Statistics on exponential averaging of periodograms

The algorithm of exponential averaging applied to subsequent periodograms of a stochastic process is used to estimate the power spectral density (PSD). For an independent process, assuming the periodogram estimates to be distributed according to a χ² distribution with two degrees of freedom, the probability density function (PDF) of the PSD estimate is derived. A closed expression is obtained for the moments of the distribution. Surprisingly, the proof of this expression features some new insights into the partitions and Euler's infinite product. For large values of the time constant of the averaging process, examination of the cumulant generating function shows that the PDF approximates the Gaussian distribution. Although restrictions for the statistics are seemingly tight, simulation of a real process indicates a wider applicability of the theory