Quick Change Detection in Discrete-Time in Presence of a Covert Adversary

Abstract: 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 $γ$.

• The paper reintroduces and clarifies Wald, Khan approximations for stop test times which become asymptotically exact due to vanishing overshoot conditions.
• The authors clarify that covertness uniquely occurs when inducing pointwise convergence without degrading expectations.
• In Gaussian and exponential settings, optimized behaviors expose strategies for maximum clandestine impact.

Problem setting and motivation

This paper studies quickest change detection (QCD) in discrete time under an adversary who is aware of the detector's false-alarm constraint. In the classical Lorden formulation, i.i.d. observations $X_1, X_2, \ldots$ have pre-change density $q_0$ and post-change density $q_1$, and the decision-maker seeks a stopping time minimizing the worst-case average detection delay subject to $E_\infty[T] \geq \gamma$. Page's CuSum rule solves this problem optimally (2601.20022), and Lorden's asymptotic result gives

$n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$

when the post-change distribution is fixed. The key departure here is that the adversary knows $\gamma$ and selects a $\gamma$-dependent post-change density $q_\gamma$ that converges pointwise to $q_0$ as $\gamma \to \infty$, so that detection becomes progressively harder precisely in the regime of large false-alarm budgets. This dependence invalidates Lorden's asymptotics and motivates a new analysis.

The paper defines covertness as the condition $q_0$0: the optimal delay grows at the same order as the largest admissible lower bound on the expected time to false alarm, rendering the CuSum procedure ineffective. It also introduces a total damage measure $q_0$1 for a strictly increasing $q_0$2 with $q_0$3, quantifying cumulative adversarial impact accrued before detection.

Main asymptotic results

The technical core rests on showing that the expected overshoots of the underlying SPRT at both boundaries vanish as $q_0$4:

$q_0$5

This behavior is not generic — classical renewal-theoretic results characterize the limiting overshoot distribution rather than its disappearance — but it holds in this regime because the log-likelihood ratio increments shrink with $q_0$6. Since these conditions involve unknown overshoot random variables, the paper provides verifiable sufficient conditions involving only the single-increment conditional expectations $q_0$7 and $q_0$8.

Under these conditions, three results are established:

• Wald-type approximations become exact: the classical Wald approximations for SPRT expected stopping times, obtained by neglecting boundary excesses, are proven to give the exact limiting behavior of $q_0$9 and $q_1$0.
• Khan's approximations become exact: similarly, Khan's approximations for CuSum run lengths are shown to be asymptotically exact:

$q_1$1

• Sharp characterization of $q_1$2: solving $q_1$3 via the Lambert $q_1$4 function yields a trichotomy governed by the product $q_1$5:

Regime	Asymptotic $q_1$6
$q_1$7	$q_1$8
$q_1$9	$E_\infty[T] \geq \gamma$0
$E_\infty[T] \geq \gamma$1	$E_\infty[T] \geq \gamma$2

where $E_\infty[T] \geq \gamma$3 satisfies $E_\infty[T] \geq \gamma$4, tending to 1 as $E_\infty[T] \geq \gamma$5 and to 0 as $E_\infty[T] \geq \gamma$6. The corresponding optimal threshold satisfies $E_\infty[T] \geq \gamma$7 in the last regime.

A corollary identifies covertness: the maximal order of $E_\infty[T] \geq \gamma$8 is attained when $E_\infty[T] \geq \gamma$9, which is exactly the covert regime $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$0. For damage with $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$1, $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$2, and $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$3, the total damage is maximized in this same transition regime, scaling as $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$4; both faster and slower convergence of $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$5 toward $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$6 yield strictly smaller orders ($n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$7).

An important caveat stated explicitly by the authors: the threshold asymptotics above do not generalize the classical $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$8 result, since they require the vanishing-overshoot conditions, which fail when $n(\gamma) \sim \frac{\log \gamma}{D(q_1 \| q_0)}$9 does not converge to $\gamma$0.

Gaussian and exponential models

For Gaussian observations with $\gamma$1 and $\gamma$2 where $\gamma$3 with $\gamma$4, the paper proves via lengthy error-function analysis that the sufficient overshoot conditions hold, yielding

$\gamma$5

Consequently, an adversary perturbing $\gamma$6 remains covert whenever $\gamma$7 or $\gamma$8. With $\gamma$9, mean-shift attacks with $\gamma$0 and variance attacks with $\gamma$1 both maximize total damage at $\gamma$2; the latter recovers the scaling reported in covert communication work against sequential detectors.

For exponential observations with rate $\gamma$3, the sufficient conditions are verified directly, giving covertness when $\gamma$4 — e.g., $\gamma$5 with $\gamma$6 — and again maximal damage $\gamma$7 when the mean difference is $\gamma$8.

Limitations and open questions

Several limitations are acknowledged. First, the main results hold only under the vanishing-overshoot conditions, whose given sufficient conditions are stronger than necessary; the authors note that a simple uniform-convergence condition on $\gamma$9 suffices but is too restrictive to cover the Gaussian and exponential models treated here. Second, the verification of the sufficient conditions for the Gaussian case relies partly on numerical interval partitioning (bounds checked over finitely many subintervals of a compact range), so parts of that argument are not purely analytic. Third, the analysis assumes i.i.d. observations, known parametric families, and a stationary post-change model; non-stationary adversarial strategies and adversaries without knowledge of $q_\gamma$0 fall outside this framework. The paper leaves open the identification of structural properties of the pair $q_\gamma$1 that would guarantee vanishing overshoots automatically, and the extension of exact asymptotics beyond the two parametric families analyzed.

Conclusion

The paper establishes that classical QCD asymptotics fail when the post-change distribution depends on the false-alarm constraint, and replaces them with rigorous trichotomous scaling laws expressed through KL divergences and the Lambert $q_\gamma$2 function. Its central findings are that Wald's and Khan's long-standing approximations become asymptotically exact in this regime, that covertness ($q_\gamma$3) occurs precisely when $q_\gamma$4, and that total adversarial damage is maximized at this detectability/covertness transition with order $q_\gamma$5. These results delineate, for Gaussian and exponential observation models, the exact perturbation rates — parameters deviating as $q_\gamma$6 with $q_\gamma$7 — at which an adversary can remain undetected indefinitely while inflicting $q_\gamma$8 cumulative damage.