0, & \mathrm{otherwise},
\end{cases}\label{eq:ExampleStrategyV}\end{aligned}$$ with $v^{*}\triangleq p^{-1}(\rho^{\frac{1}{\tau_{2}}})+1$, $\tau_{1}\triangleq\lfloor\frac{\max\{v^{*}-\overline{v},0\}\tau_{2}}{\overline{v}}\rfloor+1$, $\tau_{2}\triangleq\lfloor\max\{\log_{A}(z/w^{*}),0\}\rfloor+1$ guarantees that Assumption \[FirstAssumption\] is satisfied and the state exceeds the value $z$ with probability larger than $\rho$ at time $\tau\triangleq\tau_{1}+\tau_{2}$, i.e., $\mathbb{P}[x(\tau)>z]>\rho.$

To show this, first we define the event $E(\tau_{1},\tau_{2})\in\mathcal{F}$ by $$\begin{aligned}
E(\tau_{1},\tau_{2}) & \triangleq\big\{\omega\in\Omega\colon l(t)=1,t\in\{\tau_{1},\ldots,\tau_{1}+\tau_{2}-1\}\big\}.\end{aligned}$$ This is the event that all packet transmissions during    $t\in\{\tau_{1},\ldots,\tau_{1}+\tau_{2}-1\}$ fail. By (\[eq:ExampleStrategyV\]), we have $\mathbb{P}[E(\tau_{1},\tau_{2})]=p^{\tau_{2}}(v^{*})$. Now, since $x_{0}>0$, $A>1$, and $w^{*}>0$, we obtain $x(t)\geq w^{*},$ $t\in\mathbb{N}$. Therefore, $$\begin{aligned}
 & \mathbb{P}[x(\tau)>z]\geq\mathbb{P}[x(\tau)>z\,|\,E(\tau_{1},\tau_{2})]\mathbb{P}[E(\tau_{1},\tau_{2})]\\
 & \quad\geq\mathbb{P}[A^{\tau_{2}}x(\tau_{1})+\sum_{i=0}^{\tau_{2}-1}A^{i}w^{*}>z\,|\,E(\tau_{1},\tau_{2})]p^{\tau_{2}}(v^{*})\\
 & \quad\geq\mathbb{P}[A^{\tau_{2}}w^{*}>z\,|\,E(\tau_{1},\tau_{2})]p^{\tau_{2}}(v^{*})>1\cdot\rho^{\frac{\tau_{2}}{\tau_{2}}}=\rho.\end{aligned}$$ Furthermore, the attack strategy (\[eq:ExampleStrategyV\]) satisfies Assumption \[FirstAssumption\] with $\overline{\kappa}=0$, because $\tau_{1}\geq\frac{\max\{v^{*}-\overline{v},0\}\tau_{2}}{\overline{v}}\geq\frac{(v^{*}-\overline{v})\tau_{2}}{\overline{v}}$, and thus, $\sum_{i=0}^{\tau-1}v(i)=v^{*}\tau_{2}\leq\overline{v}(\tau_{1}+\tau_{2})=\overline{v}\tau$.

The attack strategy (\[eq:ExampleStrategyV\]) can make the state grow arbitrarily large even if the interference power bound $\overline{v}$ is very small. This attack strategy is effective, because even if the attacker initially waits for a long duration without attacking, the state never reaches a small neighborhood of zero due to the disturbance. Hence, after waiting for a while, the attacker can consecutively attack with high interference powers to cause many transmission failures and make the state norm grow to large values. This is further illustrated in Section \[sec:Illustrative-Numerical-Examples\].

Jamming Interference and Bounded Disturbance
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To ensure stability under both disturbance and jamming, the 