In a recent paper Fay and Philippe [ESAIM: PS 6 (2002) 239–258] proposed a goodness-of-fit test for long-range dependent processes which uses the logarithmic contrast as information measure. These authors established asymptotic normality under the null hypothesis and local alternatives. In the present note we extend these results and show that the corresponding test statistic is also normally distributed under fixed alternatives.

In this paper, we make use of the information measure introducedby Mokkadem (1997) for building a goodness-of-fit test forlong-range dependent processes.Our test statistic is performed in the frequency domain and writes asa non linear functional of the normalized periodogram. We establishthe asymptotic distribution of our statistic under the nullhypothesis. Under specific alternative hypotheses, we prove that the powerconverges to one. The performance of our test procedure isillustrated from different simulated series. In particular,we compare its size and its power with test of Chenand Deo.

Minimax bounds for the risk function of estimators of functionals ofthe spectral density of Gaussianfields are obtained. This result is a generalization of a previous result of Khas'minskii and Ibragimov on Gaussian processes.Efficient estimators are then constructed for these functionals. In the case of linear functionals these estimators aregiven for all dimensions. For non-linear integral functionals, theseestimators are constructed for the two and three dimensional problems.

We show that the periodogram behaves differently for a weakly dependent process with a small monotonic trend and a stationary strongly dependent process. In the former case it has a non-central -distribution with noncentrality parameter tending to 0 uniformly outside intervals . In the latter case it has λα −1 times a central -distribution, 0 < α < 1.

The model Y(t) = s(t | θ) + ε(t) is studied in the case that observations are made at scattered points τ j in a subset of Rp and θ is a finite-dimensional parameter. The particular cases of 0 = (α, β) and (α, β, ω) are considered in detail. Consistency and asymptotic normality results are developed assuming that the spatial series ε(·) and the point process {τ j} are independent, stationary and mixing. The estimates considered are equivalent to least squares asymptotically and are not generally asymptotically efficient.

Contributions of the paper include: study of the Rp case, management of irregularly placed observations, allowance for abnormal domains of observation and the discovery that aliasing complications do not arise when the point process {τ j} is mixing. There is a brief discussion of the construction and properties of maximum likelihood estimates for the spatial-temporal case.

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ < x <∞.

It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

Suppose that it is desired to determine the frequency, known to lie in a range 0 < σ ≦ ω ≦ ΩT <∞, of a periodic effect modulating the rate of occurrence of a Poisson point process. It is shown that if ΩT increases sufficiently slowly with T, the length of the observation period, then the frequency corresponding to the maximum of the Bartlett periodogram over this range provides a consistent estimate of the unknown frequency. The asymptotic covariance matrix is found for the maximum likelihood estimates of the parameters in a cyclic Poisson model, and the close analogue with frequency estimation for Gaussian processes is emphasized. It is pointed out, however, that in the point-process case periodogram estimates will be efficient only for very special models.

The paper looks at the asymptotic properties of the finite Walsh–Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.

Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function ψ (x) we shall give an estimator of whose asymptotic variance is O(n–1), where Φ(·) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.

In fitting a certain parametric family of spectral densities fθ (x) to a Gaussian stationary process with the true spectral density g (x), we propose two estimators of θ, say by minimizing two criteria D1 (·), D2(·) respectively, which measure the nearness of fθ (x) to g (x). Then we investigate some asymptotic behavior of with respect to efficiency and robustness.

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.

The asymptotic properties of the periodogram of a weakly stationary time series for the triangular array of fundamental frequencies is studied. For linear Gaussian processes, results are obtained relating the asymptotic distribution of certain Riemann sums of the periodogram of the process to those of the periodogram of the innovation process.

Conditions are given for the family of distributions of a stationary, discrete-time, Gaussian, vector-valued time-series with covariance structure given up to a finite number of parameters to satisfy the asymptotic differentiability conditions introduced by Le Cam (1969).