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

A new approach to the study of the distributions of sums of n Bernoulli variables by conditional distributions is considered, This leads to a characterization of binomial distribution, and provides a simple approach to the study of generalized binomial distributions We show that the binomial distribution and Poisson's binomial distribution can be derived as the distribution of sum of dependent Bernoulli random variables Finally the problem of Poisson approximation to the generalized binomial distribution is investigated

On the number of successes in dependent trials

In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, the Stein operators for certain compound distributions, where the random summand satisfies Panjer’s recurrence relation, are derived. A well-known perturbation approach for Stein’s method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

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On Stein operators for discrete approximations

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

Poisson and compound poisson approximations for random sums of random variables

Let be k independent gamma populations with unknown scale parameters and a common known shape parameter p > o. For the case k -2 and p a positive integer, Vellaisamy and Sharma (1988) obtained the uniformly minimum variance unbiased estimator (UMVUE) of M, the mean of the selected gamma population, and showed that the estimator dominates the natural estimator T -X(1) for squared error loss. In this note, the above two results are shown to be true for an arbitrary k and p > o, using the (u,V) -method of Bobbins (1988).

A note on the estimation op the mean op the selected gamma population

Suppose a subset of populations is selected from the given k gamma G(θ
 i,p
 ) (i = 1,2,...,k)populations, using Gupta's rule (1963, Ann. Inst. Statist. Math., 14, 199–216). The problem of estimating the average worth of the selected subset is first considered. The natural estimator is shown to be positively biased and the UMVUE is obtained using Robbins' UV method of estimation (1988, Statistical Decision Theory and Related Topics IV, Vol. 1 (eds. S. S. Gupta and J. O. Berger), 265–270, Springer, New York). A class of estimators that dominate the natural estimator for an arbitrary k is derived. Similar results are observed for the simultaneous estimation of the selected subset.

/pdf/average-worth-and-simultaneous-estimation-of-the-selected-1r8llblsl2.pdf

Palaniappan Vellaisamy

Papers

On the number of successes in dependent trials

On Stein operators for discrete approximations

Poisson and compound poisson approximations for random sums of random variables

A note on the estimation op the mean op the selected gamma population

Average worth and simultaneous estimation of the selected subset