Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This paper presents a test of independence that can be applied to the estimated residuals of any time series model that can be transformed into a model driven by independent and identically distributed errors. The first order asymptotic distribution of the test statistic is independent of estimation error provided that the parameters of the model under test can be estimated -consistently. Because of this, our method can be used as a model selection tool and as a specification test. Widely used software1 written by Dechert and LeBaron can be used to implement the test. Also, this software is fast enough that the null distribution of our test statistic can be estimated with bootstrap methods. Our method can be viewed as a nonlinear analog of the Box-Pierce Q statistic used in ARIMA analysis.

https://ecsocman.hse.ru/data/917/582/1217/article.pdf

A test for independence based on the correlation dimension

We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$. General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176347265

The Jackknife and the Bootstrap for General Stationary Observations

/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Linear and nonlinear Granger causality tests are used to examine the dynamic relation between daily Dow Jones stock returns and percentage changes in New York Stock Exchange trading volume. We find evidence of significant bidirectional nonlinear causality between returns and volume. We also examine whether the nonlinear causality from volume to returns can be explained by volume serving as a proxy for information flow in the stochastic process generating stock return variance as suggested by Clark's (1973) latent common-factor model. After controlling for volatility persistence in returns, we continue to find evidence of nonlinear causality from volume to returns.

Testing for Linear and Nonlinear Granger Causality in the Stock Price-Volume Relation

This book provides a detailed introduction to the ergodic theory of equilibrium states giving equal weight to two of its most important applications, namely to equilibrium statistical mechanics on lattices and to (time discrete) dynamical systems. It starts with a chapter on equilibrium states on finite probability spaces which introduces the main examples for the theory on an elementary level. After two chapters on abstract ergodic theory and entropy, equilibrium states and variational principles on compact metric spaces are introduced emphasizing their convex geometric interpretation. Stationary Gibbs measures, large deviations, the Ising model with external field, Markov measures, Sinai-Bowen-Ruelle measures for interval maps and dimension maximal measures for iterated function systems are the topics to which the general theory is applied in the last part of the book. The text is self contained except for some measure theoretic prerequisites which are listed (with references to the literature) in an appendix.

/pdf/equilibrium-states-in-ergodic-theory-51d4mjywmk.pdf

Equilibrium States in Ergodic Theory

We prove stability of the isolated eigenvalues of transfer operators satisfying
a Lasota-Yorke type inequality under a broad class of random and nonrandom
perturbations including Ulam-type discretizations. The results are formulated in
an abstract framework.

/pdf/stability-of-the-spectrum-for-transfer-operators-4yczskdexk.pdf

Stability of the spectrum for transfer operators

During the last decade a lot of research has been done on onedimensional dynamics. Different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered. In most of these cases the Perron-Frobenius-operator plays an important role. In this paper we try to unify these different examples, which are discussed in detail below. First we give a discription of the results proved in this paper.

Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations

We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows us to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasi-compact. (Information on the existence of a Sinai–Ruelle–Bowen measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated with smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows us to obtain easily very strong spectral stability results, which, in turn, imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam-type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite-dimensional problem.

https://iopscience.iop.org/article/10.1088/0951-7715/15/6/309/pdf

Ruelle-Perron-Frobenius spectrum for Anosov maps

Some probabilistic limit theorems for Hoeffding's U-statistic [13] and v Mises' functional are established when the underlying processes are not necessarily independent We consider absolutely regular processes [24] and processes (X
n)n≧1 which are uniformly mixing [14] as well as their time reversal (X

 −n
 )n≦−1, called uniformly mixing in both directions of time Many authors have weakened the hypothesis of independence in statistical limit theorems and considered m-dependent, Markov or weakly dependent processes; in particular for U statistics under weak dependence Sen [22] has considered *-mixing processes and derived a central limit theorem and a law of the iterated logarithm, while Yoshihara [26] proved central limit theorems and as results in the absolutely regular and uniformly mixing case Here we extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorems and as approximation by a Brownian motion Extensions to multisample versions and other extensions are briefly discussed

Gerhard Keller

Papers

Equilibrium States in Ergodic Theory

Stability of the spectrum for transfer operators

Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations

Ruelle-Perron-Frobenius spectrum for Anosov maps

On U-statistics and v. mise’ statistics for weakly dependent processes