Advanced

Random spectral measure for non Gaussian moving averages

Baxevani, Anastassia LU and Podgórski, Krzysztof LU (2018) In Communications in Statistics - Theory and Methods 47(2). p.448-462
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... (More)

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.

(Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Generalized Laplace distribution, Moving average processes, Spectral representation, Weakly stationary second-order processes
in
Communications in Statistics - Theory and Methods
volume
47
issue
2
pages
448 - 462
publisher
Marcel Dekker
external identifiers
  • scopus:85029419270
ISSN
0361-0926
DOI
10.1080/03610926.2017.1303737
language
English
LU publication?
yes
id
48e82630-5816-479a-b103-a7b4cb6fb483
date added to LUP
2017-09-29 13:50:38
date last changed
2018-02-20 11:16:16
@article{48e82630-5816-479a-b103-a7b4cb6fb483,
  abstract     = {<p>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.</p>},
  author       = {Baxevani, Anastassia and Podgórski, Krzysztof},
  issn         = {0361-0926},
  keyword      = {Generalized Laplace distribution,Moving average processes,Spectral representation,Weakly stationary second-order processes},
  language     = {eng},
  month        = {01},
  number       = {2},
  pages        = {448--462},
  publisher    = {Marcel Dekker},
  series       = {Communications in Statistics - Theory and Methods},
  title        = {Random spectral measure for non Gaussian moving averages},
  url          = {http://dx.doi.org/10.1080/03610926.2017.1303737},
  volume       = {47},
  year         = {2018},
}