There are no files associated with this item.

##### Supplementary
• Appears in Collections:

#### Conference Paper: A paradox of measure-theoretic probability

Title A paradox of measure-theoretic probability Li, GLi, VOK measure-theoretic probabilitytightness of probability measuresweak convergencesubprobability measurezero measure 2014 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? 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. Poster presentationSession 12: Probability and Statistics http://hdl.handle.net/10722/199393

DC FieldValueLanguage
dc.contributor.authorLi, Gen_US
dc.contributor.authorLi, VOKen_US
dc.date.accessioned2014-07-22T01:15:41Z-
dc.date.available2014-07-22T01:15:41Z-
dc.date.issued2014-
dc.identifier.citationThe International Congress of Mathematicians (ICM 2014), Coex, Seoul, Korea, 13-21 August 2014. In Abstract book, 2014, p. 487-488, abstract no. OP-12-0867en_US
dc.identifier.urihttp://hdl.handle.net/10722/199393-
dc.descriptionPoster presentation-
dc.descriptionSession 12: Probability and Statistics-
dc.description.abstractWe 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.languageengen_US
dc.relation.ispartofInternational Congress of Mathematicians, ICM 2014en_US
dc.subjectmeasure-theoretic probability-
dc.subjecttightness of probability measures-
dc.subjectweak convergence-
dc.subjectsubprobability measure-
dc.subjectzero measure-
dc.titleA paradox of measure-theoretic probabilityen_US
dc.typeConference_Paperen_US
dc.identifier.emailLi, G: glli@hkucc.hku.hken_US
dc.identifier.emailLi, VOK: vli@eee.hku.hken_US
dc.identifier.authorityLi, VOK=rp00150en_US
dc.identifier.hkuros230764en_US
dc.identifier.hkuros254358-
dc.identifier.spage487, abstract no. OP-12-0867-
dc.identifier.epage488-