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Weakly approaching sequences of random distributions

Part of: Distribution theory Nonparametric inference Probability theory on algebraic and topological structures Stochastic processes Parametric inference

Published online by Cambridge University Press:  14 July 2016

Yuri Belyaev  and
Sara Sjöstedt-de Luna
Yuri Belyaev*
Affiliation:
Umeå University
Sara Sjöstedt-de Luna*
Affiliation:
Umeå University
*
∗Postal address: Department of Mathematical Statistics, Umeå University, S-901 87 Umeå, Sweden
∗Postal address: Department of Mathematical Statistics, Umeå University, S-901 87 Umeå, Sweden
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Abstract

We introduce the notion of weakly approaching sequences of distributions, which is a generalization of the well-known concept of weak convergence of distributions. The main difference is that the suggested notion does not demand the existence of a limit distribution. A similar definition for conditional (random) distributions is presented. Several properties of weakly approaching sequences are given. The tightness of some of them is essential. The Cramér-Lévy continuity theorem for weak convergence is generalized to weakly approaching sequences of (random) distributions. It has several applications in statistics and probability. A few examples of applications to resampling are given.

Keywords

Weak convergenceweakly approaching sequencesresamplingbootstrapcontinuity theoremLévy metricuniform metric

MSC classification

Primary: 60B10: Convergence of probability measures 60G57: Random measures 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators 62G09: Resampling methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 807 - 822
Copyright
Copyright © Applied Probability Trust 2000 

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Footnotes

Research supported by the Bank of Sweden Tercentenary Foundation, and by MISTRA, the Foundation of Strategic Environmental Research

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