This paper studies a statistical approach to detect and diagnose a particular type of vibration impacting the control surfaces of civil aircraft. The considered phenomenon is called Limit Cycle Oscillation (LCO). It consists of an unwanted sustained oscillation of a control surface due to the combined effect of aeroelastic phenomena and an increased level of mechanical free play in the elements that connect the control surface to the aerodynamic surface. The state-of-the-art for LCO prevention is mainly based on regular free play checks performed on ground during maintenance operations. The detection is mainly achieved by the crew, and especially the pilot who can fill in a so-called "vibration reporting sheet" to describe the phenomena felt during the flight. Thus, the pilot sensitivity to vibration is still the only reference for LCO detection. In the Flight Control System (FCS) of modern aircraft there exist already several certified algorithms for the detection of vibrations of different nature, which use dedicated local sensors to monitor the control surface behaviour. The same kind of sensors have been chosen in a local approach, which eases the isolation of the vibration sources. This paper studies a new statistical approach based on the Generalized Likelihood Ratio Test (GLRT) in order to improve the state-of-the-art for LCO detection and diagnosis. The test and its theoretical performance are derived and validated. A straightforward method compliant with real-time implementation constraint for LCO prediction is proposed. A Monte Carlo test campaign is performed in order to assess the robustness and the detection/diagnosis performance of the proposed algorithm under different operating conditions.