Suppose that the attacker’s interference power process $\{v(t)\in[0,\infty)\}_{t\in\mathbb{N}_{0}}$ satisfies Assumption \[SecondAssumption\]. Then for every $\alpha_{1}\geq0$, $\alpha_{0}\geq0$ that satisfy (\[eq:alpha\_ineq\]), there exists a scalar $d\geq0$ such that $$\begin{aligned}
\sum_{j=0}^{t-2}\mathbb{E}\Big[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})\Big] & \leq d,\quad t\in\{2,3,\ldots\}.\label{eq:sum_result}\end{aligned}$$

Since (\[eq:alpha\_ineq\]) holds, it follows from Lemma \[LemmaDOne\] that $\mathbb{E}[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})]\leq\mu\theta^{(t-j-1)}$, where $\mu\geq0$ and $\theta\in(0,1)$ are scalars that depend on $\alpha_{1}$ and $\alpha_{0}$. Letting $$\begin{aligned}
d & \triangleq\mu/(1-\theta),\label{eq:DDef}\end{aligned}$$ we obtain$\sum_{j=0}^{t-2}\mathbb{E}\Big[\prod_{i=j+1}^{t-1}(\alpha_{1}l(i)+\alpha_{0})\Big]\leq\sum_{j=0}^{t-2}\mu\theta^{(t-j-1)}=\mu\sum_{i=1}^{t-1}\theta^{i}\leq\mu\sum_{i=0}^{\infty}\theta^{i}=d$, which completes the proof.

In Lemmas \[LemmaDOne\] and \[LemmaSum\], we obtained the upper-bounding inequalities (\[eq:geometric-lemma-result\]) and (\[eq:sum\_result\]) concerning the transmission failure indicator process $\{l(t)\in\{0,1\}\}_{t\in\mathbb{N}_{0}}$. In our proof of Theorem \[TheoremBoundedDisturbance\] given below, we utilize these inequalities.

*Proof of Theorem \[TheoremBoundedDisturbance\]*: By (\[eq:two-norm-ineq\]) and (\[eq:almost\_sure\_noise\_bound\]), $$\begin{aligned}
\|x(t)\|_{2} & \leq\frac{c_{2}}{c_{1}}\left(\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\right)\|x_{0}\|_{2}\\
 & \quad+\frac{c_{2}}{c_{1}}\Big(\sum_{j=0}^{t-2}\big(\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big)+1\Big)\overline{w},\end{aligned}$$ almost surely, and hence, for $t\in\mathbb{N}$, $$\begin{aligned}
\mathbb{E}[\|x(t)\|_{2}] & \leq\frac{c_{2}}{c_{1}}\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\|x_{0}\|_{2}\nonumber \\
 & \quad+\frac{c_{2}}{c_{1}}\Big(\sum_{j=0}^{t-2}\mathbb{E}\big[\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]+1\Big)\overline{w}.\label{eq:expectation_bounded_noise_case}\end{aligned}$$

Next, we apply Lemmas \[LemmaDOne\] and \[LemmaSum\] to obtain upper-bounds for the expectation terms on the right-hand side of (\[eq:expectation\_bounded\_noise\_case\]). First, since $\|A\|>1$, (\[eq:abkcond-vhat\]) implies $\|A+BK\|\in[0,1)$. Thus, we have $\zeta_{1}>0$ and $\zeta_{0}\in[0,1)$. By letting $\alpha_{1}=\zeta_{1}$ and $\alpha_{0}=\zeta_{0}$, (\[eq:abkcond-vhat\]) implies (\[eq:alpha\_ineq\]). Therefore, by Lemmas \[LemmaDOne\] and \[LemmaSum\], we have $\mathbb{E}\big[\prod_{i=0}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\leq\mu\theta^{t}$ and $\sum_{j=0}^{t-2}\mathbb{E}\big[\prod_{i=j+1}^{t-1}(\zeta_{1}l(i)+\zeta_{0})\big]\leq d$, where $\mu\geq0$, $\theta\in(0,1)$, and $d\geq0$ are scalars that depend on $\zeta_{1}$ and $\zeta_{0}$. Hence, with $$\begin{aligned}
 & \hat{\theta}\triangleq\theta,\quad\hat{\mu}\triangleq\mu c_{2}/c_{1},\quad\hat{d}\triangleq(d+1)c_{2}/c_{1},\label{eq:thetamudhat}\end{aligned}$$ the inequality (\[eq:noise\_to\_state\_one\]) follows from 