sqrt {C} \log^ {\frac {1}{2}} (T) \sqrt {T ^ {\frac {p}{2 r + p}}} + \lambda (19) \\ \end{array}
$$
where the second inequality is followed by Lemma 7. Therefore in the good event in Equation (10), for any  $\mathcal{I} = (\alpha, \beta] \subseteq (s, e]$ , it holds that
$$
a _ {\mathcal {I}} = \max _ {t = \alpha + \rho} ^ {\beta - \rho} | | \widetilde {F} _ {t, h} ^ {(\alpha , \beta ]} | | _ {L _ {2}} \leq 2 \sqrt {C} \log^ {\frac {1}{2}} (T) \sqrt {T ^ {\frac {p}{2 r + p}}} + \lambda ,
$$
$$
\begin{array}{l} 2 \sqrt {C} \log^ {\frac {1}{2}} (T) \sqrt {1 + T ^ {\frac {p}{2 r + p}}} + \lambda \\ = 2 \sqrt {C} \log^ {\frac {1}{2}} (T) \sqrt {\frac {1}{h ^ {p}} + 1} + 2 C \sqrt {\frac {\log T}{h ^ {p}}} + \frac {2 C _ {1} \sqrt {p}}{\sqrt {h ^ {p}}} + 2 C _ {2} \sqrt {T} h ^ {r}. \\ \end{array}
$$
We observe that  $\sqrt{\frac{\log(T)}{h^p}} = O\Big(\log (T)^{1 / 2}\sqrt{\frac{1}{h^p}}\Big)$ . Moreover,
$$
\sqrt {T} h ^ {r} = \sqrt {T} \left(\frac {1}{T}\right) ^ {\frac {r}{2 r + p}} \leq \left(T ^ {\frac {1}{2} - \frac {r}{2 r + p}}\right),
$$
and given that,
$$
\frac {1}{2} - \frac {r}{2 r + p} = \frac {p}{2 (2 r + p)}
$$
we get,
$$
\sqrt {T} h ^ {r} = o \left(\log^ {\frac {1}{2}} (T) \sqrt {\frac {1}{h ^ {p}}}\right).
$$
Therefore, by the choice of  $\tau$ , we will always correctly reject the existence of undetected change points, since
$$
2 \sqrt {C} \log^ {\frac {1}{2}} (T) \sqrt {T ^ {\frac {p}{2 r + p}}} + \lambda \leq \tau .
$$
Thus, by the choice of  $\tau$ , it holds that with sufficiently large constant  $C_{\tau}$
$$
a _ {\mathcal {I}} \leq \tau \quad \text {f o r a l l} \quad \mathcal {I} \subseteq (s, e ]. \tag {20}
$$
As a result, MNSBS  $((s,e],h,\tau)$  will correctly reject if  $(s,e]$  contains no undetected change points.
Step 4. Assume that there exists an undetected change point  $\eta_{\widetilde{k}} \in (s, e]$  such that
$$
\min \{\eta_ {\widetilde {k}} - s, \eta_ {\widetilde {k}} - e \} = \frac {3}{4} \Delta .
$$
Then,  $\mathcal{M}^{s,e}\neq \emptyset$ . Let  $\mathcal{I}^*$  be defined as in MNSBS  $((s,e],h,\tau)$  with
$$
\mathcal {I} ^ {*} = (\alpha^ {*}, \beta^ {*} ].
$$
To complete the induction, it suffices to show that, there exists a change point  $\eta_k \in (s, e]$  such that  $\min \{\eta_k - s, \eta_k - e\} \geq \frac{3}{4}\Delta$  and  $|b_{\mathcal{I}^*} - \eta_k| \leq \Upsilon_k$ . To this end, we consider the collection of change points of  $\{f_t\}_{t \in (\alpha^*, \beta^*]}$ . We are to ensure that the assumptions of Lemma 12 are satisfied. In the following,  $\lambda$  is used in Lemma 12. Then Equation (68) and Equation (69) are directly consequence of Equation (10), Equation (11), Equation (12). By the narrowest of  $\mathcal{I}^*$ ,
$$
\left| \mathcal {I} ^ {*} \right| \leq \left| \mathcal {I} _ {k} \right| \leq \Delta ,
$$
and by Step 1 with  $\mathcal{I}_k = (s_k, e_k]$ , it holds that
$$
\min  \left\{\eta_ {k} - s _ {k}, e _ {k} - \eta_ {k} \right\} \geq \frac {1}{1 6} \zeta_ {k} \geq c _ {2} \Delta ,
$$
Therefore for all  $k\in \{\widetilde{k}:\min \{\eta_{\widetilde{k}} - s,e - \eta_{\widetilde{k}}\} \geq c_2\Delta \}$
$$
\max_{t = \alpha^{*} + \rho}^{t = \beta^{*} - \rho}||\widetilde{F}_{t,h}^{(\alpha^{*},\beta^{*}]}||_{L_{2}}\geq \max_{t = s_{k} + \rho}^{t = e_{k} - \rho}||\widetilde{F}_{t,h}^{(s_{k},e_{k}]}||_{L_{2}}\geq c_{1}\sqrt{\Delta}\kappa_{k},
$$
where the last inequality follows from Equation (13). Therefore (70) holds in Lemma 12. Finally, Equation (71) is a direct consequence of the choices that
$$
h = C _ {h} (T) ^ {\frac {- 1}{2 r + d}} \quad \text {a n d} \quad \rho = \frac {\log (T)}{n h ^ {d}}.
$$
Thus, all the conditions in Lemma 12 are met. So that, there exists a change point  $\eta_{k}$  of  $\{f_t\}_{t\in \mathcal{I}^*}$ , satisfying
$$
\min  \left\{\beta^ {*} - \eta_ {k}, \eta_ {k} - \alpha^ {*} \right\} > c \Delta , \tag {21}
$$
$$
\begin{array}{l} \left| b _ {\mathcal {I} ^ {*}} - \eta_ {k} \right| \leq \max  \left\{C _ {3} \lambda^ {2} \kappa_ {k} ^ {- 2}, \rho \right\} \leq C _ {4} \log (T) \left(\frac {1}{h ^ {p}} + T h ^ {2 r}\right) \kappa_ {k} ^ {- 2} \\ \leq C \log (T) \left(T ^ {\frac {p}{2 r + p}}\right) \kappa_ {k} ^ {- 2} \\ \end{array}
$$
for sufficiently large constant  $C$ , where we have followed the same line of arguments as for the conclusion of (20). Observe that
i) The change points of  $\{f_t\}_{t\in \mathcal{I}^*}$  belong to  $(s,e]\cap \{\eta_k\}_{k = 1}^K$  ; and
ii) Equation (21) and  $(\alpha^{*},\beta^{*}]\subseteq (s,e]$  imply that
$$
\min  \left\{e - \eta_ {k}, \eta_ {k} - s \right\} > c \Delta \geq \Upsilon_ {\max }
$$
As discussed in the argument before Step 1, this implies that  $\eta_{k}$  must be an undetected change point of  $\{f_t\}_{t\in \mathcal{I}^*}$ .
E Proof of Theorem 2
In this section, we present the proof of theorem Theorem 2.
Proof of Theorem 2. Uniform tightness of  $\kappa_k^{2 + \frac{p}{r}}\left|\widetilde{\eta}_k - \eta_k\right|$ . Here we show a.1 and b.1. For this purpose, we will follow a series of steps. On step 1, we rewrite (6) in order to derive a uniform bound. Step 2 analyses the lower bound while Step 3 the upper bound.
Step 1: Denote  $\widetilde{r} = \widetilde{\eta}_k - \eta_k$ . Without loss of generality, suppose  $\widetilde{r} \geq 0$ . Since  $\widetilde{\eta}_k = \eta_k + \widetilde{r}$ , defined in (6), is the minimizer of  $\widehat{Q}_k(\eta)$ , it follows that
$$
\widehat {Q} _ {k} \left(\eta_ {k} + \widetilde {r}\right) - \widehat {Q} _ {k} \left(\eta_ {k}\right) \leq 0.
$$
$$
Q ^ {*} (\eta) = \sum_ {t = s _ {k} + 1} ^ {\eta} \left\| F _ {t, h _ {2}} - f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} ^ {2} + \sum_ {t = \eta + 1} ^ {e _ {k}} \left\| F _ {t, h _ {2}} - f _ {\left(\eta_ {k}, e _ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} ^ {2}, \tag {22}
$$
$$
f _ {\left(s _ {k}, \eta_ {k} \right]} = \frac {1}{\eta_ {k} - s _ {k}} \sum_ {i = s _ {k} + 1} ^ {\eta_ {k}} f _ {i}, f _ {\left(\eta_ {k}, e _ {k} \right]} = \frac {1}{e _ {k} - \eta_ {k}} \sum_ {i = \eta_ {k} + 1} ^ {e _ {k}} f _ {i}. \tag {23}
$$
Observe that,
$$
Q ^ {*} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) \leq \widehat {Q} _ {k} (\eta_ {k}) - \widehat {Q} _ {k} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) + Q ^ {*} (\eta_ {k} + \widetilde {r}). \tag {24}
$$
If  $\widetilde{r} \leq 1 / \kappa_k^{2 + \frac{p}{r}}$ , then there is nothing to show. So for the rest of the argument, for contradiction, assume that
$$
\widetilde {r} \geq \frac {1}{\kappa_ {k} ^ {2 + \frac {p}{r}}}. \tag {25}
$$
Step 2: Finding a lower bound. In this step, we will find a lower bound of the inequality (24). To this end, we observe that,
$$
\begin{array}{l} Q ^ {*} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} \\ = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} \\ - 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \langle f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}}, F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}} \\ = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \frac {1}{2} | | f _ {(s _ {k}, \eta_ {k} ]} - f _ {(\eta_ {k}, e _ {k} ]} | | _ {L _ {2}} ^ {2} - 2 | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} + f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} | | _ {L _ {2}} ^ {2} \\ - 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \langle f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}}, F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}} \\ \geq \frac {1}{2} \widetilde {r} \kappa_ {k} ^ {2} - 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} + f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} | | _ {L _ {2}} ^ {2} \\ - 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \langle f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}}, F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}} \\ \end{array}
$$
We consider,
$$
I _ {1} := 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} + f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} | | _ {L _ {2}} ^ {2}, \text {a n d ,}
$$
$$
I _ {2} := 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {\tau}} \langle f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}}, F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}}.
$$
From above, we have that,
$$
Q ^ {*} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) \geq \frac {1}{2} \widetilde {r} \kappa_ {k} ^ {2} - I _ {1} - I _ {2}.
$$
We now analyze the order of magnitude of term  $I_{1}$ . Then, we get a lower bound for the term  $-I_{1}$ . In fact  $I_{1}$ , has an upper bound of the form  $o_{p}(\widetilde{r} k_{r}^{\frac{p}{r} +2})$ , where we use that  $||f_{\eta_k}*\mathcal{K}_{h_2} - f_{\eta_k}||_{L_2} = o(1)$  and  $||f_{\eta_{k + 1}}*\mathcal{K}_{h_2} - f_{\eta_{k + 1}}||_{L_2} = o(1)$ . For the term  $I_{2}$ , we consider the random variable,
$$
Y _ {i} = \frac {\langle f _ {[ s _ {k} + 1 , \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {[ \eta_ {k} + 1 , e _ {k} ]} * \mathcal {K} _ {h _ {2}} , F _ {t , h _ {2}} - f _ {[ \eta_ {k} + 1 , e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}}}{\kappa_ {k} \mathbb {E} (| | F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}} ^ {3}) ^ {1 / 3}}.
$$
In order to use Lemma 3, we need to bound  $\mathbb{E}(|Y_i|^3)$ . For this, first we use Cauchy Schwartz inequality,
$$
\mathbb {E} (| Y _ {i} | ^ {3}) \leq \frac {(| | (f _ {\eta_ {k} + 1} - f _ {\eta_ {k}}) * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}}) ^ {3} \mathbb {E} (| | F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}} ^ {3})}{\kappa_ {k} ^ {3} \mathbb {E} (| | F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}} ^ {3})}
$$
then, by Minkowski's inequality,
$$
\begin{array}{l} \left| \left| \left(f _ {\eta_ {k} + 1} - f _ {\eta_ {k}}\right) * \mathcal {K} _ {h _ {2}} \right| \right| _ {L ^ {2}} = \left| \right| \int_ {\mathbb {R} ^ {p}} \left(f _ {\eta_ {k} + 1} - f _ {\eta_ {k}}\right) (\cdot - y) \mathcal {K} _ {h _ {2}} (y) d y \left| \right| _ {L _ {2}} \\ \leq \int_ {\mathbb {R} ^ {p}} \left| \left| \left(f _ {\eta_ {k} + 1} - f _ {\eta_ {k}}\right) (\cdot - y) \mathcal {K} _ {h _ {2}} (y) \right| \right| _ {L _ {2}} d y \\ = \left(\int_ {\mathbb {R} ^ {p}} | \mathcal {K} _ {h _ {2}} (y) | d y\right) \left| \left| \left(f _ {\eta_ {k} + 1} - f _ {\eta_ {k}}\right) (\cdot - y) \right| \right| _ {L _ {2}} \\ = \left\| f _ {\eta_ {k} + 1} - f _ {\eta_ {k}} \right\| _ {L ^ {2}} \left\| \mathcal {K} _ {h _ {2}} \right\| _ {L ^ {1}}. \\ \end{array}
$$
Therefore, by Assumption 2, we have
$$
\begin{array}{l} \frac {\left(\left|\left|\left(f _ {\eta_ {k} + 1} - f _ {\eta_ {k}}\right) * \mathcal {K} _ {h _ {2}} \right|\right| _ {L ^ {2}}\right) ^ {3} \mathbb {E} \left( \right.\left|\left| F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} \right.\right. * \mathcal {K} _ {h _ {2}} \left. \right| | _ {L ^ {2}} ^ {3}\left. \right)}{\kappa_ {k} ^ {3} \mathbb {E} \left( \right.\left|\left| F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} \right.\right. * \mathcal {K} _ {h _ {2}} \left. \right| | _ {L ^ {2}} ^ {3}\left. \right)} \\ \leq \frac {(\left| \left| f _ {\eta_ {k} + 1} - f _ {\eta_ {k}} \right| \right| _ {L ^ {2}} \left| \left| \mathcal {K} _ {h _ {2}} \right| \right| _ {L ^ {1}}) ^ {3} \mathbb {E} (\left| \left| F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} \right| \right| _ {L ^ {2}} ^ {3})}{\kappa_ {k} ^ {3} \mathbb {E} (\left| \left| F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} \right| \right| _ {L ^ {2}} ^ {3})} \\ \leq C _ {K}. \\ \end{array}
$$
for any  $t \in (\eta_k, e_k]$ . Moreover, we have that
$$
\begin{array}{l} \mathbb {E} \left(\left| \left| F _ {t, h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} \right| \right| _ {L ^ {2}} ^ {3}\right) ^ {\frac {1}{3}} = \left(\int \left(\int \left(\mathcal {K} _ {h _ {2}} (x - z) - \mathbb {E} \left(\mathcal {K} _ {h _ {2}} (x - X _ {t})\right)\right) ^ {2} d x\right) ^ {\frac {3}{2}} f _ {t} (z) d z\right) ^ {1 / 3} \\ \leq \left(\int \left(\int \left(\mathcal {K} _ {h _ {2}} (x - z)\right) ^ {2} d x\right) ^ {\frac {3}{2}} f _ {t} (z) d z\right) ^ {\frac {1}{3}} \\ = \frac {1}{\kappa^ {p / 2 r}}. \tag {26} \\ \end{array}
$$
Therefore, by Lemma 3, we have that  $I_2 = o_p\Big(\sqrt{\widetilde{r}}\kappa_k\kappa_k^{-\frac{p}{2r}}(\log (\widetilde{r}\kappa_k^{\frac{p}{r} +2}) + 1)\Big)$ . Thus,
$$
Q ^ {*} \left(\eta_ {k} + \widetilde {r}\right) - Q ^ {*} \left(\eta_ {k}\right) \geq \frac {1}{2} \widetilde {r} \kappa_ {k} ^ {2} - O _ {p} \left(\sqrt {\widetilde {r}} \kappa_ {k} \kappa_ {k} ^ {- \frac {p}{2 r}} \left(\log \left(\widetilde {r} \kappa_ {k} ^ {\frac {p}{r} + 2}\right) + 1\right)\right) - o _ {p} \left(\widetilde {r} \kappa_ {k} ^ {\frac {p}{r} + 2}\right). \tag {27}
$$
Step 3: Finding an upper bound. Now, we proceeded to get an upper bound of (24). This is, an upper bound of the following expression,
$$
\widehat {Q} _ {k} \left(\eta_ {k}\right) - \widehat {Q} _ {k} \left(\eta_ {k} + \widetilde {r}\right) - Q ^ {*} \left(\eta_ {k}\right) + Q ^ {*} \left(\eta_ {k} + \widetilde {r}\right). \tag {28}
$$
Observe that, this expression can be written as,
$$
\begin{array}{l} \widehat {Q} _ {k} (\eta_ {k}) - \widehat {Q} _ {k} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) + Q ^ {*} (\eta_ {k} + \widetilde {r}) \\ = - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {1}} - F _ {(s _ {k}, \eta_ {k} ], h _ {1}} | | _ {L _ {2}} ^ {2} + \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {1}} - F _ {(\widehat {\eta} _ {k}, e _ {k} ], h _ {1}} | | _ {L _ {2}} ^ {2} \\ + \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \left\| F _ {t, h _ {2}} - f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \left\| F _ {t, h _ {2}} - f _ {\left(\widehat {\eta} _ {k}, e _ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} ^ {2} \\ \end{array}
$$
So that,
$$
\widehat {Q} _ {k} (\eta_ {k}) - \widehat {Q} _ {k} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) + Q ^ {*} (\eta_ {k} + \widetilde {r}) = U _ {1} + U _ {2},
$$
$$
U _ {1} = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {1}} - F _ {(s _ {k}, \widehat {\eta} _ {k} ], h _ {1}} | | _ {L _ {2}} ^ {2}, \text {a n d ,}
$$
$$
U _ {2} = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {1}} - F _ {(\widehat {\eta} _ {k}, e _ {k} ], h _ {1}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2}.
$$
Now, we analyze each of the terms above. For  $U_{1}$ , observe that
$$
\begin{array}{l} \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {1}} - F _ {(s _ {k}, \widehat {\eta} _ {k} ], h _ {1}} | | _ {L _ {2}} ^ {2} \\ = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}} | | _ {L _ {2}} ^ {2} \\ + \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {1}} - F _ {(s _ {k}, \eta_ {k} ], h _ {1}} | | _ {L _ {2}} ^ {2} \\ = I _ {3} + I _ {4}, \\ \end{array}
$$
$$
I _ {3} = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}} | | _ {L _ {2}} ^ {2}, \text {a n d ,}
$$
$$
I _ {4} = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {1}} - F _ {(s _ {k}, \widehat {\eta} _ {k} ], h _ {1}} | | _ {L _ {2}} ^ {2}.
$$
To analyze  $I_{3}$ , we rewrite it as follow,
$$
\begin{array}{l} I _ {3} = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}} | | _ {L _ {2}} ^ {2} - 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \langle f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}}, F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}} \\ = I _ {3, 1} + I _ {3, 2}, \\ \end{array}
$$
$$
I _ {3, 1} = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}} | | _ {L _ {2}} ^ {2}, \text {a n d ,}
$$
$$
I _ {3, 2} = - 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \langle f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - F _ {(s _ {k}, \eta_ {k} ], h _ {2}}, F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}}.
$$
Now, we will get an upper bound for each of the terms above. The term  $I_{3,1} = O_p\left(\widetilde{r}\frac{1}{T}\frac{\log(T)}{\kappa_k^{\frac{p}{r}}}\right)$ , which is followed by the use of Remark 1. Even more, by Assumption 4, we get
$$
I _ {3, 1} = o _ {p} \left(\widetilde {r} \kappa_ {k} ^ {\frac {p}{r} + 2}\right). \tag {29}
$$
For the term  $I_{3,2}$ , by Cauchy Schwartz inequality and triangle inequality,
$$
\left\langle f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {\kappa} - F _ {\left(s _ {k}, \eta_ {k} \right], \kappa}, F _ {t, h _ {2}} - f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}}
$$
$$
\begin{array}{l} \leq \left\| f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} - F _ {\left(s _ {k}, \eta_ {k} \right], h _ {2}} \right\| _ {L _ {2}} \left\| F _ {t, h _ {2}} - f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} \\ \leq \left| \left| f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} - F _ {\left(s _ {k}, \eta_ {k} \right], h _ {2}} \right| \right| _ {L _ {2}} \left(\left| \left| F _ {t, h _ {2}} - f _ {[ \eta_ {k} + 1, e _ {k} ]} * \mathcal {K} _ {h _ {2}} \right| \right| _ {L _ {2}} + \left| \left| f _ {[ \eta_ {k} + 1, e _ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {[ s _ {k} + 1, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} \right| \right| _ {L _ {2}}\right) \\ \end{array}
$$
for any  $t \in (\eta_k, \eta_k + \widetilde{r}]$ . By the Remark 1, we have that
$$
\left| \left| f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} - F _ {\left(s _ {k}, \eta_ {k} \right], h _ {2}} \right| \right| _ {L _ {2}} = O _ {p} \Big (\frac {1}{\sqrt {T}} \sqrt {\frac {\log (T)}{\kappa_ {k} ^ {\frac {p}{r}}}} \Big)
$$
and using basic properties of integrals  $||f_{[\eta_k + 1,e_k]}*\mathcal{K}_{h_2} - f_{[s_k + 1,\eta_k]}*\mathcal{K}_{h_2}||_{L_2} = O(\kappa_k)$ . Therefore,
$$
I _ {3, 2} \leq O _ {p} \Big (\frac {1}{\sqrt {T}} \sqrt {\frac {\log (T)}{\kappa_ {k} ^ {\frac {p}{r}}}} \Big) \Big (O (\widetilde {r} \kappa_ {k}) + \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {[ \eta_ {k} + 1, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} \Big)
$$
Now, we need to get a bound of the magnitude of
$$
\sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {[ \eta_ {k} + 1, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}},
$$
in order to get an upper for  $I_{3,2}$ . This is done similarly to  $I_{2}$ . We consider the random variable
$$
\widetilde {Y} _ {i} = \frac {\langle F _ {t , h _ {2}} - f _ {(\eta_ {k} , e _ {k} ]} * \mathcal {K} _ {h _ {2}} , F _ {t , h _ {2}} - f _ {(\eta_ {k} , e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}} ^ {\frac {1}{2}} - \mathbb {E} (| | F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}})}{\mathbb {E} (| | F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}} ^ {3}) ^ {\frac {1}{3}}}.
$$
In order to use Lemma 3, we observe that since  $||F_{t,h_2} - f_{\eta_{k + 1}}*\mathcal{K}_{h_2}||_{L^2}\geq 0$
$$
\mathbb {E} (| \widetilde {Y _ {i}} | ^ {3}) \leq \frac {\mathbb {E} (| | F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}} ^ {3})}{\mathbb {E} (| | F _ {t , h _ {2}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} | | _ {L ^ {2}} ^ {3})} = 1.
$$
Therefore, using Lemma 3 and that  $\mathbb{E}(||F_{t,h_2} - f_{\eta_{k + 1}}*\mathcal{K}_{h_2}||_{L^2}) = O(\kappa_k^{\frac{-p}{2r}})$  by (26), we get that
$$
\sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {[ \eta_ {k} + 1, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} = O _ {p} (\sqrt {\widetilde {r} \kappa_ {k} ^ {- \frac {p}{r}}} (\log (\widetilde {r} \kappa_ {k} ^ {\frac {p}{r} + 2}) + 1)) + O _ {p} (\widetilde {r} \kappa_ {k} ^ {\frac {- p}{2 r}}).
$$
Thus, by Assumption 4 and above,
$$
I _ {3, 2} \leq O _ {p} \left(\frac {1}{\sqrt {T}} \sqrt {\frac {\log (T)}{\kappa_ {k} ^ {\frac {p}{r}}}}\right) \left(O (\widetilde {r} \kappa_ {k}) + O _ {p} \left(\sqrt {\widetilde {r} \kappa_ {k} ^ {- \frac {p}{r}}} \left(\log \left(\widetilde {r} \kappa_ {k} ^ {\frac {p}{r} + 2}\right) + 1\right)\right) + O _ {p} \left(\widetilde {r} \kappa_ {k} ^ {\frac {- p}{2 r}}\right)\right) = o _ {p} \left(\widetilde {r} \kappa_ {k} ^ {\frac {p}{r} + 2}\right). \tag {30}
$$
Consequently,  $I_{3}$  has been bounded, and we only need to go over the term  $I_{4}$ , to finalize the analysis for  $U_{1}$ . To analyze  $I_{4}$ , we observe that