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

Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)

https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9868.2011.00777.x

An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach

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

Probability and Measure

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

/pdf/the-fractional-poisson-process-and-the-inverse-stable-39qnffrd1h.pdf

The Fractional Poisson Process and the Inverse Stable Subordinator

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

/pdf/fractional-cauchy-problems-on-bounded-domains-3a64wrtfy7.pdf

Fractional Cauchy problems on bounded domains

Distributed-order fractional diffusions on bounded domains

Retrospective assessment of adverse childhood experiences is widely used in research, although there are concerns about its validity. In particular, recall bias is assumed to produce significant artifacts. Data from a longitudinal cohort (the British National Child Development Study; N=7710) and the retrospective Mainz Adverse Childhood Experiences Study (N=1062, Germany) were compared on 10 adverse childhood experiences and psychological adjustment at age 42 yr. Between the two methods, no significant differences in risk effects were detected. Results held for bivariate analyses on all 10 childhood adversities and a multivariate model; the latter comprises the childhood adversities which show significant long-term sequelae (not always with natural parent, chronically ill parent, financial hardship, and being firstborn) and three covariates. In conclusion, the present data did not show any bias in the retrospective assessment.

Palaniappan Vellaisamy

Papers

The Fractional Poisson Process and the Inverse Stable Subordinator

Fractional Cauchy problems on bounded domains

Distributed-order fractional diffusions on bounded domains

Sequelae of Prospective versus Retrospective Reports of Adverse Childhood Experiences

Inverse Tempered Stable Subordinators