Random or Nonrandom Signal in High-Dimensional Regimes

This paper proposes a new hypothesis test to check the randomness and nonrandomness of the contaminated high dimensional signal. Specifically, for a signal plus noise model, we propose a statistic to distinguish whether the corresponding signal is random or not. In order to analyze the performance of the proposed method, we also prove two important results for signal plus noise models: 1) No eigenvalues outside the support of the limiting spectral distribution of the noncentered and centered sample covariance matrix; and 2) Exact separation of eigenvalues of the noncentered and centered sample covariance matrix. Simulation studies demonstrate that our proposed method works well under a variety of settings.