Stationary Gaussian Sequences Research Articles

Asymptotic Expansion for the Distribution and Density Functions of the Quadratic Form of a Stationary Gaussian Process in the Large Deviation Cramer Zone

The work considers the asymptomic expansions in the large deviation Cramer zone for the distribution and its density functions of the quadratic form a stationary Gaussian sequence. To this end the general authors lemma [1], [3] for an arbitrary random variable with regular behavior of its cumulants is used.

Higher-order terms in asymptotic expansion for information loss in quantized random processes

Uniform quantization of random vectors onto e-grids eℤ n is considered. Higherorder terms in asymptotic expansions for the entropy of the e-quantized random vector and for the loss of the mutual information between two random vectors under such quantization as e→0+are obtained. The coefficients in these asymptotics are explicitly calculated for Gaussian distributed vectors. Taken for initial segments of stationary Gaussian sequences, these factors have limit average values per unit of time. For such sequences governed by state-space equations, computation of these average values is reduced to solutions of algebraic matrix Riccati and Lyapunov equations.

Adaptive Estimation of the Spectrum of a Stationary Gaussian Sequence

In this paper, we study the problem of nonparametric adaptive estimation of the spectral density f of a stationary Gaussian sequence. For this purpose, we consider a collection of finite-dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on possibly irregular grids or spaces of trigonometric polynomials). We estimate the spectral density by a projection estimator based on the periodogram and constructed on a data-driven choice of linear space from the collection. This data-driven choice is made via the minimization of a penalized projection contrast. The penalty function depends on If In,, but we give results including the estimation of this bound. Moreover, we give extensions to the case of unbounded spectral densities (long-memory processes). In all cases, we state non-asymptotic risk bounds in L2-norm for our estimator, and we show that it is adaptive in the minimax sense over a large class of Besov balls.

Distributional limit theorems over a stationary Gaussian sequence of random vectors

Let { X j } j=1 ∞ be a stationary Gaussian sequence of random vectors with mean zero. We study the convergence in distribution of a n −1∑ j=1 n (G(X j)−E[G(X j)]), where G is a real function in R d with finite second moment and { a n } is a sequence of real numbers converging to infinity. We give necessary and sufficient conditions for a n −1∑ j=1 n (G(X j)−E[G(X j)]) to converge in distribution for all functions G with finite second moment. These conditions allow to obtain distributional limit theorems for general sequences of covariances. These covariances do not have to decay as a regularly varying sequence nor being eventually nonnegative. We present examples when the convergence in distribution of a n −1∑ j=1 n (G(X j)−E[G(X j)]) is determined by the first two terms in the Fourier expansion of G( x).

Algorithms for accelerated convergence of adaptive PCA

We derive and discuss new adaptive algorithms for principal component analysis (PCA) that are shown to converge faster than the traditional PCA algorithms due to Oja, Sanger, and Xu. It is well known that traditional PCA algorithms that are derived by using gradient descent on an objective function are slow to converge. Furthermore, the convergence of these algorithms depends on appropriate choices of the gain sequences. Since online applications demand faster convergence and an automatic selection of gains, we present new adaptive algorithms to solve these problems. We first present an unconstrained objective function, which can be minimized to obtain the principal components. We derive adaptive algorithms from this objective function by using: 1) gradient descent; 2) steepest descent; 3) conjugate direction; and 4) Newton-Raphson methods. Although gradient descent produces Xu's LMSER algorithm, the steepest descent, conjugate direction, and Newton-Raphson methods produce new adaptive algorithms for PCA. We also provide a discussion on the landscape of the objective function, and present a global convergence proof of the adaptive gradient descent PCA algorithm using stochastic approximation theory. Extensive experiments with stationary and nonstationary multidimensional Gaussian sequences show faster convergence of the new algorithms over the traditional gradient descent methods.We also compare the steepest descent adaptive algorithm with state-of-the-art methods on stationary and nonstationary sequences.

Asymptotic distribution of sum and maximum for Gaussian processes

Let {Xn, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = EX0Xn. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

Asymptotic distribution of sum and maximum for Gaussian processes

Let {X n , n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = E X 0 X n . Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

Identification of polyperiodic Volterra systems by means of input–output noisy measurements

Identification methods of polyperiodically time-varying Volterra systems of finite order N by input–output measurements are proposed. The presence of additive signal-independent noise on the system input and output signal measurements is taken into account. Two identification methods are presented: both assume that the input signal be amplitude-modulated cyclostationary signal. The former supposes that the modulating sequence be white at least up to 2 N-order; the latter considers a stationary Gaussian sequence. They exploit the higher-order cyclostationarity selectivity property of the input signal to reject noises and interferences. The performance analysis, carried out by computer simulations for a periodically time-varying quadratic Volterra system, shows the capability of the proposed methods to operate satisfactorily in severe noise environments.

The Law of the Iterated Logarithm over a Stationary Gaussian Sequence of Random Vectors

Let {Xj}j = 1∞ be a stationary Gaussian sequence of random vectors with mean zero. We give sufficient conditions for the compact law of the iterate logarithm of $$(n{\text{ 2 log log }}n{\text{)}}^{{\text{ - 1/2}}} \sum\limits_{j{\text{ }} = {\text{ }}1}^n {(G(X_j ) - E[{\text{ }}G} (X_j )])$$ where G is a real function defined on ℝd with finite second moment. Our result builds on Ho,(6) who proved an upper-half of the law of iterated logarithm for a sequence of random variables.

Higher-order cyclostationarity-based methods for identifying volterra systems by input–output noisy measurements

Identification methods of nonlinear Volterra systems of finite order N by input–output measurements are proposed. The input signal is assumed to be amplitude-modulated cyclostationary signal and the measures of the input and output signals are supposed to be corrupted by signal-independent additive noise. Two identification methods are proposed: the former assumes that the modulating sequence be white at least up to 2 N-order; the latter considers a stationary Gaussian sequence. They exploit the higher-order cyclostationarity selectivity property of the input signal to reject noises and interferences. The performance analysis, carried out by computer simulations for a quadratic Volterra system, shows the capability of the proposed methods to operate satisfactorily in severe noise environment.

Convergence of the clipped sample autocorrelation and autocovariance

Statistics composed of clipped binary sequences are not sensitive to the existence of outliers. We estimate the autocorrelation and autocovariance functions of a linear Gaussian stationary sequence by the clipped binary series, and show the low of the iterated logairthm and the central limit theorem for these statistics.

Properties of a fourier bootstrap method for time series

A method for bootstrapping stationary Gaussian sequences is studied. The FFT is applied to the original data, randomized in the frequency domain, and the inverse FFT is applied. The result is a sequence whose second order properties are similar to those of the original sequence. Conditions under which the method is valid are given.

On extreme-order statistics and point processes of exceedances in multivariate stationary Gaussian sequences

This paper deals with a weak convergence of extreme-order statistics and point processes of exceedances built on the base of stationary and normal sequences of random vectors. The convolution of normal distribution and a double-exponential-type distribution for the limiting extreme-order statistics, and Cox distribution for the limiting point process of exceedances are found.

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Consider the stationary sequenceX 1=G(Z 1),X 2=G(Z 2),..., whereG(·) is an arbitrary Borel function andZ 1,Z 2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z 1 Z k+1) satisfyingr(0)=1 and ∑ =1 ∞ |r(k)| m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X n }, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetF n (·) be the empirical distribution function ofX 1,...,X n . We prove that, asn→∞, the empirical processn 1/2{F n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].

MODEL SELECTION AND ORDER DETERMINATION FOR TIME SERIES BY INFORMATION BETWEEN THE PAST AND THE FUTURE

Abstract. In this paper, the information between the past and the future of a Gaussian stationary sequence is calculated either by its spectral density or by its autocovariances, and is related to the problem of model fitting. It is demonstrated that the criterion of minimum mutual information is the generalization of that of maximum entropy. By employing the above information quantity, we propose a procedure, which is called LIC for simplicity, to obtain consistent estimate of the order of the Bloomfield model or the autoregressive model. In Monte Carlo studies, we illustrate the LIC procedure by several examples, and also estimate the spectral density of time series by the Bloomfield model and LIC method.

On identification of parallel block-cascade nonlinear models

The class of nonlinear systems studied in this paper is assumed to be modelled by parallel block-cascades. Such models are composed of parallel branches where each branch has a linear block in cascade with a zero-memory nonlinear block followed by another linear block. These types of models are extensively used to represent nonlinear dynamic systems and are known in the literature as Wiener-Hammerstein models. Using a zero-mean stationary white gaussian sequence as an input to such models, a structure identification criterion is developed, utilizing the bispectrum estimate of the output sequence only. The application of this criterion is shown by several simulation examples. Also, impulse response estimation of an example of such a model is considered to show the effectiveness of the proposed identification technique.

Nonparametric Estimation of Smooth Spectral Densities of Gaussian Stationary Sequences

The problem of spectral density estimation in the. Hilbert space norm of $L_2 ( - \pi ,\pi )$ is considered for a Gaussian stationary sequence. On the basis of the criterion involving the unbiased estimate for mean square risk of linear estimates we construct the class of nonlinear estimates for spectral density which are locally asymptotically minimax on the neighborhoods of smooth functions.

Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors

Limit theorems for functions of stationary mean-zero Gaussian sequences of vectors satisfying long range dependence conditions are considered. Depending on the rate of decay of the coefficients, the limit law can be either Gaussian or the law of a multiple Ito-Wiener integral. We prove the bootstrap of these limit theorems in the case when the limit is normal. A sufficient bracketing condition for these limit theorems to happen uniformly over a class of functions is presented.

Crack closure measurements and analysis of fatigue crack propagation under variable amplitude loading

This paper presents the results of an experimental programme conducted on a 316L stainless steel to analyze the behaviour of surface and embedded metallurgical defects in components. Crack closure measurements have been performed under plane stress and plane strain conditions, under constant amplitude loading and after overloads. The crack opening load was measured by four different methods including surface, center and bulk measurements. Fatigue crack growth rates were determined, at different R ratios, under constant amplitude and under random loading conditions, using a stationary Gaussian sequence (I = 99 %). The paper presents the experimental results and numerical analysis of the crack closure phenomenon.

A Strong Invariance Principle Concerning the $J$-Upper Order Statistics for Stationary Gaussian Sequences

It is shown that in the case of stationary Gaussian processes, the $J$th $(J \geq 1)$ record times $\{T_n, n \geq 1\}$ and the corresponding $J$-upper order statistics $\{X_{T_n - J + 1, T_n}, \ldots, X_{T_n, T_n}\}$ can almost surely be identified via a translation of the time index $n$ to the corresponding elements defined on a sequence of independent and identically distributed random variables. A construction method for approximating sequences of record times and the corresponding upper order statistics introduced by Haiman (1987a, b) for the case $J = 1$ is extended and applied under weaker conditions concerning the covariance function, and also under different sets of new hypotheses.