In the paper, we discuss the properties of Cowan and Mecke's Markov chain P in general form. We give some criteria for determining whether the chain P is transient, recurrent null or recurrent non-null, and for the chain P to have invariant measures.

In the theory of autoregressive model fitting it is of interest to know the asymptotic behaviour, for large sample size, of the coefficients fitted. A significant role is played in this connection by the moments of the norms of the inverse sample covariance matrices. We establish uniform boundedness results for these, first under generally weak conditions and then for the special case of (infinite order) processes. These in turn imply corresponding ergodic theorems for the matrices in question.

This paper considers estimators of parameters of the Boolean model which are obtained by means of the method of intensities. For an estimator of the intensity of the point process of germ points the asymptotic normality is proved and the corresponding variance is given. The theory is based on a study of second-order characteristics of the point process of lower-positive tangent points of the Boolean model. An estimator of the distribution of a typical grain is also discussed.

The continuous autoregressive and minification stationary non-negative time series models discussed by Chernick et al. (1988) are generalized to model marginal distributions which have atoms of mass at zero. The reversibility theorem relating these processes with exponential marginal distributions is extended to the case where the marginal distribution has exponential tail.

Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

The aim of this paper is to extend the existing theory of second-order self-similar processes as defined by Cox (1984) from the univariate case to higher dimensions. Multivariate self-similar processes defined in terms of second-order theory for stationary time series can be used as models for long-range dependent observations when the marginal observations are long-range dependent. An interesting question concerns the correlation structure within the processes when the marginal processes are correlated. We show that the self-similarity requirement, as defined in this article, implies a cross-correlation structure similar to that for the marginal processes. This occurs both in the time domain and in the frequency domain. This fact can be used to obtain generalized least squares estimates for the long-range dependence parameters. We discuss some difficulties concerning estimation based on simulations.

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

This paper is concerned with the problem of estimation for the diffusion coefficient of a diffusion process on R, in a non-parametric situation. The drift function can be unknown and considered as a nuisance parameter. We propose an estimator of σ based on discrete observation of the diffusion X throughout a given finite time interval. We describe the asymptotic behaviour of this estimator when the step of discretization tends to zero. We prove consistency and asymptotic normality, the rate of convergence to the normal law being a random variable linked to the local time of the diffusion or to its suitable discrete approximation. This can also be interpreted as a convergence to a mixture of normal law.

This paper considers the joint limiting behavior of sums and maxima of stationary discrete-valued processes. The asymptotic behavior is a cross between a central limit theorem and asymptotic bounds for the distribution of the maxima. Some applications and simulations are also included.

For two-dimensional spatial data, a spatial unilateral autoregressive moving average (ARMA) model of first order is defined and its properties studied. The spatial correlation properties for these models are explicitly obtained, as well as simple conditions for stationarity and conditional expectation (interpolation) properties of the model. The multiplicative or linear-by-linear first-order spatial models are seen to be a special case which have proved to be of practical use in modeling of two-dimensional spatial lattice data, and hence the more general models should prove to be useful in applications. These unilateral models possess a convenient computational form for the exact likelihood function, which gives proper treatment to the border cell values in the lattice that have a substantial effect in estimation of parameters. Some simulation results to examine properties of the maximum likelihood estimator and a numerical example to illustrate the methods are briefly presented.

We propose an AR(1) model that can be used to generate logistic processes. The proposed model has simple probability and correlation structure that can accommodate the full range of attainable correlation. The correlation structure and the joint distribution of the proposed model are given, as well as their conditional mean and variance.

The first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process is introduced and its properties are studied. As an illustration, we discuss the special case of the first-order autoregressive Mittag-Leffler (MLAR(1)) process.

Using a simple characterization of the Linnik distribution, discrete-time processes having a stationary Linnik distribution are constructed. The processes are structurally related to exponential processes introduced by Arnold (1989), Lawrance and Lewis (1981) and Gaver and Lewis (1980). Multivariate versions of the processes are also described. These Linnik models appear to be viable alternatives to stable processes as models for temporal changes in stock prices.

We examine the main properties of the Markov chain Xt = T(Xt– 1) + σ(Xt– 1)ε t. Under general and tractable assumptions, we derive bounds for the tails of the stationary density of the process {Xt} in terms of the common density of the ε t's.

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk-valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.

The paper gives a spectral representation for a class of random fields which are bounded in mean square almost surely. A characterisation of the corresponding spectral measure in the representation is obtained based on Beurling's duality theory and generalised Fourier transforms. A representation for the covariance function of asymptotically stationary random fields is also derived.

Two recent papers by Petruccelli and Woolford (1984) and Chan et al. (1985) showed that the key element governing ergodicity of a threshold AR(1) model is the joint behavior of the two linear AR(1) pieces falling in the two boundary threshold regimes. They used essentially the necessary and sufficient conditions for ergodicity of a general Markov chain of Tweedie (1974), (1975) in a rather clever manner. However, it is difficult to extend the results to the more general threshold ARMA models. Besides, irreducibility is also required to apply Tweedie's results. In this paper, instead of pursuing the ideas in Tweedie's results, we shall develop a criterion similar in spirit to the technique used by Beneš (1967) in the context of continuous-time Markov chains. Consequently, we derive a necessary and sufficient condition for existence of a strictly stationary solution of a general non-linear ARMA model to be introduced in Section 2 of this paper. This condition is then applied to the threshold ARMA(1, q) model to yield a sufficient condition for strict stationarity which is identical to the condition given by Petruccelli and Woolford (1984) for the threshold AR(1). Hence, the conjecture that the moving average component does not affect stationarity is partially verified. Furthermore, under an additional irreducibility assumption, ergodicity of a non-linear ARMA model is established. The paper then concludes with a necessary condition for stationarity of the threshold ARMA(1, q) model.

The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞. It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

Discrete minification processes are introduced and it is proved that the discrete first-order autoregression of McKenzie (1986) and the discrete minification process are mutually time-reversible if and only if they have common marginal geometric distribution, corresponding to a result for continuous processes given by Chernick et al. (1988). It is also proved that a discrete minification process is time-reversible if and only if it has marginal Bernoulli distribution.