$$
\begin{array}{l} \left| \frac {\sqrt {n}}{\sigma_ {P ^ {(m)}}} W _ {3, P ^ {(m)}} ^ {(m)} \right| = \left| \mathbb {E} _ {P ^ {(m)}} \left[ \ell \left(\int \hat {g} ^ {(m)} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x), Y\right) - \ell \left(\int g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x), Y\right) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \right| \\ \leq \frac {\sqrt {n}}{\sigma_ {P (m)}} \mathbb {E} _ {P ^ {(m)}} \left[ \left| \ell \left(\int \hat {g} ^ {(m)} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x), Y\right) - \ell \left(\int g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x), Y\right) \right| \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\sigma_ {P (m)}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \left\| \int \hat {g} ^ {(m)} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x) - \int g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x) \right\| _ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ = \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \left\| \int \hat {g} ^ {(m)} (x, Z _ {1}) - g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x) \right\| _ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \int \left\| \hat {g} ^ {(m)} (x, Z _ {1}) - g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) \right\| _ {2} d Q _ {X | Z} ^ {*} (x) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ = \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \int \left\| \hat {g} ^ {(m)} (x, Z _ {1}) - g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) \right\| _ {2} \frac {d Q _ {X | Z} ^ {*}}{d P _ {X | Z}} (x) d P _ {X | Z} (x) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ = \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \left\| \hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right) - g _ {P ^ {(m)}} ^ {*} \left(X _ {1}, Z _ {1}\right) \right\| _ {2} \frac {d Q _ {X | Z} ^ {*}}{d P _ {X | Z}} \left(X _ {1}\right) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \left(\mathbb {E} _ {P ^ {(m)}} \left[ \left\| \hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right) - g _ {P ^ {(m)}} ^ {*} \left(X _ {1}, Z _ {1}\right) \right\| _ {2} ^ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \mathbb {E} _ {P ^ {(m)}} \left[ \left(\frac {d Q _ {X | Z} ^ {*}}{d P _ {X | Z}} \left(X _ {1}\right)\right) ^ {2} \right]\right) ^ {1 / 2} \\ = \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \left(\mathbb {E} _ {P ^ {(m)}} \left[ \left\| \hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right) - g _ {P ^ {(m)}} ^ {*} \left(X _ {1}, Z _ {1}\right) \right\| _ {2} ^ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \mathbb {E} _ {P ^ {(m)}} \left[ \int \frac {d Q _ {X | Z} ^ {*}}{d P _ {X | Z}} (x) d Q _ {X | Z} ^ {*} (x) \right]\right) ^ {1 / 2} \\ = \frac {L \sqrt {n}}{\sigma_ {P (m)}} \cdot \left(\mathbb {E} _ {P (m)} \left[ \left\| \hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right) - g _ {P (m)} ^ {*} \left(X _ {1}, Z _ {1}\right) \right\| _ {2} ^ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \mathbb {E} _ {P (m)} \left[ \chi^ {2} \left(Q _ {X | Z} ^ {*} \| P _ {X | Z}\right) + 1 \right]\right) ^ {1 / 2} \\ \leq \frac {L R \sqrt {n}}{\inf  _ {P \in \mathcal {P}} \sigma_ {P}} \cdot \left(\mathbb {E} _ {P ^ {(m)}} \left[ \left\| \hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right) - g _ {P ^ {(m)}} ^ {*} \left(X _ {1}, Z _ {1}\right) \right\| _ {2} ^ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right]\right) ^ {1 / 2} \\ = \frac {L R}{\inf  _ {P \in \mathcal {P}} \sigma_ {P}} \cdot \left(o (m ^ {\gamma}) \mathcal {O} _ {p} (m ^ {- \gamma})\right) ^ {1 / 2} \\ = o _ {p} (1) \\ \end{array}
$$