Abstract
The Linear Gaussian white noise process is an independent and identically distributed (iid) sequence with zero mean and finite variance with distribution N (0, σ2 ) . Hence, if X1, x2, …, Xn is a realization of such an iid sequence, this paper studies in detail the covariance structure of X1d, X2d, …, Xnd, d=1, 2, …. By this study, it is shown that: 1) all powers of a Linear Gaussian White Noise Process are iid but, not normally distributed and 2) the higher moments (variance and kurtosis) of Xtd, d=2, 3, … can be used to distinguish between the Linear Gaussian white noise process and other processes with similar covariance structure.
Highlights
IntroductionThe objective of estimation procedures is to produce residuals (the estimated noise sequence) with no apparent deviations from stationarity, and in particular with no dependence among these residuals
The objective of estimation procedures is to produce residuals with no apparent deviations from stationarity, and in particular with no dependence among these residuals
The specific objective of this paper is to investigate if powers of Xt,t ∈ Z are iid and to determine the distribution o= f Yt2 − E2 (Yt)
Summary
The objective of estimation procedures is to produce residuals (the estimated noise sequence) with no apparent deviations from stationarity, and in particular with no dependence among these residuals. If there is no dependence among these residuals, we can regard them as observations of independent random variables; there is no further modeling to be done except to estimate their mean and variance. If there is significant dependence among the residuals, we need to look for the noise sequence that accounts for the dependence [1]. We examine the covariance structure of powers of the noise sequence when the noise sequence is assumed to be independent and identically distributed normal (Gaussian) random variates with mean zero and finite va-. Some simple tests for checking the hypothesis that the residuals and their powers are observed values of independent and identically distributed random variables are considered. Considered are tests for normality of the residuals and their powers
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