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Conference Paper: A paradox of measuretheoretic probability
Title  A paradox of measuretheoretic probability 

Authors  
Keywords  measuretheoretic probability tightness of probability measures weak convergence subprobability measure zero measure 
Issue Date  2014 
Citation  The International Congress of Mathematicians (ICM 2014), Coex, Seoul, Korea, 1321 August 2014. In Abstract book, 2014, p. 487488, abstract no. OP120867 How to Cite? 
Abstract  We report a paradox of measuretheoretic probability. Denote by B(R) and B(R) the 
algebras corresponding to the real line R and extended real line R, respectively. Let R be
equipped with a topology induced by a fixed metric. Let ; ; n; n; n = 1; 2; be probability
measures on (R; B(R)), such that ; 1; 2; are distributions of i.i.d. random variables
X;X1;X2; , respectively, with (f0g) = (f1g) = n(f0g) = n(f1g) = 1/2
and (f 1g) = n(f 1g) = 0, and n are distributions of Zn = maxfi : Xi =
Yn; i 2 f1; 2; ; ngg where Yn = maxfXi : i 2 f1; 2; ; ngg. The weak limit
of n is trivially , i.e., n ) ; n ! 1. Although n(f 1g) = 0 for all n, since
(Zn)n 1 is nondecreasing, n ) where (f1g) = 1. So ( n)n 1 is not tight on
(R; B(R)). As a probability measure on (R; B(R)), is the Dirac measure concentrated at
f1g. However, when restricted to B(R), is a subprobability measure on (R; B(R)) with
(R) = 0. Tightness of probability measures on (R; B(R)) is a condition to prevent mass
from ‘escaping to infinity’. Although ( n)n 1 is trivially tight on (R; B(R)), the definition
of Zn implies that ( )n 1 is intrinsically connected to ( n)n 1. We show that, from such connection, (f0g) = (R) = 0 is deducible, though it contradicts (f0g) = 1/2. See
http://www.eee.hku.hk/research/doc/tr/TR2014001.pdf for more details. 
Description  Poster presentation Session 12: Probability and Statistics 
Persistent Identifier  http://hdl.handle.net/10722/199393 
DC Field  Value  Language 

dc.contributor.author  Li, G  en_US 
dc.contributor.author  Li, VOK  en_US 
dc.date.accessioned  20140722T01:15:41Z   
dc.date.available  20140722T01:15:41Z   
dc.date.issued  2014   
dc.identifier.citation  The International Congress of Mathematicians (ICM 2014), Coex, Seoul, Korea, 1321 August 2014. In Abstract book, 2014, p. 487488, abstract no. OP120867  en_US 
dc.identifier.uri  http://hdl.handle.net/10722/199393   
dc.description  Poster presentation   
dc.description  Session 12: Probability and Statistics   
dc.description.abstract  We report a paradox of measuretheoretic probability. Denote by B(R) and B(R) the  algebras corresponding to the real line R and extended real line R, respectively. Let R be equipped with a topology induced by a fixed metric. Let ; ; n; n; n = 1; 2; be probability measures on (R; B(R)), such that ; 1; 2; are distributions of i.i.d. random variables X;X1;X2; , respectively, with (f0g) = (f1g) = n(f0g) = n(f1g) = 1/2 and (f 1g) = n(f 1g) = 0, and n are distributions of Zn = maxfi : Xi = Yn; i 2 f1; 2; ; ngg where Yn = maxfXi : i 2 f1; 2; ; ngg. The weak limit of n is trivially , i.e., n ) ; n ! 1. Although n(f 1g) = 0 for all n, since (Zn)n 1 is nondecreasing, n ) where (f1g) = 1. So ( n)n 1 is not tight on (R; B(R)). As a probability measure on (R; B(R)), is the Dirac measure concentrated at f1g. However, when restricted to B(R), is a subprobability measure on (R; B(R)) with (R) = 0. Tightness of probability measures on (R; B(R)) is a condition to prevent mass from ‘escaping to infinity’. Although ( n)n 1 is trivially tight on (R; B(R)), the definition of Zn implies that ( )n 1 is intrinsically connected to ( n)n 1. We show that, from such connection, (f0g) = (R) = 0 is deducible, though it contradicts (f0g) = 1/2. See http://www.eee.hku.hk/research/doc/tr/TR2014001.pdf for more details.  en_US 
dc.language  eng  en_US 
dc.relation.ispartof  International Congress of Mathematicians, ICM 2014  en_US 
dc.subject  measuretheoretic probability   
dc.subject  tightness of probability measures   
dc.subject  weak convergence   
dc.subject  subprobability measure   
dc.subject  zero measure   
dc.title  A paradox of measuretheoretic probability  en_US 
dc.type  Conference_Paper  en_US 
dc.identifier.email  Li, G: glli@hkucc.hku.hk  en_US 
dc.identifier.email  Li, VOK: vli@eee.hku.hk  en_US 
dc.identifier.authority  Li, VOK=rp00150  en_US 
dc.identifier.hkuros  230764  en_US 
dc.identifier.hkuros  254358   
dc.identifier.spage  487, abstract no. OP120867   
dc.identifier.epage  488   