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pkg
science forum beginner
Joined: 05 Jan 2006
Posts: 30
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 Posted: Tue Apr 25, 2006 7:18 am Post subject:
Prohorov metric and total variation distance
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Can anyone tell me whether the topology on a measure space induced by
the Prohorov metric (which generates the topology of weak convergence)
on the one hand and the topology induced by the total variation
distance are equivalent. In other words does the total variation
distance induce the topology of weak convergence (the measure theoretic
weak convergence!).
Many thanks for answers ...
pkg |
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user@domain.invalid
science forum beginner
Joined: 08 Oct 2005
Posts: 6
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| Posted: Tue Apr 25, 2006 7:38 am Post subject:
Re: Prohorov metric and total variation distance
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pkg wrote:
| Quote: | Can anyone tell me whether the topology on a measure space induced by
the Prohorov metric (which generates the topology of weak convergence)
on the one hand and the topology induced by the total variation
distance are equivalent. In other words does the total variation
distance induce the topology of weak convergence (the measure theoretic
weak convergence!).
Many thanks for answers ...
pkg
Definitely not. |
delta_{1/n} converges to delta_0 in the weak topology, but not with
respect to the total variation metric.
Dieter |
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pkg
science forum beginner
Joined: 05 Jan 2006
Posts: 30
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| Posted: Tue Apr 25, 2006 11:17 am Post subject:
Re: Prohorov metric and total variation distance
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Yes, that's right ... Thanks. pkg |
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Herman Rubin
science forum Guru
Joined: 25 Mar 2005
Posts: 730
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| Posted: Tue Apr 25, 2006 4:43 pm Post subject:
Re: Prohorov metric and total variation distance
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In article <1145949483.675644.326210@y43g2000cwc.googlegroups.com>,
pkg <Patric.Gloede@t-online.de> wrote:
| Quote: | Can anyone tell me whether the topology on a measure space induced by
the Prohorov metric (which generates the topology of weak convergence)
on the one hand and the topology induced by the total variation
distance are equivalent. In other words does the total variation
distance induce the topology of weak convergence (the measure theoretic
weak convergence!).
Many thanks for answers ...
pkg
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Definitely not. In the Prohorov metric, which in the case
of R_1 is equivalent to the usual notion of weak
convergence of distributions, a sequence of discrete
distributions often converges to a continuous distribution
(Central Limit Theorem) for example.
Or even simpler, convergence in probability.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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