attacks need to be restricted in a way that high jamming interference powers at consecutive times are not allowed. To this end, we consider the following assumption.

\[SecondAssumption\] There exist scalars $\hat{\kappa},\hat{v}\geq0$ such that $$\begin{aligned}
\mathbb{P}\big[\sum_{i=t_{1}}^{t_{2}-1}v(i)\leq\hat{\kappa}+\hat{v}(t_{2}-t_{1})\big] & =1,\label{eq:attack_assumption_two}\end{aligned}$$ for all $t_{1},t_{2}\in\mathbb{N}_{0}$ with $t_{1}<t_{2}$.

Notice that (\[eq:attack\_assumption\_two\]) implies (\[eq:attack\_assumption\_one\]) (with $\overline{\kappa}=\hat{\kappa}$ and $\overline{v}=\hat{v}$), but the converse is not true. Assumption \[SecondAssumption\] is thus more restrictive than Assumption \[FirstAssumption\]. In particular, under Assumption \[SecondAssumption\], the attacker can attack with a jamming interference power $v^{*}>\hat{v}$ consecutively for at most $\lfloor\hat{\kappa}/(v^{*}-\hat{v})\rfloor$ time steps; thus, the destabilizing attacks discussed in Example \[ExampleStrategy\] are avoided.

Assumption \[SecondAssumption\] is related to other characterizations of malicious attacks in the literature. In particular, in the continuous-time deterministic denial-of-service attack characterization of [@de2015inputtran], the number of attacks in a given time frame as well as the total duration of those attacks are bounded by certain ratios of the length of that time frame. Under that characterization, the maximum possible length of a continuous attack duration is bounded, which enables analysis of input-to-state stability under disturbance. The restriction on jamming through Assumption \[SecondAssumption\] is similar, since long consecutive emissions of high powered interference signals are not allowed. We note, however, that 