$$
\begin{array}{l} \mathbb {E} \left[ \min  _ {b \in [ b _ {t} ]} r _ {t} ^ {[ b ]} \mid \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] \leq \mathbb {E} \left[ \frac {1}{b _ {t}} \sum_ {b = 1} ^ {b _ {t}} \left[ 2 \left(\omega c _ {t} \sigma_ {t - 1} (\overline {{\boldsymbol {x}}} _ {t}) + (1 - \omega) c _ {t} ^ {\prime} \sigma_ {t - 1} ^ {\prime} (\overline {{\boldsymbol {x}}} _ {t})\right) \right. \right. \\ + \omega c _ {t} \sigma_ {t - 1} \left(\boldsymbol {x} _ {t} ^ {[ b ]}\right) + (1 - \omega) c _ {t} ^ {\prime} \sigma_ {t - 1} ^ {\prime} \left(\boldsymbol {x} _ {t} ^ {[ b ]}\right) \Bigg ] \left| \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] + 2 \widetilde {B} \mathbb {P} \left[ \overline {{E ^ {f _ {t}} (t)}} \cup \overline {{E ^ {g _ {t}} (t)}} | \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] \\ \leq \mathbb {E} \left[ \frac {1}{b _ {t}} \sum_ {b = 1} ^ {b _ {t}} \left[ \frac {2}{p ^ {2} - 1 / t ^ {2}} \left(\omega c _ {t} \sigma_ {t - 1} \left(\boldsymbol {x} _ {t} ^ {[ b ]}\right) + (1 - \omega) c _ {t} ^ {\prime} \sigma_ {t - 1} ^ {\prime} \left(\boldsymbol {x} _ {t} ^ {[ b ]}\right)\right) \right. \right. \\ \left. + \omega c _ {t} \sigma_ {t - 1} (\boldsymbol {x} _ {t} ^ {[ b ]}) + (1 - \omega) c _ {t} ^ {\prime} \sigma_ {t - 1} ^ {\prime} (\boldsymbol {x} _ {t} ^ {[ b ]}) \right] \Big | \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \Bigg ] + 2 \widetilde {B} \frac {1}{t ^ {2}} \\ \leq \omega c _ {t} \left(1 + \frac {2 8}{p ^ {2}}\right) \mathbb {E} \left[ \frac {1}{b _ {t}} \sum_ {b = 1} ^ {b _ {t}} \sigma_ {t - 1} \left(\boldsymbol {x} _ {t} ^ {[ b ]}\right) \mid \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] \\ + (1 - \omega) c _ {t} ^ {\prime} \left(1 + \frac {2 8}{p ^ {2}}\right) \mathbb {E} \left[ \frac {1}{b _ {t}} \sum_ {b = 1} ^ {b _ {t}} \sigma_ {t - 1} ^ {\prime} (\boldsymbol {x} _ {t} ^ {[ b ]}) | \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] + \frac {2 \widetilde {B}}{t ^ {2}} \\ \leq \omega c _ {t} e ^ {C} \left(1 + \frac {2 8}{p ^ {2}}\right) \mathbb {E} \left[ \frac {1}{b _ {t}} \sqrt {C _ {2} b _ {t} \mathbb {I} (f ; \boldsymbol {y} _ {t} | \boldsymbol {y} _ {1 : t - 1})} | \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] \\ + (1 - \omega) c _ {t} ^ {\prime} e ^ {C} \left(1 + \frac {2 8}{p ^ {2}}\right) \mathbb {E} \left[ \frac {1}{b _ {t}} \sqrt {C _ {2} b _ {t} \mathbb {I} \left(g ; \boldsymbol {y} _ {t} ^ {\prime} | \boldsymbol {y} _ {1 : t - 1} ^ {\prime}\right)} | \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] + \frac {2 \widetilde {B}}{t ^ {2}} \\ = e ^ {C} \left(1 + \frac {2 8}{p ^ {2}}\right) \left[ \omega c _ {t} \mathbb {E} \left[ \frac {1}{b _ {t}} \sqrt {C _ {2} b _ {t} \mathbb {I} \left(f ; \boldsymbol {y} _ {t} | \boldsymbol {y} _ {1 : t - 1}\right)} \mid \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] \right. \\ + (1 - \omega) c _ {t} ^ {\prime} \mathbb {E} \left[ \frac {1}{b _ {t}} \sqrt {C _ {2} b _ {t} \mathbb {I} \left(g ; \boldsymbol {y} _ {t} ^ {\prime} \mid \boldsymbol {y} _ {1 : t - 1} ^ {\prime}\right)} \left| \mathcal {F} _ {t - 1}, \mathcal {F} _ {t - 1} ^ {\prime} \right] \right] + \frac {2 \widetilde {B}}{t ^ {2}}. \tag {41} \\ \end{array}
$$