  In this paper, we investigate the stationarity of stochastic processes in the
fractional Fourier domains. We study the stationarity of a stochastic process
after performing fractional Fourier transform (FRFT), and discrete fractional
Fourier transform (DFRT) on both continuous and discrete stochastic processes,
respectively. Also we investigate the stationarity of the fractional Fourier
series (FRFS) coefficients of a continuous time stochastic process, and the
stationarity of the discrete time fractional Fourier transform (DTFRFT) of a
discrete time stochastic process. Closed formulas of the input process
autocorrelation function and pseudo-autocorrelation function after performing
the fractional Fourier transform are derived given that the input is a
stationary stochastic process. We derive a formula for the output
autocorrelation as a function of the $a^{th}$ power spectral density of the
input stochastic process, also we derived a formula for the input fractional
power spectral density as a function of the fractional Fourier transform of the
output process autocorrelation function. We proved that, the input stochastic
process must be zero mean to satisfy a necessary but not a sufficient condition
of stationarity in the fractional domains. Closed formulas of the resultant
statistics are also shown. It is shown that, in case of real input process, the
output process is stationary if and only if the input process is white. On the
other hand, if the input process is a complex process, it should be proper
white process to obtain a stationary output process.
