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1984 | Buch

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Convergence of Stochastic Processes

verfasst von: David Pollard

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

A more accurate title for this book might be: An Exposition of Selected Parts of Empirical Process Theory, With Related Interesting Facts About Weak Convergence, and Applications to Mathematical Statistics. The high points are Chapters II and VII, which describe some of the developments inspired by Richard Dudley's 1978 paper. There I explain the combinatorial ideas and approximation methods that are needed to prove maximal inequalities for empirical processes indexed by classes of sets or classes of functions. The material is somewhat arbitrarily divided into results used to prove consistency theorems and results used to prove central limit theorems. This has allowed me to put the easier material in Chapter II, with the hope of enticing the casual reader to delve deeper. Chapters III through VI deal with more classical material, as seen from a different perspective. The novelties are: convergence for measures that don't live on borel a-fields; the joys of working with the uniform metric on D[O, IJ; and finite-dimensional approximation as the unifying idea behind weak convergence. Uniform tightness reappears in disguise as a condition that justifies the finite-dimensional approximation. Only later is it exploited as a method for proving the existence of limit distributions. The last chapter has a heuristic flavor. I didn't want to confuse the martingale issues with the martingale facts.

Inhaltsverzeichnis

Frontmatter

Chapter I. Functionals on Stochastic Processes

Abstract

Functions analyzed as points of abstarct spaces of functions appear in many branches of mathematics. Geometric intuitions about distance (or approximation, or convergence, or orthogonality, or any other ideas learned from the study of euclidean space) carry over to those abstract spaces, lending familiarity to operations carried out on the functions. We enjoy similar benefits in the study of stochastic processes if we analyze them as random elements of spaces of functions.

David Pollard

Chapter II. Uniform Convergence of Empirical Measures

Abstract

For independent sampling from a distribution function F, the strong law of large numbers tells us that the proportion of points in an interval (−∞, t] converges almost surely to F(t). The classical Glivenko-Cantelli theorem strengthens the result by adding that the convergence holds uniformly over all t. The strong law also tells us that the proportion of points in any fixed set converges almost surely to the probability of that set. The strengthening of this result, to give uniform convergence over classes of sets more interesting than intervals on the real line, and its further generalization to classes of functions, will be the main concern of this chapter.

David Pollard

Chapter III. Convergence in Distribution in Euclidean Spaces

Abstract

Convergence is distribution of a sequence X_n of real random variable is traditionally defined to mean convergence of distribution functions at each continuity point of the limit distribution function:

$$\mathbb{P}\{ X_n \leqslant x\} \to \mathbb{P}\{ X \leqslant x\} \,\text{whenever}\,\mathbb{P}\{ X = x\} = 0$$

David Pollard

Chapter IV. Convergence in Distribution in Metric Spaces

Abstract

We write a statistic as a functional on the sample paths of a stochastic process in order to break an analysis of the statistic into two parts: the study of continuity properties of the functional; the study of the stochastic process as a random element of a space of functions. The method has its greatest appeal when many different statistics can be written as functionals on the same process, or when the process has a form that suggests a simple approximation, as in the goodness-of-fit example from Chapter I. There we expressed various statistics as functionals on the empirical process U_n, which defines a random element of D[0, 1]. Doob’s heuristic argument suggested that U_n should behave like a brownian bridge, in some distributional sense.

David Pollard

Chapter V. The Uniform Metric on Spaces of Cadlag Functions

Abstract

The theory developed in Chapter IV justifies its existence by what it has to say about the limiting distributions of functional defined on sequences of stochastic processes. Processes X_n(t):t ∈ T are identified with random elements of some space X of functions on T, a space large enough to contain the sample paths of every X_n; the functional are maps defined on X to which the Continuous Mapping Theorem can be applied. In this chapter we shall specialize the general theory to the particular function space D[0, 1], under its uniform metric. It will turn out that most applications, especially those that come up with brownian bridges and brownian motions as limit processes, require no fancier setting than this.

David Pollard

Chapter VI. The Skorohod Metric on D[0, ∞)

Abstract

The uniform metric on D[0, 1] is the best choice for applications where the limit distribution concentrates on C[0, 1], or on some other separable subset of D[0,1]. It is well suited for convergence to brownian motion, brownian bridge, and the gaussian processes that appear as limits in the Empirical Central Limit Theorem. But it excludes, for example, poisson processes and other non-gaussian processes with independent increments, whose jumps are not constrained to lie in a fixed, countable subset of [0, 1]. To analyze such processes, Skorohod (1956) introduced four new metrics, all weaker than the uniform metric. Of these, the J₁ metric has since become the most popular. (Too popular in my opinion—too often it is dragged into problems for which the uniform metric would suffice.) But Skorohod’s J₁ metric on D[0, 1] will not be the main concern of this chapter. Instead we shall investigate a sort of J₁ convergence on compacta for D[0, ∞], the space where the interesting applications live.

David Pollard

Chapter VII. Central Limit Theorems

Abstract

Much asymptotic theory boils down to careful application of Taylor’s theorem. To bound remainder terms we impose regularity conditions, which add rigor to informal approximation arguments, but usually at the cost of increased technical detail. For some asymptotics problems, especially those concerned with central limit theorems for statistics defined by maximization or minimization of a random process, many of the technicalities can be drawn off into a single stochastic equicontinuity condition. This section shows how. Empirical process methods for establishing stochastic equicontinuity will be developed later in the chapter.

David Pollard

Chapter VIII. Martingales

Abstract

Martingale theory must surely be the most successful of all the attempts to extend the classical theory for sums of independent random variables to cover dependent variables. Many of the classical limit theorems have martingale analogues that rival them for elegance and far exceed them in diversity of application. We shall explore two of these martingale theorems in this chapter.

David Pollard

Backmatter

Titel: Convergence of Stochastic Processes
verfasst von: David Pollard
Verlag: Springer New York
Electronic ISBN: 978-1-4612-5254-2
Print ISBN: 978-1-4612-9758-1
DOI: https://doi.org/10.1007/978-1-4612-5254-2

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter I. Functionals on Stochastic Processes

Chapter II. Uniform Convergence of Empirical Measures

Chapter III. Convergence in Distribution in Euclidean Spaces

Chapter IV. Convergence in Distribution in Metric Spaces

Chapter V. The Uniform Metric on Spaces of Cadlag Functions

Chapter VI. The Skorohod Metric on D[0, ∞)

Chapter VII. Central Limit Theorems

Chapter VIII. Martingales

Backmatter