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We discuss the use of the Hilbert transform for the analysis of periodically non-stationary random signals (PNRSs), whose carrier harmonics are modulated by jointly stationary high-frequency narrow-band random processes. PNRS of this type are suitable models for numerous natural and man-made phenomena, including the vibration of a damaged mechanism. We show that the auto-covariance function of the signal and its Hilbert transform are the same, and that their cross-covariance functions differ only in their sign, meaning that the sum of squares of the signal and its Hilbert transform cannot be considered a 'squared envelope' and no new information is contained compared with the variance of the raw signal. A representation of the signal in the form of a superposition of high-frequency components is obtained and it is shown that these components are jointly periodically non-stationary random processes. The properties of the band-pass filtered signals are examined, and it is shown that band-pass filtering can reduce both the number of signal variance cyclic harmonics and their amplitudes. We show that it is possible to extract the quadratures of narrow-band high-frequency modulation processes using the Hilbert transform. The results obtained here theoretically substantiate the use of the Hilbert transform for the analysis of high-frequency modulation which occurs when a fault appears. They offer a new way to consider the traditional approach to vibration diagnosis. A processing technique that can be considered an alternative to envelope analysis is described, and its use in the analysis of a vibration signal is discussed., Competing Interests: Declaration of Competing Interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Roman Yuzefovych reports was provided by Karpenko Physico-mechanical institute of NAS of Ukraine. Roman Yuzefovych reports a relationship with Karpenko Physico-mechanical institute of NAS of Ukraine that includes: employment., (Copyright © 2023 ISA. Published by Elsevier Ltd. All rights reserved.)