This paper considers a distributed detection setup where agents in a network want to detect a time-varying signal embedded in temporally correlated noise. The signal of interest is the impulse response of an ARMA (auto-regressive moving average) filter, and the noise is the output of yet another ARMA filter which is fed white Gaussian noise. For this extended problem setup, which can prompt novel behaviour, we propose a comprehensive solution. First, we extend the well-known running consensus detector (RCD) to this correlated setup; then, we design an efficient implementation of the RCD by exploiting the underlying ARMA structures; and, finally, we derive the theoretical asymptotic performance of the RCD in this ARMA setup. It turns out that the error probability at each agent exhibits one of two regimes: either (a) the error probability decays exponentially fast to zero or (b) it converges to a strictly positive error floor. While regime (a) spans staple results in large deviation theory, regime (b) is new in distributed detection and is elicited by the ARMA setup. We fully characterize these two scenarios: we give necessary and sufficient conditions, phrased in terms of the zero and poles of the underlying ARMA models, for the emergence of each regime, and provide closed-form expressions for both the decay rates of regime (a) and the positive error floors of regime (b). Our analysis also shows that the ARMA setup leads to two novel features: (1) the threshold level used in RCD can influence the asymptotics of the error probabilities and (2) some agents might be weakly informative, in the sense that their observations do not improve the asymptotic performance of RCD and, as such, can be safely muted to save sensing resources. Numerical simulations illustrate and confirm the theoretical findings.

翻译：本文研究一种分布式检测场景，其中网络中的智能体希望检测嵌入在时间相关噪声中的时变信号。目标信号是ARMA（自回归滑动平均）滤波器的脉冲响应，而噪声是另一个以白高斯噪声为输入的ARMA滤波器的输出。针对这种可能引发新行为的扩展问题设置，我们提出了一套全面的解决方案。首先，我们将著名的运行共识检测器（RCD）扩展到这一相关场景；其次，通过利用底层ARMA结构，我们设计了RCD的高效实现；最后，我们推导了ARMA设置下RCD的理论渐近性能。结果表明，每个智能体的错误概率呈现两种模式之一：要么（a）错误概率以指数速度衰减至零，要么（b）收敛至严格为正的错误下限。模式（a）符合大偏差理论中的经典结论，而模式（b）是分布式检测中的新现象，由ARMA设置引发。我们完整刻画了这两种情形：给出了每种模式出现的充要条件（以底层ARMA模型的零点和极点形式表述），并提供了模式（a）的衰减率和模式（b）的正错误下限的闭式表达式。我们的分析还表明，ARMA设置带来两个新特性：（1）RCD中使用的阈值水平会影响错误概率的渐近行为；（2）某些智能体可能是弱信息性的，即其观测不会改善RCD的渐近性能，因此可安全静默以节省感知资源。数值仿真验证并支持了理论发现。