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

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

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

We establish necessary and sufficient conditions for a sequence of d-dimensional vectors of multiple stochastic integrals \(\mathbf{F}_{d}^{k} = (F_{1}^{k}, \dots, F_{d}^{k})\), \(k\geq 1\), to converge in distribution to a d-dimensional Gaussian vector \(\mathbf{N}_{d} = (N_{1}, \dots, N_{d}) \). In particular, we show that if the covariance structure of F d k converges to that of N d , then componentwise convergence implies joint convergence. These results extend to the multidimensional case the main theorem of [10].

Gaussian Limits for Vector-valued Multiple Stochastic Integrals

In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail.

Stochastic evolution equations with fractional Brownian motion

We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the Gaussian theory of regularity and supremum estimation to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of H\"{o}lder-regularity (any \emph{Hurst} parameter) We prove existence and strong consistency of the MLE for linear and nonlinear equations We\ also prove that a basic discretized version of the MLE, is still a strongly consistent estimator

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Statistical aspects of the fractional stochastic calculus

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Major (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Ito multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.

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Analysis of the Rosenblatt process

https://bibos.math.uni-bielefeld.de/preprints/05-03-177.pdf

Ciprian A. Tudor

Papers

Gaussian Limits for Vector-valued Multiple Stochastic Integrals

Stochastic evolution equations with fractional Brownian motion

Statistical aspects of the fractional stochastic calculus

Analysis of the Rosenblatt process

On bifractional Brownian motion