Statistics for Spatial Data.

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

Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.

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Stochastic Geometry for Wireless Networks

Representation of generalized additive models (GAM's) using penalized regression splines allows GAM's to be employed in a straightforward manner using penalized regression methods. Not only is inference facilitated by this approach, but it is also possible to integrate model selection in the form of smoothing parameter selection into model fitting in a computationally efficient manner using well founded criteria such as generalized cross-validation. The current fitting and smoothing parameter selection methods for such models are usually effective, but do not provide the level of numerical stability to which users of linear regression packages, for example, are accustomed. In particular the existing methods cannot deal adequately with numerical rank deficiency of the GAM fitting problem, and it is not straightforward to produce methods that can do so, given that the degree of rank deficiency can be smoothing parameter dependent. In addition, models with the potential flexibility of GAM's can also present ...

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Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models

Using a representation formula expressing the mixed cumulants of realvalued random variables by corresponding moments, sufficient conditions are given for the normal convergence of suitably standardized shot noise assuming that the generating stationary point process is independently marked and Brillinger mixing and that its intensity tends to oo. Furthermore, estimates for the rate of this normal convergence are obtained by exploiting a general lemma on probabilities of large deviations and on the rate of normal convergence.

Normal convergence of multidimensional shot noise and rates of this convergence

We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

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Central limit theorem for a class of random measures associated with germ-grain models

In this paper we derive a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most $q$. By using this rank condition we establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system. Our proofs also comes with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles.

denote positive constants (not depending on N). 0 stands for a complex number with [0l=<l which "may be differ from one expression to another and Xa denotes the indicator function of the event A. [] indicates the end of a proof. The main purpose of this paper is to provide a general method for the derivation of limit theorems for sums of m-dependent

A method for the derivation of limit theorems for sums of m -dependent random variables

The main purpose of this paper is to present cer~trai limit tileoreins including functional limit theorems for empirical factorial moment measures and kernel-type product density estimators when the underlying point process is a regular infinitely divisible one.The requied moment conditions are minimal, they are necessary to ensure finite variances of the estimators under consideration. In the special case of a stationary poisson process the obtained results are used to construct a goodness-of-fit test for the function λK(t), 0≤t≤T, denoting the mean number of points within a sphere with radius t around a typical point of the process.

Lothar Heinrich

Papers

Normal convergence of multidimensional shot noise and rates of this convergence

Central limit theorem for a class of random measures associated with germ-grain models

Central limit theorem for a class of random measures associated with germ-grain models

A method for the derivation of limit theorems for sums of m -dependent random variables

Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary poisson cluster processes