Dominique Dehay

Spectral analysis of the covariance of the almost periodically correlated processes

This paper deals with the spectrum of the almost periodically correlated (APC) processes defined on . It is established that the covariance kernel of such a process admits a Fourier series decomposition, K(s+t, s) , whose coefficient functions b[alpha] are the Fourier transforms of complex measures m[alpha], [alpha][set membership, variant], which are absolutely continuous with respect to the measure mo. Considering the APC strongly harmonizable processes, the spectral covariance of the process can be expressed in terms of these complex measures m[alpha]. The usual estimators for the second order situation can be modified to provide consistent estimators of the coefficient functions b[alpha] from a sample of the process. Whenever the measures m[alpha] are absolutely continuous with respect to the Lebesgue measure, so m[alpha](d[lambda])=f[alpha]([lambda]) d[lambda], the estimation of the corresponding density functions f[alpha] is considered. Under hypotheses on the covariance kernel K and on the coefficient functions b[alpha], we establish rates of convergence in quadratic mean and almost everywhere of these estimators.

Component statistical analysis of second order hidden periodicities

The component method is applied to define estimators of the periods for Gaussian periodically correlated random processes (mathematical model of stochastic oscillations). The properties of these period estimators are obtained using some small parameter method and the rate of convergence is shown to be optimal. Specific results for the simplest models of periodically correlated process are presented. Finally the method is illustrated with a simulated sequence and a real life vibration signal.

Testing stationarity for stock market data

Abstract Numerous examples in recent research on the volatility of asset returns show that the data frequently show a lack of covariance stationarity. This paper introduces a general method of testing stock market data for covariance stationarity. The test presented is based on the result of Dehay and Leśkow and allows us to detect non-stationarity for general time-series models.

On a class of asymptotically stationary harmonizable processes

We prove that every harmonizable process with [sigma]-finite bimeasure is asymptotically stationary and we give its associated spectral measure.

Empirical determination of the frequencies of an almost periodic time series

This article deals with the problem of the determination of the finite or countable set of frequencies belonging to any arbitrary almost periodic (in the sense of Bohr) time series. For this purpose, we present a simple computation procedure based on the local maxima of the modulus of a weighted Fourier transform from finite observation of the time series, computed at frequencies in a finite uniform grid of [0, 2π). We study the convergence of this algorithm as the length of the observation goes to infinity. First non-random signals are considered. Then we tackle the case of a signal disturbed by an additive noise. Finally we show how the algorithm can be applied to almost periodically correlated random time series.

Central Limit Theorem in the Functional Approach

The central limit theorem is proved within the framework of the functional approach for signal analysis. In this framework, a signal is modeled as a single function of time rather than a stochastic process. Distribution function, expectation, and all the familiar probabilistic parameters are built starting from this single function of time by resorting to the concept of relative measure. Furthermore, the concept of independence among functions of time can be introduced. In the paper it is shown that if a sequence of independent signals fulfills some mild regularity assumptions, then the asymptotic distribution of the appropriately scaled average of such signals has a limiting normal distribution. The approach is shown to be useful when only one realization of a signal is available and no ensemble of realizations is observed or exists. The obtained results also allow one to rigorously justify stochastic models for signals and channels that up to now have been derived starting from a deterministic description of phenomena and for which the inferred stochastic model is built invoking a not proved ergodicity property. An application to the statistical characterization of the output signal of a multipath Doppler channel is presented.

Strongly harmonizable approximations of bounded continuous random fields

Every continuous and bounded random field on k is the limit of a sequence of strongly harmonnizable random fields, uniformly on compact subsets of k. These harmonizable fields are obtained from the given random field by nonstationary linear filterings.

Block Bootstrap for Poisson‐Sampled Almost Periodic Processes

Let be an almost periodically correlated process and {N(t),t≥0} be a homogeneous Poisson process and {T,k≥1} be its jump moments. We assume that and {N(t),t≥0} are independent. Moreover, the process is not observed continuously but only in the time moments {T,k≥1}; In this paper, we focus on the estimation of the cyclic means of . The asymptotic normality of the rescaled error of the estimator is shown. Additionally, the bootstrap method based on the circular block bootstrap is proposed. The consistency of the bootstrap technique is proved, and the bootstrap pointwise and simultaneous confidence intervals for the cyclic means are constructed. The results are illustrated by a simulated data example.

Empirical determination of the frequencies of an almost periodic sequence

This note is concerned with the problem of determination of the countable set /spl Lambda/={/spl lambda//sub 1/, /spl lambda//sub 2/, ...} of frequencies belonging to an almost periodic sequence by methods in which a finite number of frequencies {/spl lambda//sub 1//sup (n)/, /spl lambda//sub 2//sup (n)/, ..., L(K/sub n/)/sup (n)/}=/spl Lambda//sub n/ are produced at each stage n. We seek algorithms for which /spl Lambda//sub n/ converges to /spl Lambda/ but yet each /spl Lambda//sub n/ is not too big.

On the product of two harmonizable time series

In order to state sufficient conditions for the harmonizability of the product time series of two harmonizable time series, the notion of Lp-harmonizable time series is introduced for 1 [less-than-or-equals, slant] p [less-than-or-equals, slant] + [infinity]. Then, the problem of the product of two stochastic measures is tackled and Fubini type theorems are deduced. We derive sufficient conditions for the harmonizability of a weighted convolution time series of two harmonizable time series. As an application to nonlinear prediction theory, asymptotically unbiased estimors for values of the cross spectral bimeasure of two harmonizable time series are given. The L1-convergence of these estimators towards some random variables is established from the law of large numbers stated for Lp-harmonizable series. Sufficient conditions for the a.e. convergence are obtained from the strong law of large numbers. The case of two jointly stationary harmonizable series is also considered. The results apply to the estimation of the asymptotic spectral measure of some asymptotically stationary series.