Abstract: Explicit formulae for the autocorrelation functions and the fluctuation spectra with respect to complex functions of the irreducible three-state Markov processes were investigated. By the theory of cubic and quintic equations, the necessary and sufficient condition for the fluctuation spectra with respect to complex functions to be nonmonotonic on [0,+∞) was presented when there existed coinciding eigenvalues for the transition rate matrix, and the sufficient conditions for the fluctuation spectra with respect to complex functions to be nonmonotonic on [0,+∞)were given when there existed distinct nonzero eigenvalues.

Key words: autocorrelation function, nonequilibrium, steady state, three-state Markov processes, power spectral density, autocorrelation function, nonequilibrium, steady state, three-state Markov processes