$$
\begin{array}{l} \mathbb {E} \left[ \| F _ {v _ {k}} (\hat {x} _ {k}) - F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} \right] \leq (1 + \alpha) \mathbb {E} \left[ \| F (\hat {x} _ {k}) - F (\hat {x} _ {k - 1}) \| ^ {2} \right] \\ + (1 + \alpha^ {- 1}) \mathbb {E} \left[ \| F _ {v _ {k}} (\hat {x} _ {k}) - F (\hat {x} _ {k}) \right. \\ \left. - \left(F _ {v _ {k - 1}} \left(\hat {x} _ {k - 1}\right) - F \left(\hat {x} _ {k - 1}\right)\right) \| ^ {2} \right] \\ \leq (1 + \alpha) L ^ {2} \mathbb {E} \left[ \| \hat {x} _ {k} - \hat {x} _ {k - 1} \| ^ {2} \right] \\ + 2 (1 + \alpha^ {- 1}) \mathbb {E} \left[ \left\| F _ {v _ {k}} (\hat {x} _ {k}) - F (\hat {x} _ {k}) \right\| ^ {2} \right. \\ + \left\| F _ {v _ {k - 1}} \left(\hat {x} _ {k - 1}\right) - F \left(\hat {x} _ {k - 1}\right) \right\| ^ {2} \bigg ] \\ \stackrel {(8)} {\leq} (1 + \alpha) L ^ {2} \mathbb {E} \left[ \| \hat {x} _ {k} - x _ {k} + x _ {k} - x _ {k - 1} + x _ {k - 1} - \hat {x} _ {k - 1} \| ^ {2} \right] \\ + 2 (1 + \alpha^ {- 1}) \delta \mathbb {E} \left[ \| \hat {x} _ {k} - x ^ {*} \| ^ {2} + \| \hat {x} _ {k - 1} - x ^ {*} \| ^ {2} \right] \\ + 8 (1 + \alpha^ {- 1}) \sigma_ {*} ^ {2} \\ \leq (1 + \alpha) L ^ {2} \mathbb {E} \left[ \| (\gamma + \omega) F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) - \gamma F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \mathbb {E} \left[ \| x _ {k} - x ^ {*} \| ^ {2} + \| x _ {k - 1} - x ^ {*} \| ^ {2} \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \gamma^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} + \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + 8 (1 + \alpha^ {- 1}) \sigma_ {*} ^ {2} \\ = (1 + \alpha) L ^ {2} (\gamma + \omega) ^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} \right] \\ + (1 + \alpha) L ^ {2} \gamma^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ - 2 (1 + \alpha) L ^ {2} \gamma (\gamma + \omega) \mathbb {E} \left[ \left\langle F _ {v _ {k - 1}} \left(\hat {x} _ {k - 1}\right), F _ {v _ {k - 2}} \left(\hat {x} _ {k - 2}\right) \right\rangle \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \mathbb {E} \left[ \| x _ {k} - x ^ {*} \| ^ {2} + \| x _ {k - 1} - x ^ {*} \| ^ {2} \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \gamma^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} + \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + 8 (1 + \alpha^ {- 1}) \sigma_ {*} ^ {2} \\ = (1 + \alpha) L ^ {2} (\gamma + \omega) ^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} \right] \\ + (1 + \alpha) L ^ {2} \gamma^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ - (1 + \alpha) L ^ {2} \gamma (\gamma + \omega) \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} + \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + (1 + \alpha) L ^ {2} \gamma (\gamma + \omega) \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) - F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \mathbb {E} \left[ \| x _ {k} - x ^ {*} \| ^ {2} + \| x _ {k - 1} - x ^ {*} \| ^ {2} \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \gamma^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} + \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + 8 (1 + \alpha^ {- 1}) \sigma_ {*} ^ {2} \\ = (1 + \alpha) L ^ {2} \omega (\gamma + \omega) \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} \right] \\ - (1 + \alpha) L ^ {2} \gamma \omega \mathbb {E} \left[ \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + (1 + \alpha) L ^ {2} \gamma (\gamma + \omega) \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) - F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \mathbb {E} \left[ \| x _ {k} - x ^ {*} \| ^ {2} + \| x _ {k - 1} - x ^ {*} \| ^ {2} \right] \\ + 4 (1 + \alpha^ {- 1}) \delta \gamma^ {2} \mathbb {E} \left[ \| F _ {v _ {k - 1}} (\hat {x} _ {k - 1}) \| ^ {2} + \| F _ {v _ {k - 2}} (\hat {x} _ {k - 2}) \| ^ {2} \right] \\ + 8 (1 + \alpha^ {- 1}) \sigma_ {*} ^ {2} \\ \end{array}
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