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Stochastic Limit TheoryAn Introduction for Econometricians$
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James Davidson

Print publication date: 1994

Print ISBN-13: 9780198774037

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198774036.001.0001

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Stochastic Convergence

Stochastic Convergence

Chapter:
(p.281) 18 Stochastic Convergence
Source:
Stochastic Limit Theory
Author(s):

James Davidson

Publisher:
Oxford University Press
DOI:10.1093/0198774036.003.0018

The modes of convergence introduced in Ch. 12 are studied in detail. Conditions for almost‐sure convergence are derived via the Borel‐Cantelli lemma. Convergence in probability is contrasted, and then a number of results for convergence of transformed series are given. Convergence in LP ‐norm is introduced as a sufficient condition for convergence in probability. Examples are given, and the chapter concludes with a preliminary look at the laws of large numbers.

Keywords:   almost‐sure convergence, Borel's normal number theorem, Borel‐Cantelli lemma, convergence in LP‐norm, convergence in mean square, convergence in probability, law of large numbers, Slutsky's theorem

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