SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} Q ^ {*} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} \\ = \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} \\ - 2 \sum_ {t = \eta_ {k} + 1} ^ {\eta_ {k} + \widetilde {r}} \langle f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}}, F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}} \\ \underline {{\mathcal {D}}} \sum_ {t = 1} ^ {\widetilde {r}} 2 \Big \langle F _ {h _ {2}, \eta_ {k} + t} - f _ {\eta_ {k} + t} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k + 1}} - f _ {\eta_ {k}}) * \mathcal {K} _ {h _ {2}} \Big \rangle_ {L _ {2}} + \widetilde {r} | | (f _ {\eta_ {k + 1}} - f _ {\eta_ {k}}) * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2}. \\ \end{array}
$$
When  $\widetilde{r} < 0$  and  $T\to \infty$  , we have that
$$
\begin{array}{l} Q ^ {*} (\eta_ {k} + \widetilde {r}) - Q ^ {*} (\eta_ {k}) = \sum_ {t = \eta_ {k} + \widetilde {r}} ^ {\eta_ {k} - 1} | | F _ {t, h _ {2}} - f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} - \sum_ {t = \eta_ {k} + \widetilde {r}} ^ {\eta_ {k} - 1} | | F _ {t, h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} \\ = \sum_ {t = \eta_ {k} + \widetilde {r}} ^ {\eta_ {k} - 1} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} \\ - 2 \sum_ {t = \eta_ {k} + \widetilde {r}} ^ {\eta_ {k} - 1} \left\langle f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} - f _ {\left(\eta_ {k}, e _ {k} \right]} * \mathcal {K} _ {h _ {2}}, F _ {t, h _ {2}} - f _ {\left(\eta_ {k}, e _ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}} \\ \end{array}
$$
$$
\underline {{\mathcal {D}}} \sum_ {t = \widetilde {r} + 1} ^ {0} 2 \Big \langle F _ {h _ {2}, \eta_ {k} + t} - f _ {\eta_ {k} + t} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} \Big \rangle_ {L _ {2}} + \widetilde {r} | | (f _ {\eta_ {k + 1}} - f _ {\eta_ {k}}) * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2}.
$$
Therefore, using Slutsky's theorem and the Argmax (or Argmin) continuous mapping theorem (see 3.2.2 Theorem van der Vaart and Wellner, 1996) we conclude
$$
\left(\widetilde {\eta} _ {k} - \eta_ {k}\right) \kappa_ {k} ^ {\frac {p}{r} + 2} \underset {\widetilde {r} \in \mathbb {Z}} {\mathcal {D}} \arg \min  _ {P _ {k}} (\widetilde {r}) \tag {39}
$$
Vanishing regime. Vanishing regime. Let  $m = \kappa_k^{-2 - \frac{p}{r}}$ , and we have that  $m \to \infty$  as  $T \to \infty$ . Observe that for  $\widetilde{r} > 0$ , we have that
$$
\begin{array}{l} Q _ {k} ^ {*} \Big (\eta_ {k} + \widetilde {r} m \Big) - Q _ {k} ^ {*} \Big (\eta_ {k} \Big) = \sum_ {t = \eta_ {k}} ^ {\eta_ {k} + \widetilde {r} m - 1} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2} \\ - 2 \sum_ {t = \eta_ {k}} ^ {\eta_ {k} + \widetilde {r} m - 1} \left\langle f _ {\left(s _ {k}, \eta_ {k} \right]} * \mathcal {K} _ {h _ {2}} - f _ {\left(\eta_ {k}, e _ {k} \right]} * \mathcal {K} _ {h _ {2}}, F _ {t, h _ {2}} - f _ {\left(\eta_ {k}, e _ {k} \right]} * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}} \\ \end{array}
$$
Following the Central Limit Theorem for  $\alpha$ -mixing, see Lemma 4, we get
$$
\frac {1}{\sqrt {m}} \sum_ {t = \eta_ {k}} ^ {\eta_ {k} + r m - 1} \frac {\langle f _ {(s _ {k} , \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k} , e _ {k} ]} * \mathcal {K} _ {h _ {2}} , F _ {t , h _ {2}} - f _ {(\eta_ {k} , e _ {k} ]} * \mathcal {K} _ {h _ {2}} \rangle_ {L _ {2}}}{\kappa_ {k} ^ {\frac {p}{2 r} + 1}} \xrightarrow {\mathcal {D}} \kappa_ {k} ^ {- \frac {p}{r}} \widetilde {\sigma} _ {\infty} (k) \mathbb {B} (\widetilde {r}),
$$
where  $\mathbb{B}(\widetilde{r})$  is a standard Brownian motion and  $\widetilde{\sigma}(k)$  is the long-run variance given in (8). Therefore, it holds that when  $T \to \infty$
$$
Q _ {k} ^ {*} \Big (\eta_ {k} + \widetilde {r} m \Big) - Q _ {k} ^ {*} \Big (\eta_ {k} \Big) \stackrel {\mathcal {D}} {\to} \kappa_ {k} ^ {- \frac {p}{r}} \widetilde {\sigma} _ {\infty} (k) \mathbb {B} _ {1} (r) + \widetilde {r} \kappa_ {k} ^ {- \frac {p}{r} - 2} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2}.
$$
Similarly, for  $\widetilde{r} < 0$ , we have that when  $n \to \infty$
$$
Q _ {k} ^ {*} \Big (\eta_ {k} + r m \Big) - Q _ {k} ^ {*} \Big (\eta_ {k} \Big) \stackrel {\mathcal {D}} {\to} \kappa_ {k} ^ {- \frac {p}{r}} \widetilde {\sigma} _ {\infty} (k) \mathbb {B} _ {1} (- \widetilde {r}) - \widetilde {r} \kappa_ {k} ^ {- \frac {p}{r} - 2} | | f _ {(s _ {k}, \eta_ {k} ]} * \mathcal {K} _ {h _ {2}} - f _ {(\eta_ {k}, e _ {k} ]} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} ^ {2}.
$$
Then, using Slutsky's theorem and the Argmax (or Argmin) continuous mapping theorem (see 3.2.2 Theorem in van der Vaart & Wellner (1996)), and the fact that,  $\mathbb{E}(||f_{(s_k,\eta_k]}*\mathcal{K}_{h_2} - f_{(\eta_k,e_k]}\ast$ $\mathcal{K}_{h_2}||_{L_2}^2) = O(\kappa_k^2)$ , we conclude that
$$
\kappa_ {k} ^ {2 + \frac {p}{r}} \left(\widetilde {\eta} _ {k} - \eta_ {k}\right) \xrightarrow {\mathcal {D}} \underset {r \in \mathbb {Z}} {\arg \min } \widetilde {\sigma} _ {\infty} (k) B (\widetilde {r}) + | \widetilde {r} |,
$$
which completes the proof of b.2.
In this section, we present the proof of theorem 3.
Proof of Theorem 3. First, letting  $h_2 = c_\kappa \kappa_k^{\frac{1}{r}}$  and  $R = O\left(\frac{T^{\frac{2r}{2r + p}}}{\kappa_k^{\frac{p}{2r} + \frac{3}{2}}}\right)$ , we consider
$$
\check {\sigma} _ {\infty} ^ {2} (k) = \frac {1}{R} \sum_ {r = 1} ^ {R} \left(\frac {1}{\sqrt {S}} \sum_ {i \in S _ {r}} \check {Y} _ {i}\right) ^ {2}, \text {w h e r e ,} \check {Y} _ {i} = \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}}. \tag {40}
$$
We will show that
(i)  $\left|\widehat{\sigma}_{\infty}^{2}(k) - \check{\sigma}_{\infty}^{2}(k)\right| \xrightarrow{P} 0, \quad T \to \infty$ , and
(ii)  $\left|\breve{\sigma}_{\infty}^{2}(k) - \widetilde{\sigma}_{\infty}^{2}(k)\right| \xrightarrow{P} 0, \quad T \to \infty$
in order to conclude the result. For (i), we use  $a^2 - b^2 = (a + b)(a - b)$ , to write,
$$
\begin{array}{l} \left| \hat {\sigma} _ {\infty} ^ {2} (k) - \check {\sigma} _ {\infty} ^ {2} (k) \right| = \left| \frac {1}{R} \sum_ {r = 1} ^ {R} \left(\frac {1}{\sqrt {S}} \sum_ {i \in S _ {r}} \check {Y} _ {i}\right) ^ {2} - \frac {1}{R} \sum_ {r = 1} ^ {R} \left(\frac {1}{\sqrt {S}} \sum_ {i \in S _ {r}} Y _ {i}\right) ^ {2} \right| \\ = \left| \frac {1}{R} \sum_ {r = 1} ^ {R} \left(\frac {1}{\sqrt {S}} \sum_ {i \in \mathcal {S} _ {r}} \check {Y} _ {i} - Y _ {i}\right) \left(\frac {1}{\sqrt {S}} \sum_ {i \in \mathcal {S} _ {r}} \check {Y} _ {i} + Y _ {i}\right) \right| \\ = \left| \frac {1}{R} \sum_ {r = 1} ^ {R} I _ {1} I _ {2} \right| \\ \end{array}
$$
Then, we bound each of the terms  $I_{1}$  and  $I_{2}$ . For  $I_{1}$ , we observe that,
$$
I _ {1} = \left| \frac {1}{\sqrt {S}} \sum_ {i \in \mathcal {S} _ {r}} \breve {Y} _ {i} - Y _ {i} \right| \leq \frac {1}{\sqrt {S}} \sum_ {i \in \mathcal {S} _ {r}} \left| \breve {Y} _ {i} - Y _ {i} \right|.
$$
Then, adding and subtracting,  $\widehat{\kappa}_{k}^{\frac{p}{2r} -1}\Big\langle F_{h_1,i} - f_i*\mathcal{K}_{h_1},(f_{\eta_k} - f_{\eta_{k + 1}})*\mathcal{K}_{h_2}\Big\rangle_{L_2}$  and
$$
\widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} - F _ {h _ {1}, i} + f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}},
$$
we get that,
$$
\begin{array}{l} \left| \breve {Y} _ {i} - Y _ {i} \right| \\ = \Bigl | \kappa_ {k} ^ {\frac {p}{2 r} - 1} \Bigl \langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} \Bigr \rangle_ {L _ {2}} - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \Bigl \langle F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Bigr \rangle_ {L _ {2}} \Bigr | \\ = \left| \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}} - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}, \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}} \right. \\ + \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} \Big \rangle_ {L _ {2}} - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \Big | \\ + \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} - F _ {h _ {1}, i} + f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \\ - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} - F _ {h _ {1}, i} + f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \\ \end{array}
$$
which can be written as,
$$
\begin{array}{l} \left| \left\langle \kappa_ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}) - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}), (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} - (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \right. \\ + \Big \langle \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}) - \kappa_ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}), (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} - (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} \Big \rangle_ {L _ {2}} \\ + \Big \langle \kappa_ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}), (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} - (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \\ + \Big \langle \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}) - \kappa_ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}), (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \Big |. \\ \end{array}
$$
Now, we bound the expression above. For this purpose, by triangle inequality, it is enough to bound each of the terms above. Then, we use Hölder's inequality. First,
$$
\begin{array}{l} \left| \left\langle \kappa_ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}) - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}), (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} - (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \right| \\ \leq | \kappa_ {k} ^ {\frac {p}{2 r} - 1} - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} | | | F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} - F _ {h _ {1}, i} + f _ {i} * \mathcal {K} _ {h _ {1}} | | _ {L _ {2}} | | (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} - (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} | | _ {L _ {2}}. \\ \end{array}
$$
Then, using (54), we have that  $|\kappa_k^{\frac{p}{2r} -1} - \widehat{\kappa}_k^{\frac{p}{2r} -1}| = O_p\bigg(\frac{1}{\kappa_k^{2 - \frac{p}{2r}}}\bigg(\bigg(\frac{\log(T)}{\Delta}\bigg)^{\frac{2r}{2r + p}} + \frac{T^{\frac{p}{2r + p}}\log(T)}{\kappa\Delta}\bigg)^{\frac{1}{2}}\bigg)$
and using Remark 2, it follows that
$$
\begin{array}{l} \left\| F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} - F _ {h _ {1}, i} + f _ {i} * \mathcal {K} _ {h _ {1}} \right\| _ {L _ {2}} \leq \left\| F _ {h _ {2}, i} - F _ {h _ {1}, i} \right\| _ {L _ {2}} + \left\| f _ {i} * \mathcal {K} _ {h _ {1}} - f _ {i} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} \\ = O _ {p} \left(\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \left(\left(\frac {\log (T)}{\Delta}\right) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta}\right) ^ {\frac {1}{2}}\right) \\ \end{array}
$$
$$
\begin{array}{l} \left| \left| \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} - \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} \right| \right| _ {L _ {2}} \\ \leq \left| \left| f _ {\eta_ {k}} * \mathcal {K} _ {h _ {2}} - f _ {\widehat {\eta} _ {k}} * \mathcal {K} _ {h _ {1}} \right| \right| _ {L _ {2}} + \left| \left| f _ {\widehat {\eta} _ {k + 1}} * \mathcal {K} _ {h _ {1}} - f _ {\eta_ {k + 1}} * \mathcal {K} _ {h _ {2}} \right| \right| _ {L _ {2}} \\ = O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac {1}{2}} \Big). \\ \end{array}
$$
$$
\begin{array}{l} \left| \left\langle \kappa_ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}) - \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}), (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} - (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \right| \\ = O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {2 - \frac {p}{2 r}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac {1}{2}} \Big) O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac {1}{2}} \Big) \\ \end{array}
$$
$$
\cdot O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac {1}{2}} \Big).
$$
Now, in a similar way, we observe that
$$
\begin{array}{l} \left\langle \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left(F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}\right), \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} - \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \\ \leq \left|\left| \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left(F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}\right)\right|\right| _ {L _ {2}} \left|\right|\left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} - \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} \left. \right| | _ {L _ {2}} \\ = O _ {p} \left(\kappa_ {k} ^ {\frac {p}{2 r} - 1} \kappa_ {k} ^ {- \frac {p}{2 r}}\right) O _ {p} \left(\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \left(\left(\frac {\log (T)}{\Delta}\right) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta}\right) ^ {\frac {1}{2}}\right) \\ \end{array}
$$
where equality is followed by noticing that
$$
\begin{array}{l} \left\| F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} \leq \left\| F _ {h _ {2}, i} \right\| _ {L _ {2}} + \left\| f _ {i} * \mathcal {K} _ {h _ {2}} \right\| _ {L _ {2}} (41) \\ = O \left(\kappa_ {k} ^ {- \frac {p}{2 r}}\right) + O (1), (42) \\ \end{array}
$$
and then using Remark 2 and Assumption 1. Finally,
$$
\begin{array}{l} \left\langle \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \left(F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}\right) - \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left(F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}\right), \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \\ \leq \left| \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} - \kappa_ {k} ^ {\frac {p}{2 r} - 1} \right| | | \left(F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}\right) - \left(F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}\right) | | _ {L _ {2}} | | \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} | | _ {L _ {2}} \\ = O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {2 - \frac {p}{2 r}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac {1}{2}} \Big) O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac {1}{2}} \Big) \kappa_ {k} \\ \end{array}
$$
where equality is followed by Remark 2, Assumption 1 and Minkowski's inequality. Therefore,
$$
\begin{array}{l} I _ {1} \leq \frac {1}{\sqrt {S}} \sum_ {i \in S _ {r}} \left| \breve {Y} _ {i} - Y _ {i} \right| \\ \sqrt {S} \Big (O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {2 - \frac {p}{2 r}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) \Big) \\ \cdot O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac {1}{2}} \Big) O _ {p} \Big (\frac {1}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \Big (\Big (\frac {\log (T)}{\Delta} \Big) ^ {\frac {2 r}{2 r + p}} + \frac {T ^ {\frac {p}{2 r + p}} \log (T)}{\kappa \Delta} \Big) ^ {\frac { 1}{2}} \Big) \\ + O _ {p} \left(\kappa_ {k} ^ {\frac {p}{2 r} - 1} \kappa_ {k} ^ {- \frac {p}{2 r}}\right) O _ {p} \left(\frac {T ^ {- \frac {r}{2 r + p}}}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \log^ {\frac {r}{2 r + p}} (T)\right) + O _ {p} \left(\frac {T ^ {- \frac {r}{4 r + 2 p}}}{\kappa_ {k} ^ {2 - \frac {p}{2 r}}} \log^ {\frac {r}{2 r + p}} (T)\right) O _ {p} \left(\frac {T ^ {- \frac {r}{2 r + p}}}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \log^ {\frac {r}{2 r + p}} (T)\right) \kappa_ {k}\left. \right) \\ = \sqrt {S} \Big (O _ {p} (\frac {T ^ {- \frac {2 r}{2 r + p}}}{\kappa_ {k} ^ {2 - \frac {p}{2 r}}} \log (T) ^ {\frac {2 r}{2 r + p}}) O _ {p} (\frac {T ^ {- \frac {2 r}{2 r + p}}}{\kappa_ {k} ^ {\frac {p}{r} + 1}} \log (T) ^ {\frac {2 r}{2 r + p}}) + O _ {p} (\kappa_ {k} ^ {- 1}) O _ {p} (\frac {T ^ {- \frac {r}{2 r + p}}}{\kappa_ {k} ^ {\frac {p}{2 r} + \frac {1}{2}}} \log (T) ^ {\frac {r}{2 r + p}}) \Big). \\ \end{array}
$$
To bound the  $I_{2}$  term, we add and subtract  $\kappa_{k}^{\frac{p}{2r} -1}\Bigl {\langle}F_{h_{2},i} - f_{i}*\mathcal{K}_{h_{2}},(f_{\widehat{\eta}_{k}} - f_{\widehat{\eta}_{k + 1}})*\mathcal{K}_{h_{1}}\Bigr{\rangle}_{L_{2}}$  , to get
$$
\begin{array}{l} \left| \breve {Y} _ {i} + Y _ {i} \right| \\ = \left| \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}} + \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \right| \\ = \left| \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}} - \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \right. \\ + \kappa_ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} + \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \Big | \\ = \left| \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}} + \kappa_ {k} ^ {\frac {p}{2 r} - 1} \left\langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} \right\rangle_ {L _ {2}} \right. \\ + \Big \langle \widehat {\kappa} _ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {1}, i} - f _ {i} * \mathcal {K} _ {h _ {1}}) - \kappa_ {k} ^ {\frac {p}{2 r} - 1} (F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}), (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \Big |. \\ \end{array}
$$
Then, as before, we bound each of the terms above using Hölder's inequality. We start with the term
$$
\begin{array}{l} \left| \kappa_ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} \Big \rangle_ {L _ {2}} \right| \\ \leq \kappa_ {k} ^ {\frac {p}{2 r} - 1} | | F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} | | \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} \\ \leq \kappa_ {k} ^ {\frac {p}{2 r} - 1} O _ {p} (\kappa_ {k} ^ {- \frac {p}{2 r}}) \kappa_ {k} = O _ {p} (1) \\ \end{array}
$$
where the second inequality is followed by (41). Similarly,
$$
\begin{array}{l} \left| \kappa_ {k} ^ {\frac {p}{2 r} - 1} \Big \langle F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}}, (f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}) * \mathcal {K} _ {h _ {1}} \Big \rangle_ {L _ {2}} \right| \\ \leq \kappa_ {k} ^ {\frac {p}{2 r} - 1} | | F _ {h _ {2}, i} - f _ {i} * \mathcal {K} _ {h _ {2}} | | _ {L _ {2}} | | \left(f _ {\widehat {\eta} _ {k}} - f _ {\widehat {\eta} _ {k + 1}}\right) * \mathcal {K} _ {h _ {1}} | | _ {L _ {2}} \\ \leq \kappa_ {k} ^ {\frac {p}{2 r} - 1} O _ {p} (\kappa_ {k} ^ {- \frac {p}{2 r}}) \kappa_ {k} = O _ {p} (1) \\ \end{array}
$$
where the second inequality is followed by the (41). Finally, the term