Jiří Anděl

Autoregressive series with random parameters

In the paper an authoregressive model is introduced and investigated, the parameters of which are random variables. The necessary and sufficient conditions for stationarity are derived. It is shown that the covariance function of a stationary autorgressive series with random parameters satisifies the same Yule-Walker equations as in the usual autoregressive model with fixed parameters. The inverse variance matrix in stationary case is also given.

Nonlinear nonnegative AR(1) processes

Let {et} be a nonnegative strict white noise and Xtan independent nonnegative random variable. For t ≧ 2 define where g is a nonnegative function and b ≧ 0. Then is an estimator for b . Under some general conditions, which do not include stationarity, it is proved that is strongly consistent.

Marginal Distributions of Autoregressive Processes

Consider a stationary autoregressive process {Xs} defined by the relation Xs=ς Xs−1+Ys, where {Ys} is a white noise. Assume that Ys are independent and have the same distribution (normal, rectangular, Laplace or Cauchy). For these cases the distribution of Xs is calculated. Conversely, if a distribution of Xs is given (normal, exponential, gamma, Laplace, rectangular or Cauchy), the corresponding distribution of Ys is derived. The results are applicable in Monte Carlo methods for constructing dependent random variables with a given univariate marginal distribution.

Measures of dependence in discrete stationary processes

Usually, the dependence in stationary processes is described by a set of coefficients. In this paper, a measure of dependence is proposed which can be used instead of the autocorrelation function, and another measure for the dependence between two processes instead of cross-correlation function and coherence coefficients. In the end, an improvement of extrapolation of a process is investigated which is caused by the knowledge of another related process.

Bernoulli Processes with Non-Positive Correlations

We deal with sequences of dependent Bernoulli variables with non positive correlations. Some special models for 1-dependent and 2-dependent 0–1 valued variables are analyzed, namely Bernoulli variables obtained by clipping a linear combination of iid variables. Formulas describing dependence of their correlation function on the clipping parameters are derived. The lower bound for the sum of autocorrelations of Bernoulli variables obtained by clipping a Gaussian process is provided.

On Stationary Distributions of some Time Series Models

If a time series model and its stationary distribution are given, then the hard problem is to find the corresponding distribution of a strict white noise used in the model. We solve a version of this problem when some moments of the stationary distribution are given.

An Autoregressive Representation of ARMA Processes

Let {Xt} be a p-dimensional stationary invertible autoregressive-moving average process given by

On the Backward Extrapolation of Non-Stationary Autoregressive Series

\mathop {\min }\limits_x f(x)/g(x){\text{ subject to }}h(x) \leqslant 0