2019/11/16 by Anastassia Baxevani, Baxevani, Anastassia, Krzysztof Podgórski +1
Engineering · Physics and Astronomy · #Control Systems and Identification #FOS: Mathematics #Fault Detection and Control Systems #Probability (math.PR) #Scientific Research and Discoveries
paper · pdf · doi:10.48550/arxiv.1911.07061
openalex publication_date 2019/11/16 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
It has been observed that an interesting class of non-Gaussian stationary\nprocesses is obtained when in the harmonics of a signal with random amplitudes\nand phases, frequencies can also vary randomly. In the resulting models, the\nstatistical distribution of frequencies determines the process spectrum while\nthe distribution of amplitudes governs the process distributional properties.\nSince decoupling the distributional and spectral properties can be advantageous\nin applications, we thoroughly investigate a variety of properties exhibited by\nthese models. We extend previous work that represented processes as finite sum\nof harmonics, by conveniently embedding them into the class of harmonizable\nprocesses. Harmonics are integrated with respect to independently scattered\nsecond order non-Gaussian random measures. The proposed approach provides with\na proper mathematical framework that allows to study spectral, distributional,\nand ergodic properties. The mathematical elegance of these representations\navoids serious conceptual and technical difficulties with limiting behavior of\nthe models while at the same time facilitates derivation of their fundamental\nproperties. In particular, the multivariate distributions are obtained and the\nasymptotic behavior of time averages is formally derived through the strong\nergodic theorem. Several deficiencies following from the previous approaches\nare resolved and some of the results appearing in the literature are corrected\nand extended. It is shown that due to the lack of ergodicity processes exhibit\nan interesting property of non-trivial randomness remaining in the limit of\ntime averages. This feature maybe utilized to modelling signals observed in the\npresence of influential and variable random factors.\n