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Main Directions in the Theory of Probability Metrics Read chapter Read first chapter

2013 | OriginalPaper | Chapter

11. Uniformity in Weak and Vague Convergence

Authors : Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Published in: The Methods of Distances in the Theory of Probability and Statistics

Publisher: Springer New York

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Abstract

The goals of this chapter are to: • Extend the notion of uniformity, • Study the metrization of weak convergence, • Describe the notion of vague convergence, • Consider the question of its metrization.

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previous chapter Moment Distances
next chapter Glivenko–Cantelli Theorem and Bernstein–Kantorovich Invariance Principle
Footnotes
1
See Definition 4.4.1 in Chap.​ 4.
 
2
See, for example, Hennequin and Tortrat [1965].
 
3
See (8.3.5)–(8.3.7) in Chap.​ 8.
 
4
See Billingsley (1999, Sect. 5).
 
5
See Kallenberg [1975] and Kerstan et al. [1978].
 
6
See Kallenberg [1975], Szasz [1975], and Kerstan et al. [1978].
 
7
See Example 3.3.2 in Chap.​ 3.
 
Literature
Billingsley P (1999) Convergence of probability measures, 2nd edn. Wiley, New York
Fortet R, B Mourier (1953) Convergence de la réparation empirique vers la répétition theorétique. Ann Sci Ecole Norm 70:267–285
Hennequin PL, Tortrat A (1965) Théorie des probabilités et quelques applications. Masson, Paris
Kallenberg O (1975) Random measures. Akademie, Berlin
Kerstan J, Matthes K, Mecke J (1978) Infinitely divisible point processes. Wiley, New York
Prokhorov YuV (1956) Convergence of random processes and limit theorems in probability theory. Theor Prob Appl 1:157–214
Ranga RR (1962) Relations between weak and uniform convergence of measures with applications. Ann Math Statist 33:659–680
Szasz D (1975) On a metrization of the vague convergence. Studio Sci Math Hungarica 9:219–222
Metadata
Title
Uniformity in Weak and Vague Convergence
Authors
Svetlozar T. Rachev
Lev B. Klebanov
Stoyan V. Stoyanov
Frank J. Fabozzi
Copyright Year
2013
Publisher
Springer New York
Book
The Methods of Distances in the Theory of Probability and Statistics

Print ISBN: 978-1-4614-4868-6

Electronic ISBN: 978-1-4614-4869-3

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-4869-3_11