We study the problem of covert quickest change detection in a discrete-time setting, where a sequence of observations undergoes a distributional change at an unknown time. Unlike classical formulations, we consider a covert adversary who has knowledge of the detector's false alarm constraint parameter $γ$ and selects a stationary post-change distribution that depends on it, seeking to remain undetected for as long as possible. Building on the theoretical foundations of the CuSum procedure, we rigorously characterize the asymptotic behavior of the average detection delay (ADD) and the average time to false alarm (AT2FA) when the post-change distribution converges to the pre-change distribution as $γ\to \infty$. Our analysis establishes exact asymptotic expressions for these quantities, extending and refining classical results that no longer hold in this regime. We identify the critical scaling laws governing covert behavior and derive explicit conditions under which an adversary can maintain covertness, defined by ADD = $Θ(γ)$, whereas in the classical setting, ADD grows only as $\mathcal{O}(\log γ)$. In particular, for Gaussian and Exponential models under adversarial perturbations of their respective parameters, we asymptotically characterize ADD as a function of the Kullback--Leibler divergence between the pre- and post-change distributions and $γ$.

翻译：本文研究离散时间设置下的隐蔽快速变化检测问题，其中观测序列在未知时刻发生分布变化。与经典模型不同，我们考虑一个隐蔽对手，该对手知晓检测器虚警约束参数$γ$，并据此选择一个依赖于该参数的静态变化后分布，旨在尽可能长时间不被检测。基于CuSum程序的理论基础，我们严格刻画了当变化后分布随$γ\to \infty$收敛于变化前分布时，平均检测延迟（ADD）与平均虚警时间（AT2FA）的渐近行为。我们的分析建立了这些量的精确渐近表达式，扩展并完善了在此机制下不再成立的经典结论。我们揭示了控制隐蔽行为的关键标度律，并推导出对手能够维持隐蔽性的显式条件——此时ADD满足$Θ(γ)$，而在经典设定中ADD仅以$\mathcal{O}(\log γ)$增长。特别地，针对高斯模型与指数模型在参数受对抗性扰动的情形，我们通过变化前后分布的Kullback--Leibler散度与$γ$的函数关系，渐近刻画了ADD的特性。