We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. For general non-Gaussian noise, assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio (LR) of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold. The threshold is optimal in the sense that the reliable detection is possible by a transformed principal component analysis (PCA) above it. From the mean and the variance of the limiting Gaussian for the log-LR, we compute the limit of the sum of the Type-I error and the Type-II error of the likelihood ratio test. We also prove similar results for a rank-one spiked IID model where the noise is asymmetric but the signal is symmetric.

翻译：我们考虑在秩一尖峰Wigner模型中检测信号存在的问题。针对一般非高斯噪声，假设信号服从Rademacher先验分布，我们证明当信噪比低于特定阈值时，尖峰模型相对于零模型的对数似然比（LR）收敛于高斯分布。该阈值具有最优性：在其之上通过变换主成分分析（PCA）可实现可靠检测。根据对数似然比极限高斯的均值与方差，我们计算了似然比检验第一类错误与第二类错误之和的极限。对于噪声非对称但信号对称的秩一尖峰独立同分布模型，我们也证明了类似结论。