High frequency spectra and analyticity properties of time series in classical fluids

It is well known that spectra in classical fluids at equilibrium decay faster than any power of ω for high frequency. It is suggested that the ultimate decay is exponential for finite systems but slower than any exponential in infinite fluids. This renders time series in finite systems Wiener-predictable but Wiener-unpredictable in infinite fluids. It is further suggested, that time series of this kind and their autocorrelation functions are infinitely differentiable but nowhere analytic in t. This means that the Taylor series diverges everywhere and the series cannot be analytically continued into any part of the complex t-plane. Typical time series where these arguments should apply are the density n(k,t) of the fluid or the velocity and dipole moment of a test particle.