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© 2017 The Authors. Published by Elsevier Ltd. In nonlinear dynamical systems the determination of stable and unstable periodic orbits as part of phase space prediction is problematic in particular if perturbed by noise. Fourier spectra of the time series or its autocorrelation function have shown to be of little use if the dynamic process is not strictly wide-sense stationary or if it is nonlinear. To locate unstable periodic orbits of a chaotic attractor in phase space the least stable eigenvalue can be determined by approximating locally the trajectory via linearisation. This approximation can be achieved by employing a Gaussian kernel estimator and minimising the summed up distances of the measured time series i.e. its estimated trajectory (e.g. via Levenberg-Marquardt). Noise poses a significant problem here. The application of the Wiener-Khinchin theorem to the time series in combination with recurrence plots, i.e. the Fourier transform of the recurrence times or rates, has been shown capable of detecting higher order dynamics (period-2 or period-3 orbits), which can fail using classical FouRiER-based methods. However little is known about its parameter sensitivity, e.g. with respect to the time delay, the embedding dimension or perturbations. Here we provide preliminary results on the application of the recurrence time spectrum by analysing the Hénon and the Rössler attractor. Results indicate that the combination of recurrence time spectra with a nonlinearly filtered plot of return times is able to estimate the unstable periodic orbits. Owing to the use of recurrence plot based measures the analysis is more robust against noise than the conventional Fourier transform.

We illustrate how Basu's theorem can be used to derive the spatial version of the Wiener-Hopf factorization for a specific class of piecewise-deterministic Markov processes. The classical factorization results for both random walks and Lévy processes follow immediately from our result. The approach is particularly elegant when used to establish the factorization for spectrally one-sided Lévy processes.

A functional central limit theorem is proved for certain random fields whose domain has both a temporal and a spadai component. These processes are made up of dependent summands which are measurable with respect to an increasing filtration. Temporal limit theory for semi-martingales is utilized to provide spatial finite-dimensional convergence. Consequently the limiting random fields have independent increments in time, and can be thought of as evolving random fields. In deriving a tightness result the notion of majorizing measures is employed to allow local spatial variability. Thus a functional central limit theorem for evolving random fields with a rather general dependence structure is given here for the first time. Comparison with results available for empirical processes suggest that this result is close to optimal. Introduction. In this paper we consider conditions for weak convergence of a sequence of random fields which are individually evolving over time. For motivation though, let's first examine a given sequence of random fields at some fixed time point. Let X be an arbitrary index set and for each ? > 1, let {Yn>i(x) : l ? . Let *?(*> = S y*.-w for xsXi l,xejr}isa sequence of random fields. We are interested in general conditions under which S^ converges weakly. We wish to allow dependence among the individual summands, the Y^i's. We do this by considering conditions of the martingale type, involving only conditional first and second moments. (These are more desirable than mixing type conditions which involve the entire distribution.) In addition, we must constrain our indexing family X to satisfy a complexity condition involving the notion of majorizing measures (or, alternatively, metric entropy). There will be a natural topology on X associated with our problem. In two papers Goldie and Greenwood (1986a, b) found conditions for weak convergence of sequences of set-indexed random fields. An unexpectedly difficult aspect of the problem was the characterization of the limiting distribution in

The notion of sampling for second-order q -variate processes is defined. It is shown that if the components of a q -variate process (not necessarily stationary) admits a sampling theorem with some sample spacing, then the process itself admits a sampling theorem with the same sample spacing. A sampling theorem for q -variate stationary processes, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloy's work. This sampling theorem is used to show that if a q -variate stationary process admits a sampling theorem, then each of its components will admit a sampling theorem too.

In [2] a central limit theorem [CLT] for sequences of univariate random linear forms was proved. That result is extended in this note to a multivariate CLT for $q$-dimensional linear forms of constant $q$-vectors with real-valued random weight coefficients. Some applications are indicated in Section 3.

A time series is the realization of a stochastic process. An estimate of the process average can be taken as the equally weighted average of observed values taken from the time series. Normally, the sequence of observations consists of observations taken at equal intervals of time. In certain applications, it is more appropriate to consider a sequence of observations spaced by intervals which are randomly and independently chosen from an exponential distribution. Let $\lbrack x_t, t \varepsilon T\rbrack$ be a real stochastic process that takes on values in a space $X, T$ being restricted to the real line. Let $\Omega$ be a space of points $\omega, F_T = \beta(x_t, t \varepsilon T)$ be the Borel field of $\omega$-sets generated by the class of sets of the form $\lbrack x_t(\omega) \varepsilon A\rbrack$ where $t \varepsilon T$ and $A$ is any Borel set, $P$ be the probability measure of $F_T$-sets, $\alpha$ be the Borel field of $X$-sets and for every real $t, x_t(\cdot)$ be a function from $\Omega$ to $X$ such that $\lbrack x_t(\omega) \varepsilon A\rbrack$ is an $F_T$-set for $A \varepsilon \alpha$. Assume that the process satisfies, for all $t \varepsilon T$, the conditions \begin{align*}\tag{1.1}\quad E\{x_t\} &= \bar x \quad \text{exists and is independent of} \quad t \\ \tag{1.2}E\{(x_t - \bar x^2\} &= \operatorname{var} x \quad \text{exists and is independent of} \quad t \\ \tag{1.3} E\{x_t - \bar x) (x_{t + \tau} - \bar x)\} &= r \quad (\tau) \operatorname{var} x \quad \text{depends only on} \quad \tau \quad \text{and} \\ r \quad (\tau) \text{is harmonizable}. \end{align*} A stochastic process satisfying Conditions (1.1) to (1.3) is often referred to as being stationary in the wide sense [2]. \begin{equation*}\tag{1.4}\lim_{\tau\rightarrow\infty} \tau r(\tau) \rightarrow 0.\end{equation*} Let $\{\xi\}$ be a measurable space of mutually independent random variables $\xi_i$ identically distributed with probability $p$ of having value 1 and $(1 - p)$ of having value 0, all independent of the $x_t$-process. The sample space $\Omega$ contains all possible realizations of the stochastic process. Given a scheme for observing the values of $x_t$ in time and some form of estimate of process mean from the observed values, expected values of such an estimate and of its square can be determined. A measure of the efficiency of the sampling scheme may be defined as the ratio of the variance of the estimator to that which would be obtained if the observations taken were independent of each other. The limit in probability of this measure of sampling efficiency for observations spaced by exponentially distributed mutually independent intervals is obtained simply in terms of the mean observation rate and the spectral density component at zero frequency. The limit theorem considered is a weak version. By augmenting Condition (1.4) with relations between sample size $N$, mean observation rate $\lambda$, length of time series $T$ and sufficiently rapid decay of $r(\tau)$ with $\tau$, sharper probability statements can be made.