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

https://link.springer.com/content/pdf/10.1007/BF02056908.pdf

Statistics of random processes I: General theory

Consider an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. It is an interesting problem to find criteria for whether the process is stable or has a unit root, given a finite sample of observations. Recently, various asymptotic distributions for estimators of the drift parameter have been developed. We illustrate through computer simulations and through a Stein's bound that these asymptotic distributions are inadequate approximations of the finite-sample distribution for moderate values of the drift and the sample size. We propose a new model to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power.

/pdf/hypothesis-testing-in-a-fractional-ornstein-uhlenbeck-model-474eov6suw.pdf

Hypothesis Testing in a Fractional Ornstein-Uhlenbeck Model

This paper discusses analytical and numerical results for nonharmonic, undamped, single-well, stochastic oscillators driven by additive noises. It focuses on average kinetic, potential, and total energies together with the corresponding distributions under random drivings, involving Gaussian white, Ornstein-Uhlenbeck, and Markovian dichotomous noises. It demonstrates that insensitivity of the average total energy to the single-well potential type, V(x)∝x^{2n}, under Gaussian white noise does not extend to other noise types. Nevertheless, in the long-time limit (t→∞), the average energies grow as power law with exponents dependent on the steepness of the potential n. Another special limit corresponds to n→∞, i.e., to the infinite rectangular potential well, when the average total energy grows as a power law with the same exponent for all considered noise types.

pdf/energetics-of-single-well-undamped-stochastic-oscillators-3vfc32l7lz.pdf

Energetics of single-well undamped stochastic oscillators.

The objective of the paper is to identify and investigate all possible types of asymptotic behavior for the maximum likelihood estimators of the unknown parameters in the second-order linear stochastic ordinary differential equation driven by Gaussian white noise. The emphasis is on the non-ergodic case, when the roots of the corresponding characteristic equation are not both in the left half-plane.

Second-order continuous-time non-stationary Gaussian autoregression

While consistency of the maximum likelihood estimator of the drift matrix in a multi-dimensional continuous time Ornstein-Uhlenbeck pro- cess holds under rather general conditions, little is known about the rate of convergence and the limiting distribution of the estimator when the un- derlying process is not ergodic. The objective of the paper is to investigate these questions for an important example of a non-ergodic two-dimensional Ornstein-Uhlenbeck process: an undamped harmonic oscillator.

/pdf/undamped-harmonic-oscillator-driven-by-ad-ditive-gaussian-oke60xi32p.pdf

Undamped harmonic oscillator driven by ad ditive gaussian white noise: a statistical analysis

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Undamped harmonic oscillator driven by ad ditive gaussian white noise: a statistical analysis