Strong Consistency and CLT for the Random Decrement Estimator

The random decrement technique (RDT), introduced in the sixties by Cole [NASA CR-2005, 1973], is a very performing method of analysis for vibration signature of a structure under ambientloading. But the real nature of the random decrement signature has been misunderstood until now.Moreover, the various interpretations were made in continuous time setting, while real experimentaldata are obtained in discrete time. In this paper, the really implemental discrete time algorithms arestudied. The asymptotic analysis as the number of triggering points go to infinity is achieved, anda Law of Large Numbers as well as a Central Limit Theorem is proved. Moreover, the limit as thediscretization time step goes to zero is computed, giving more tractable formulas to approximate therandom decrement. This is a new approach of the famous "Kac-Slepian paradox" [Ann. Math. Stat.,30, 1215-1228 (1959)]. The main point might be that the RDT is a very efficient functional estimator ofthe correlation function of a stationary ergodic Gaussian process. Very fast, it is to classical estimatorswhat Fast Fourier Transform (FFT) is to ordinary Discrete Fourier Transforms.