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

The measurement of meaning

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).

The cultural origins of human cognition

Generalizing the definition of the memory parameter d in terms of the differentiated series, we showed in Velasco (Non-stationary log-periodogram regression, Forthcoming J. Economet., 1997) that it is possible to estimate consistently the memory of non-stationary processes using methods designed for stationary long-range-dependent time series. In this paper we consider the Gaussian semiparametric estimate analysed by Robinson (Gaussian semiparametric estimation of long range dependence. Ann. Stat. 23 (1995), 1630–61) for stationary processes. Without a priori knowledge about the possible non-stationarity of the observed process, we obtain that this estimate is consistent for d∈ (−½, 1) and asymptotically normal for d∈ (−½,¾) under a similar set of assumptions to those in Robinson's paper. Tapering the observations, we can estimate any degree of non-stationarity, even in the presence of deterministic polynomial trends of time. The semiparametric efficiency of this estimate for stationary sequences also extends to the non-stationary framework.

/pdf/gaussian-semiparametric-estimation-of-non-stationary-time-37f5lkeqdg.pdf

Gaussian Semiparametric Estimation of Non-stationary Time Series

Non-stationary log-periodogram regression

Digital sensory marketing: Integrating new technologies into multisensory online experience

This article provides an overview of the recent literature on the use of internet-based testing to address important questions in perception research. Our goal is to provide a starting point for the perception researcher who is keen on assessing this tool for their own research goals. Internet-based testing has several advantages over in-lab research, including the ability to reach a relatively broad set of participants and to quickly and inexpensively collect large amounts of empirical data, via services such as Amazon’s Mechanical Turk or Prolific Academic. In many cases, the quality of online data appears to match that collected in lab research. Generally-speaking, online participants tend to be more representative of the population at large than those recruited for lab based research. There are, though, some important caveats, when it comes to collecting data online. It is obviously much more difficult to control the exact parameters of stimulus presentation (such as display characteristics) with online research. There are also some thorny ethical elements that need to be considered by experimenters. Strengths and weaknesses of the online approach, relative to others, are highlighted, and recommendations made for those researchers who might be thinking about conducting their own studies using this increasingly-popular approach to research in the psychological sciences.

Conducting perception research over the internet: a tutorial review.

This article examines consistent estimation of the long-memory parameters of stock-market trading volume and volatility. The analysis is carried out in the frequency domain by tapering the data instead of detrending them. The main theoretical contribution of the article is to prove a central limit theorem for a multivariate two-step estimator of the memory parameters of a nonstationary vector process. Using robust semiparametric procedures, the long-memory properties of trading volume for the 30 stocks in the Dow Jones Industrial Average index are analyzed. Two empirical results are found. First, there is strong evidence that stock-market trading volume exhibits long memory. Second, although it is found that volatility and volume exhibit the same degree of long memory for most of the stocks, there is no evidence that both processes share the same long-memory component.

/pdf/long-memory-in-stock-market-trading-volume-4f0p7k62va.pdf

Carlos Velasco

Papers

Gaussian Semiparametric Estimation of Non-stationary Time Series

Non-stationary log-periodogram regression

Digital sensory marketing: Integrating new technologies into multisensory online experience

Conducting perception research over the internet: a tutorial review.

Long Memory in Stock-Market Trading Volume