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Conference Paper: A paradox of measure-theoretic probability
Title | A paradox of measure-theoretic probability |
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Authors | |
Keywords | measure-theoretic 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, 13-21 August 2014. In Abstract book, 2014, p. 487-488, abstract no. OP-12-0867 How to Cite? |
Abstract | We report a paradox of measure-theoretic 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 non-decreasing, 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/TR-2014-001.pdf for more details. |
Description | Poster presentation Session 12: Probability and Statistics |
Persistent Identifier | http://hdl.handle.net/10722/199393 |
DC Field | Value | Language |
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dc.contributor.author | Li, G | en_US |
dc.contributor.author | Li, VOK | en_US |
dc.date.accessioned | 2014-07-22T01:15:41Z | - |
dc.date.available | 2014-07-22T01:15:41Z | - |
dc.date.issued | 2014 | - |
dc.identifier.citation | The International Congress of Mathematicians (ICM 2014), Coex, Seoul, Korea, 13-21 August 2014. In Abstract book, 2014, p. 487-488, abstract no. OP-12-0867 | 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 measure-theoretic 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 non-decreasing, 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/TR-2014-001.pdf for more details. | en_US |
dc.language | eng | en_US |
dc.relation.ispartof | International Congress of Mathematicians, ICM 2014 | en_US |
dc.subject | measure-theoretic probability | - |
dc.subject | tightness of probability measures | - |
dc.subject | weak convergence | - |
dc.subject | subprobability measure | - |
dc.subject | zero measure | - |
dc.title | A paradox of measure-theoretic 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. OP-12-0867 | - |
dc.identifier.epage | 488 | - |