Timing analysis is a powerful tool with which to shed light on the still obscure emission physics and geometry of the prompt emission of GRBs. Fourier power density spectra (PDS) characterise time series as stochastic processes and can be used to search for coherent pulsations and to investigate the dominant variability timescales. Because of the limited duration and of the statistical properties, modelling the PDS of individual GRBs is challenging, and only average PDS of large samples have been discussed in the literature. We characterise the individual PDS of GRBs in terms of a stochastic process, and carry out for the first time a systematic search for periodic signals and for a link between the PDS and other observables. We present a Bayesian procedure that uses a Markov chain Monte Carlo technique and apply it to study 215 bright long GRBs detected with the Swift Burst Alert Telescope from January 2005 to May 2015. The PDS are modelled with a power-law either with or without a break. Two classes of GRBs emerge: with or without a unique dominant timescale. A comparison with active galactic nuclei (AGNs) reveals similar distributions of PDS slopes. Unexpectedly, GRBs with subsecond-dominant timescales and duration longer than a few tens of seconds in the source frame appear to be either very rare or altogether absent. Three GRBs are found with possible evidence for a periodic signal at ~3 sigma (Gaussian) significance, corresponding to a multitrial chance probability of ~1%. Thus, we found no compelling evidence for periodic signals. The analogy between the PDS of GRBs and of AGNs could tentatively indicate similar stochastic processes that rule BH accretion across different BH mass scales and objects. In addition, we find evidence that short dominant timescales and duration are not completely independent of each other, in contrast with commonly accepted paradigms (abridged).
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