Random spectral measure for non Gaussian moving averages

ABSTRACT

We study the distribution of phases and amplitudes for the spectral representation of weighted moving averages of a general noise measure. The simple independent structure, known for the Gaussian case, and involving Rayleigh amplitude and uniform phase distributions, is lost for the non Gaussian noise case. We show that the amplitude/phase distributions exhibit a rich and more complex structure depending not just on the covariance of the process but specifically on the form of the kernel and the noise distribution. We present a theoretical tool for studying these distributions that follows from a proof of the spectral theorem that yields an explicit expression for the spectral measure. The main interest is in noise measures based on second-order Lévy motions since such measures are easily available through independent sampling. We approximate the spectral stochastic measure by independent noise increments which allows us to obtain amplitude/phase distributions that is of fundamental interest for analyzing processes in the frequency domain. For the purpose of approximating the moving average process through sums of trigonometric functions, we assess the mean square error of discretization of the spectral representation. For a specified accuracy, the approximation is explicitly given. We illustrate the method for the moving averages driven by the Laplace motion.

KEYWORDS:

• Generalized Laplace distribution
• Moving average processes
• Spectral representation
• Weakly stationary second-order processes

Mathematics Subject Classification:

• 60G10
• 60G12
• 60G51
• 60G57

Funding

The research was supported by the Riksbankens Jubileumsfond Grant Dnr: P13-1024:1 and the Swedish Research Council Grant Dnr: 2013-5180.