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

Convergence of Probability Measures

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Table Of Integrals Series And Products

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).

Overview.- An Overview of Empirical Processes.- Overview of Semiparametric Inference.- Case Studies I.- Empirical Processes.- to Empirical Processes.- Preliminaries for Empirical Processes.- Stochastic Convergence.- Empirical Process Methods.- Entropy Calculations.- Bootstrapping Empirical Processes.- Additional Empirical Process Results.- The Functional Delta Method.- Z-Estimators.- M-Estimators.- Case Studies II.- Semiparametric Inference.- to Semiparametric Inference.- Preliminaries for Semiparametric Inference.- Semiparametric Models and Efficiency.- Efficient Inference for Finite-Dimensional Parameters.- Efficient Inference for Infinite-Dimensional Parameters.- Semiparametric M-Estimation.- Case Studies III.

Introduction to empirical processes and semiparametric inference

Let (X k) k≥1 be a sequence of random variables with common distribution function F(x) = P(X 1 ≤ x). Define the empirical distribution function 

$$ {{F}_{n}}(x) = \frac{1}{n}\# \{ 1 \leqslant i \leqslant n:{{X}_{1}} \leqslant x\} , $$

and the empirical process by $ \sqrt {n} ({{F}_{n}}(x) - F(x)) $ In this chapter we provide a survey of classical as well as modern techniques in the study of empirical processes of dependent data. We begin with a sketch of the early roots of the field in the theory of uniform distribution mod 1, of sequences defined by X k = {n k ω}, ω ∈ [0, 1], dating back to Weyl’s celebrated 1916 paper. In the second section we provide the essential tools of empirical process theory, and we prove Donsker’s classical empirical process invariance principle for i.i.d. processes. The third section provides an introduction to the subject of weakly dependent random variables. We introduce a variety of mixing concepts, provide necessary technical tools like correlation and moment inequalities, and prove central limit theorems for partial sums. The empirical process of weakly dependent data is investigated in the fourth section, where we put special emphasis on almost sure approximation techniques. The fifth section is devoted to the empirical distribution of U-statistics, defined as 

$$ Un(x) = {{\left( {\begin{array}{*{20}{c}} n \\ 2 \\ \end{array} } \right)}^{{ - 1}}}\# \{ 1 \leqslant i < j \leqslant n:h({{X}_{i}},{{X}_{j}}) \leqslant x\} $$

for some symmetric kernel h. We give some applications, e.g., to dimension estimation in the analysis of time series, and prove weak convergence of the corresponding empirical process. Empirical processes of long-range dependent data are the topic of the sixth section. We give an introduction to the area of long-range dependent processes, provide important technical tools for the study of their partial sums and investigate the limit behavior of the empirical process. It turns out that the limit process is of a completely different type as in the case of independent or weakly dependent data, and that this has important consequences for various functionals of the empirical process. The final section is devoted to pair correlations, i.e., U-statistics empirical processes over short intervals associated with the kernel h(x, y) = |x − y|

Empirical Process Techniques for Dependent Data

Let $(X_j)^\infty_{j = 1}$ be a stationary, mean-zero Gaussian process with covariances $r(k) = EX_{k + 1} X_1$ satisfying $r(0) = 1$ and $r(k) = k^{-D}L(k)$ where $D$ is small and $L$ is slowly varying at infinity. Consider the two-parameter empirical process for $G(X_j),$ $\bigg\{F_N(x, t) = \frac{1}{N} \sum^{\lbrack Nt \rbrack}_{j = 1} \lbrack 1\{G(X_j) \leq x\} - P(G(X_1) \leq x) \rbrack; // -\infty < x < + \infty, 0 \leq t \leq 1\bigg\},$ where $G$ is any measurable function. Noncentral limit theorems are obtained for $F_N(x, t)$ and they are used to derive the asymptotic behavior of some suitably normalized von Mises statistics and $U$-statistics based on the $G(X_j)$'s. The limiting processes are structurally different from those encountered in the i.i.d. case.

/pdf/the-empirical-process-of-some-long-range-dependent-sequences-2kn28lmnaq.pdf

The Empirical Process of some Long-Range Dependent Sequences with an Application to $U$-Statistics

In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

/pdf/limit-theorems-for-functionals-of-mixing-processes-with-2dvrkclt2g.pdf

Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation

We propose a new test against a change in correlation at an unknown point in time based on cumulated sums of empirical correlations. The test does not require that inputs are independent and identically distributed under the null. We derive its limiting null distribution using a new functional delta method argument, provide a formula for its local power for particular types of structural changes, give some Monte Carlo evidence on its finite-sample behavior, and apply it to recent stock returns.

/pdf/testing-for-a-change-in-correlation-at-an-unknown-point-in-1ejo0r95lw.pdf

Testing for a change in correlation at an unknown point in time using an extended functional delta method

We obtain the almost sure approximation of the partial sums of random variables with values in a separable Hilbert space and satisfying a strong mixing condition by a suitable Brownian motion. This is achieved by a modification of the proof of a similar result by Kuelbs and Philipp (1980) on $\phi$-mixing Banach space valued random variables. As by-products we get almost sure invariance principles for sums of absolutely regular sequences of random variables with values in a Banach space and necessary and sufficient conditions for the almost sure invariance principle for sums of independent, identically distributed random variables.

https://projecteuclid.org/download/pdf_1/euclid.aop/1176993777

Herold Dehling

Papers

Empirical Process Techniques for Dependent Data

The Empirical Process of some Long-Range Dependent Sequences with an Application to $U$-Statistics

Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation

Testing for a change in correlation at an unknown point in time using an extended functional delta method

Almost Sure Invariance Principles for Weakly Dependent Vector-Valued Random Variables