Difference between revisions of "Weak convergence of probability measures"
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The general setting for weak convergence of probability measures is that of a complete separable [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971801.png" /> (cf. also [[Complete space|Complete space]]; [[Separable space|Separable space]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971802.png" /> being the metric, with probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971804.png" /> defined on the Borel sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971805.png" />. It is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971806.png" /> converges weakly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971807.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971808.png" /> if for every bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971809.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718010.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718011.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718012.png" />. If random elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718014.png" /> taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718015.png" /> are such that the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718016.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718018.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718019.png" />, and says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718020.png" /> converges in distribution to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718022.png" /> converges weakly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718023.png" /> (cf. also [[Convergence in distribution|Convergence in distribution]]). | The general setting for weak convergence of probability measures is that of a complete separable [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971801.png" /> (cf. also [[Complete space|Complete space]]; [[Separable space|Separable space]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971802.png" /> being the metric, with probability measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971804.png" /> defined on the Borel sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971805.png" />. It is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971806.png" /> converges weakly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971807.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971808.png" /> if for every bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w0971809.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718010.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718011.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718012.png" />. If random elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718014.png" /> taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718015.png" /> are such that the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718016.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718018.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718019.png" />, and says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718020.png" /> converges in distribution to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718022.png" /> converges weakly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097180/w09718023.png" /> (cf. also [[Convergence in distribution|Convergence in distribution]]). |
Revision as of 07:27, 19 August 2012
2020 Mathematics Subject Classification: Primary: 60B10 [MSN][ZBL]
See also Convergence of measures.
The general setting for weak convergence of probability measures is that of a complete separable metric space (cf. also Complete space; Separable space),
being the metric, with probability measures
,
defined on the Borel sets of
. It is said that
converges weakly to
in
if for every bounded continuous function
on
one has
as
. If random elements
,
taking values in
are such that the distribution of
is
,
one writes
, and says that
converges in distribution to
if
converges weakly to
(cf. also Convergence in distribution).
The metric spaces in most common use in probability are ,
-dimensional Euclidean space,
, the space of continuous functions on
, and
, the space of functions on
which are right continuous with left-hand limits.
Weak convergence in a suitably rich metric space is of considerably greater use than that in Euclidean space. This is because a wide variety of results on convergence in distribution on can be derived from it with the aid of the continuous mapping theorem, which states that if
in
and the mapping
is continuous (or at least is measurable and
, where
is the set of discontinuities of
), then
. In many applications the limit random element is Brownian motion, which has continuous paths with probability one.
One of the most fundamental weak convergence results is Donsker's theorem for sums ,
, of independent and identically-distributed random variables
with
,
. This can be framed in
by setting
and
,
, where
denotes the integer part of
. Then Donsker's theorem asserts that
, where
is standard Brownian motion. Application of the continuous mapping theorem then readily provides convergence-in-distribution results for functionals such as
,
,
, and
, where
is the indicator function and
if
and
otherwise.
References
[B] | P. Billingsley, "Convergence of probability measures" , Wiley (1968) pp. 9ff MR0233396 Zbl 0172.21201 |
Weak convergence of probability measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_convergence_of_probability_measures&oldid=26973