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We investigate the performance of Neyman-Pearson detection of a stationary Gaussian process in noise, using a large wireless sensor network (WSN). In our model, each sensor compresses its observation sequence using a linear precoder and a final decision is taken by a fusion center (FC) based on the compressed information. Two families of precoders are studied: random i.i.d. precoders and orthogonal precoders. We analyse their performance under a regime where both the number of sensors k and the number of samples n per sensor tend to infinity at the same rate, that is, k/n→ c ∈ [0,1]. Contributions are as follows. 1) Using results from random matrix theory and large Toeplitz matrices, we prove that, when the above families of precoders are used, the miss probability of the Neyman-Pearson detector converges exponentially to zero. Closed form expressions of the corresponding error exponents are derived. 2) In particular, we propose a practical orthogonal precoding strategy, the Principal Frequencies Strategy (PFS), which achieves the best error exponent among all orthogonal strategies, and which requires very little signaling overhead between the central processor and the nodes of the network. 3) When the PFS is used, a simplified low-complexity testing procedure can be implemented at the FC. We show that the proposed suboptimal test enjoys the same error exponent as the Neyman-Pearson test, which indicates a similar asymptotic behavior of the performance. We illustrate our findings by numerical experiments on several examples.