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Let (X"n) be a sequence of d-dimensional stationary Gaussian vectors, and let M"n denote the partial maxima of {X"k,1@?k@?n}. Suppose that there are missing data in each component of X"k and let M@?"n denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector (M@?"n,M"n) where the correlation and cross-correlation satisfy some dependence conditions.

In this paper we derive the asymptotic normality and a Berry–Esseen type bound for the kernel conditional density estimator proposed in Ould-Saïd and Cai (2005) [26] when the censored observations with multivariate covariates form a stationary α-mixing sequence.

In this paper we study the asymptotic joint behavior of the maximum and the partial sum of a multivariate Gaussian sequence. The multivariate maximum is defined to be the coordinatewise maximum. Results extend univariate results of McCormick and Qi. We show that, under regularity conditions, if the maximum has a limiting distribution it is asymptotically independent of the partial sum. We also prove that the maximum of a stationary sequence, when normalized in a special sense which includes subtracting the sample mean, is asymptotically independent of the partial sum (again, under regularity conditions). The limiting distributions are also obtained.

We study dependence orderings for functionals of k-variate point processes Φ and Ψ. We view the first process as a collection of counting measures, whereas the second as the sequences of interpoint distances. Subsequently, we establish regularity properties of stationary sequences which generalize known results for iid case. The theoretical results are illustrated by many special cases including comparison of multivariate sums and products, comparison of multivariate shock models and queueing systems.

Let {Xn, n⩾1} be a sequence of stationary negatively associated random variables, Sj(l)=∑li=1Xj+i, Sn=∑ni=1Xi. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums∑j=1nfSj(l)l,n⩾1,are investigated. Applications to limiting distributions of estimators of VarSn are also discussed.

For a nonnegative strictly stationary random sequence satisfying the “minimal” dependence condition necessary and sufficient conditions for the relative stability are found. As an application the well-known Khinchine stability result for i.i.d. random variables is proved for uniformly strong mixing sequences.

In this paper we propose a new approach for estimating the unknown parameter in the stochastic linear regressive model with stationary ergodic sequence of covariates. Under mild conditions on the joint distribution of the covariate and the error, the estimator constructed is shown to be strongly consistent in two important special cases: (1) The sequence of (variate, covariate) is independent identically distributed (i.i.d.), and (2) the sequence of variates is a stationary autoregressive series. The asymptotical normality is also discussed under more assumptions on the distribution of the covariate.

Let { X n , n ≥ 1} be a stationary sequence of ρ-mixing random variables satisfying EX n = μ, EX 2 n Sn / n → σ 2 > 0. This paper presents a class of estimators of σ and investigates their weak consistency as well as their asymptotic normality. Applications to the self-normalizing central limit theorem and confidence for the sample mean are also discussed.

When Z is a random L2H-valued measure, where H is a Hilbert space, we prove that there exists an L2Cq-valued measure, which may depend on constraints and which best sums up the random measure Z according to a stationary criterion. Then a technique to reduce a random function is deduced from the above result. The random function is defined on a locally compact abelian group and is stationary and continuous. This work generalizes Brillinger′s results on stationary time series.

Burg′s maximum entropy spectrum given the first p + 1 autocovariances is the spectrum of an autoregressive process of order p. The purpose of this paper is to present a multivariate version of Burg′s spectrum.

The notion of continuity for continuous-time information sources which was introduced by Pinsker has found numerous applications in information theory. Continuity in probability is an important concept in the theory of continuous-time stochastic processes. It is shown that these two forms of continuity are equivalent for stationary processes whose state space is a separable metric space.

Under weak regularity conditions of the covariance sequence, it is shown that the joint limiting distribution of the maxima on each coordinate of a stationary Gaussian multivariate sequence is that of independent random variables with marginal Gumbel distributions.

Necessary and sufficient conditions for an arbitrary q-variate stationary sequence xt, t ∈ Z, to be deterministic are presented. A characterization of the rank r(x) of xt, t ∈ Z, and a method to construct the Wold-Cramér decomposition for xt, t ∈ Z, are given. Subordination of q-variate bounded orthogonally scattered vector measures is considered.

This article is motivated by a central limit theorem of Ibragimov for strictly stationary random sequences satisfying a mixing condition based on maximal correlations. Here we show that the mixing condition can be weakened slightly, and construct a class of stationary random sequences covered by the new version of the theorem but not Ibragimov's original version. Ibragimov's theorem is also extended to triangular arrays of random variables, and this is applied to some kernel-type estimates of probability density.

The object of the present investigation is to show that the elegant asymptotic almost-sure representation of a sample quantile for independent and identically distributed random variables, established by Bahadur [1] holds for a stationary sequence of φ-mixing random variables. Two different orders of the remainder term, under different φ-mixing conditions, are obtained and used for proving two functional central limit theorems for sample quantiles. It is also shown that the law of iterated logarithm holds for quantiles in stationary φ-mixing processes.

A distributional mixing condition is introduced for stationary sequences of random vectors to study their extremes. For a sequence satisfying the condition, the following topics which concern the weak limit F of properly normalized partial maxima are studied: (1) To obtain characterizations of F. (2) To study a condition under which the partial maxima behave as they would if the sequence were i.i.d. (3) To consider problems in connection with the independence of the margins of F.

We consider a stationary time series {Xt} given byXt=∑∞k=−∞ψkZt−k, where {Zt} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {Xt} is squared integrable andmτ∑|k|⩾mψ2k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionmτ∑|k|⩾mψ2k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm∑|k|⩾mψ2k→0.

A necessary and sufficient condition is given for the existence of stationary probability distributions of a non-Markovian model with linear transition rule. Similar to the Markovian case, stationary probability distributions are characterized as eigenvectors of nonnegative matrices. The model studied includes as special cases the Markovian model as well as the linear learning model and has applications in psychological and biological research, in control theory, and in adaption theory.

A nearly unstable sequence of stationary spatial autoregressive processes is investigated, when the sum of the absolute values of the autoregressive coefficients tends to one. It is shown that after an appropriate norming the least squares estimator for these coefficients has a normal limit distribution. If none of the parameters equals zero than the typical rate of convergence is n., 26 pages To appear in: J. Multivariate Anal

Let ( X j , j ≥ 1) be a strictly stationary sequence of uniformly mixing random variables with zero mean, unit variance and finite fourth moment. Form the vector S n = Σ j = 1 n α nj X j where α nj = ( α nj 1 , α nj 2 ) t , α nj 1 , α nj 2 ∈ R 1 and ∥ α nj 1 ∥ ≤ 1, ∥ α nj 2 ∥ ≤ 1. We estimate the rate at which S n converges to normality. The extension of this result to bounded R s -valued weights ( s ≥ 1) is immediate.

A stationary random field is often more complicated than a univariate stationary time series, since dependence for a random field extends in all directions, while there is only the natural distinction of past and future at any instant in a univariate time series. In this paper we start from a simple correlation structure, derive a class of stationary random fields with the simple correlation function and the simple spectral density function by using linear combinations of separable spatial correlation functions, and discuss a problem of embedding a lattice model into a continuous domain model.

Given some regularity conditions on the distribution F(·) of a random X1, ..., Xn emanating from a strictly stationary sequence of random variables satisfying a strong mixing condition, it is shown that the sequence of quantile processes {n12f(F−1(s))(F−1n(s) − F−1(s)); 0 < s < 1} behaves like a sequence of Brownian bridges {Bn(s); 0

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.

In this paper we define the quasi-Markov property and give a complete characterization of a stationary Gaussian quasi-Markov process. We show that the only stationary Gaussian quasi-Markov process on the whole real line is a Markov process, but there exists a non-Markovian singular stationary Gaussian quasi-Markov process on the set of all the integers. The Markov property implies the quasi-Markov property, and a sufficient condition for a stationary quasi-Markov process to be Markov is given. Following Rozanov we define a generalized notion of boundary, called an L-boundary, and the L-Markov property, and we give a discussion of the uniqueness of a probability distribution corresponding to a family of conditional distributions for the stationary Gaussian L-Markov case. A characterization of the stationary Gaussian L-field is given in terms of the spectral distribution as well as a discussion of the relationship between L-Markov and L-field. We study the existence of a properly nonsingular-stationary Gaussian L-field and show that it can exist only in three or higher dimensions. We also discuss the technique of the best linear interpolation in mean-square and show how this technique can be used in studying the above problems.