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

Preface. Preface to First Edition. 1. Financial Time Series and Their Characteristics. 2. Linear Time Series Analysis and Its Applications. 3. Conditional Heteroscedastic Models. 4. Nonlinear Models and Their Applications. 5. High-Frequency Data Analysis and Market Microstructure. 6. Continuous-Time Models and Their Applications. 7. Extreme Values, Quantile Estimation, and Value at Risk. 8. Multivariate Time Series Analysis and Its Applications. 9. Principal Component Analysis and Factor Models. 10. Multivariate Volatility Models and Their Applications. 11. State-Space Models and Kalman Filter. 12. Markov Chain Monte Carlo Methods with Applications. Index.

http://inis.jinr.ru/sl/M_Mathematics/MV_Probability/MVspf_Stochastics%20in%20finance/Tsay%20Analysis.pdf

Analysis of Financial Time Series

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).

Summary. The availability of intraday data on the prices of speculative assets means that we can use quadratic variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods.

/pdf/econometric-analysis-of-realized-volatility-and-its-use-in-43r6n1l7fn.pdf

Econometric analysis of realized volatility and its use in estimating stochastic volatility models

http://neumann.hec.ca/~p283/gii/Longin_Solnik.pdf

Is the correlation in international equity returns constant: 1960–1990?

Threshold heteroskedastic models

This book provides a comprehensive and systematic approach to understanding GARCH time series models and their applications whilst presenting the most advanced results concerning the theory and practical aspects of GARCH. The probability structure of standard GARCH models is studied in detail as well as statistical inference such as identification, estimation and tests. The book also provides coverage of several extensions such as asymmetric and multivariate models and looks at financial applications.Key features:- Provides up-to-date coverage of the current research in the probability, statistics and econometric theory of GARCH models. - Numerous illustrations and applications to real financial series are provided. - Supporting website featuring R codes, Fortran programs and data sets. - Presents a large collection of problems and exercises. This authoritative, state-of-the-art reference is ideal for graduate students, researchers and practitioners in business and finance seeking to broaden their skills of understanding of econometric time series models.

GARCH Models: Structure, Statistical Inference and Financial Applications

We prove the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of pure generalized autoregressive conditional heteroscedastic (GARCH) processes, and of autoregressive moving-average models with noise sequence driven by a GARCH model. Results are obtained under mild conditions.

/pdf/maximum-likelihood-estimation-of-pure-garch-and-arma-garch-4kbj5wo342.pdf

Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes

This paper attempts to enlarge the class of Threshold Heteroscedastic Models (TARCH) introduced by Zakoian (1991a). We show that it is possible to relax the positivity constraints on the parameters of the conditional variance. Unconstrained models provide a greater generality of the paths allowing for nonlinearities in the volatility. Cyclical behaviour is permitted as well as different relative impacts of positive and negative shocks on volatility, depending on their size. We give empirical evidence using French stock returns.

https://www.jstor.org/stable/pdfplus/2285109.pdf

Jean-Michel Zakoian

Papers

Threshold heteroskedastic models

GARCH Models: Structure, Statistical Inference and Financial Applications

Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes

Threshold arch models and asymmetries in volatility

Stationarity of multivariate Markov–switching ARMA models