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

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

http://www.long-memory.com/volatility/Giraitis-etal2003.pdf?origin=publication_detail

Rescaled variance and related tests for long memory in volatility and levels

This paper studies a broad class of nonnegative ARCH(∞) models Sufficient conditions for the existence of a stationary solution are established and an explicit representation of the solution as a Volterra type series is found Under our assumptions, the covariance function can decay slowly like a power function, falling just short of the long memory structure A moving average representation in martingale differences is established, and the central limit theorem is proved

Stationary arch models: dependence structure and central limit theorem

The paper studies the change-point problem and the cross-covariance function for ARCH models. Bounds for the cross-covariance function are derived and explicit formulae are obtained in special cases. Consistency of a cusum type change-point estimator is proved and its rate of convergence is established. A Hajek-Renyi type inequality is also proved. Results are obtained under weak moment assumptions.

/pdf/change-point-estimation-in-arch-models-ytb2y6h767.pdf

Change-point estimation in ARCH models

We extend the class of fractional ARIMA models to the class of fractional ARUMA models, which describe long-memory time series with long-range periodical behavior at a finite number of spectrum frequencies. The exact asymptotics of the covariance function and the spectrum at the points of peaks and zeros are given. To obtain asymptotic expansions, Gegenbauer polynomials are used. Consistent parameter estimation is discussed using Whittle's estimate.

A generalized fractionally differencing approach in long-memory modeling

The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharya et al. (1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modified R/S test. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool the R/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger than N -1/2 , where N is the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.

https://rmgsc.cr.usgs.gov/outgoing/threshold_articles/Giraitisetal2001.pdf

Remigijus Leipus

Papers

Rescaled variance and related tests for long memory in volatility and levels

Stationary arch models: dependence structure and central limit theorem

Change-point estimation in ARCH models

A generalized fractionally differencing approach in long-memory modeling

Testing for long memory in the presence of a general trend