On the construction of Wold decomposition for multivariate stationary processes

A method to construct the Wold decomposition for multivariate stationary stochastic processes x_k, k Z, is presented. The method is based on orthogonal decompositions for x_k, k Z, obtained by forming orthogonal projections of x_k, k Z, onto its component processes , k Z, j = 1, …, q. The method does not give a complete solution to the Wold decomposition problem.     

Critical multitype branching processes with infinite variance

The exponential limit law for the critical multitype Bienaymé-Galton-Watson process is extended to a class of offspring distributions some or all of whose second moments are infinite. Several asymptotic consequences pertaining to transition probabilities and invariant measures are derived.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

Trimmed portmanteau test for linear processes with infinite variance

In this article, we consider a model check test for linear processes with infinite variance. As a test statistic, we employ the portmanteau test with trimmed residuals. It is shown that the limiting null distribution of the test is a chi-square distribution. Simulation results are provided for illustration.     

Quantile inference for nonstationary processes with infinite variance innovations

Based on the quantile regression,we extend Koenker and Xiao (2004) and Ling and McAleer (2004)'s works from finite-variance innovations to infinite-variance inn...     

Properties of spectral covariance for linear processes with infinite variance

We consider a measure of dependence for symmetric α-stable random vectors, which was introduced by the second author in 1976. We demonstrate that this measure of dependence, which we suggest to call the spectral covariance, can be extended to random vectors in the domain of normal attraction of general stable vectors. We investigate the asymptotic of the spectral covariance function for linear stable (Ornstein–Uhlenbeck, log-fractional, linear-fractional) processes with infinite variance and show that, in comparison with the results on the properties of codifference of these processes, obtained two decades ago, the results for the spectral variance are obtained under more general conditions and calculations are simpler.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

Bounds for the variance of certain stationary point processes

For the variance of stationary renewal and alternating renewal processes N_n(·) the paper establishes upper and lower bounds of the form

$\text{?B}{}_1\text{?varN}{}_8\text{(0,x–Aλx?B}{}_2\text{(0<x<∞)}$

, where λ=EN₈(0,1), with constants A, B₁ and B₂ that depend on the first three moments of the interval distributions for the processes concerned. These results are consistent with the value of the constant A for a general stationary point process suggested by Cox in 1963 [1].     

One-step prediction forP n-weakly stationary processes

The one-step prediction problem is studied in the context ofP _n-weakly stationary stochastic processes , where is an orthogonal polynomial sequence defining a polynomial hypergroup on . This kind of stochastic processes appears when estimating the mean of classical weakly stationary processes. In particular, it is investigated whether these processes are asymptoticP _n-deterministic, i.e. the prediction mean-squared error tends to zero. Sufficient conditions on the covariance function or the spectral measure are given for being asymptoticP _n-deterministic. For Jacobi polynomialsP _n(x) the problem of being asymptoticP _n-deterministic is completely solved.     

Robust prediction and interpolation for vector stationary processes

Summary Robust multivariate prediction and interpolation problems for statistically contaminated vector valued second order stationary processes are considered. The statistical contamination is modeled by requiring that the spectral density matrices of the processes lie within certain nonparametric classes. Both prediction and interpolation are then formalized as games whose saddle point solutions are sought. Finally, such solutions are found and analyzed, for two specific multivariate spectral classes.Research supported by the Air Force Office of Scientific Research under Grants AFOSR-83-0229 and AFOSR-82-0030     

Some remarks on filtering and prediction of stationary processes

This note contains a some remarks concerning filtering and prediction theory. One of them is a solution to an old question of H. Furstenberg which indicates an unexpected phenomenon arising from the lack of integrability. Another gives some general results on the possibility of constructing two valued universal guessing schemes for distinguishing between classes of stochastic processes.     

A test for independence of two stationary infinite order autoregressive processes

This paper considers the independence test for two stationary infinite order autoregressive processes. For a test, we follow the empirical process method and construct the Cramér-von Mises type test statistics based on the least squares residuals. It is shown that the proposed test statistics behave asymptotically the same as those based on true errors. Simulation results are provided for illustration.     

The decomposition and measurement of the interdependency between second-order stationary processes

Summary For a given pair of multivariate stationary processes, the process of one-way effect is extracted from each of the processes. Each process is decomposed into two orthogonal processes, namely, into the process generated by the one-way effect of the other process and the process orthogonal to it. Based on the decomposition, three measures characterizing the interdependency of the pair of processes are introduced. They are the measure of association, the measure of one-way effect and the measure of reciprocity. Each of the measures is defined as overall as well as frequencywise measure. The paper shows that the measure of association is equal to the sum of the others. It discusses the relationships of those measures to the ones proposed by Gel'fand-Yaglom and by Geweke.This paper is partially supported by the Japanese Ministry of Education Grant for Scientific Research No. C02630010     

Estimation for regression with infinite variance errors

This paper addresses the problem of modelling time series with nonstationarity from a finite number of observations. Problems encountered with the time varying parameters in regression type models led to the smoothing techniques. The smoothing methods basically rely on the finiteness of the error variance, and thus, when this requirement fails, particularly when the error distribution is heavy tailed, the existing smoothing methods due to [1], are no longer optimal. In this paper, we propose a penalized minimum dispersion method for time varying parameter estimation when a regression model generated by an infinite variance stable process with characteristic exponent α ε (1, 2). Recursive estimates are evaluated and it is shown that these estimates for a nonstationary process with normal errors is a special case.     

Convergence conditions for variance estimator of Gaussian stationary stochastic processes in Orlicz spaces

Conditions are derived ensuring convergence of the variance estimator of a Gaussian stationary stochastic process in Orlicz space norm. Confidence intervals are constructed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 113–116, 1992.     

Functional central limit theorems for self-normalized least squares processes in regression with possibly infinite variance data

Based on an R²-valued random sample {(y_i,x_i),1≤i≤n} on the simple linear regression model y_i=x_iβ+α+ε_i with unknown error variables ε_i, least squares processes (LSPs) are introduced in D[0,1] for the unknown slope β and intercept α, as well as for the unknown β when α=0. These LSPs contain, in both cases, the classical least squares estimators (LSEs) for these parameters. It is assumed throughout that {(x,ε),(x_i,ε_i),i≥1} are i.i.d. random vectors with independent components x and ε that both belong to the domain of attraction of the normal law, possibly both with infinite variances. Functional central limit theorems (FCLTs) are established for self-normalized type versions of the vector of the introduced LSPs for (β,α), as well as for their various marginal counterparts for each of the LSPs alone, respectively via uniform Euclidean norm and sup–norm approximations in probability. As consequences of the obtained FCLTs, joint and marginal central limit theorems (CLTs) are also discussed for Studentized and self-normalized type LSEs for the slope and intercept. Our FCLTs and CLTs provide a source for completely data-based asymptotic confidence intervals for β and α.