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{{MSC|60B10}}
 
{{MSC|60B10}}
  
Weak convergence or convergence in variation, and defined as follows. A sequence of distributions (probability measures) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335601.png" /> on the Borel sets of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335602.png" /> is called weakly convergent to a distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335603.png" /> if
+
Weak convergence or convergence in variation, and defined as follows. A sequence of distributions (probability measures) $  \{ P _ {n} \} $
 +
on the Borel sets of a metric space $  S $
 +
is called weakly convergent to a distribution $  P $
 +
if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\lim\limits _ { n }  \int\limits _ { S } f  dP _ {n}  = \int\limits _ { S } f  dP
 +
$$
  
for any real-valued bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335605.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335606.png" />. Weak convergence is a basic type of convergence considered in probability theory. It is usually denoted by the sign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335607.png" />. The following conditions are equivalent to weak convergence:
+
for any real-valued bounded continuous function $  f $
 +
on $  S $.  
 +
Weak convergence is a basic type of convergence considered in probability theory. It is usually denoted by the sign $  \Rightarrow $.  
 +
The following conditions are equivalent to weak convergence:
  
1) (*) holds for any bounded uniformly-continuous real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335608.png" />;
+
1) (*) holds for any bounded uniformly-continuous real-valued function $  f $;
  
2) (*) holds for any bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d0335609.png" />-almost-everywhere continuous real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356010.png" />;
+
2) (*) holds for any bounded $  P $-
 +
almost-everywhere continuous real-valued function $  f $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356011.png" /> for any closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356012.png" />;
+
3) $  \lim\limits _ {n} \sup  P _ {n} ( F) \leq  P ( F) $
 +
for any closed set $  F \subset  S $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356013.png" /> for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356014.png" />;
+
4) $  \lim\limits _ {n} \inf  P _ {n} ( G) \geq  P ( G) $
 +
for any open set $  G \subset  S $;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356015.png" /> for any Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356016.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356018.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356019.png" />;
+
5) $  \lim\limits _ {n}  P _ {n} ( A) = P ( A) $
 +
for any Borel set $  A \subset  S $
 +
with $  P ( \partial  A) = 0 $,  
 +
where $  \partial  A $
 +
is the boundary of $  A $;
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356021.png" /> is the [[Lévy–Prokhorov metric|Lévy–Prokhorov metric]].
+
6) $  \lim\limits _ {n}  p ( P _ {n} , P) = 0 $,  
 +
where $  p $
 +
is the [[Lévy–Prokhorov metric|Lévy–Prokhorov metric]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356022.png" /> be a class of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356023.png" />, closed under intersection and such that every open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356024.png" /> is a finite or countable union of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356025.png" />. Then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356027.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356028.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356031.png" /> are the distribution functions corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356033.png" /> respectively, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356034.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356035.png" /> at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356036.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356037.png" /> is continuous.
+
Let $  U $
 +
be a class of subsets of $  S $,  
 +
closed under intersection and such that every open set in $  S $
 +
is a finite or countable union of sets in $  U $.  
 +
Then if $  \lim\limits _ {n}  P _ {n} ( A) = P ( A) $
 +
for all $  A \in U $,  
 +
it follows that $  P _ {n} \Rightarrow P $.  
 +
If $  S = \mathbf R  ^ {k} $
 +
and $  F _ {n} $,  
 +
$  F $
 +
are the distribution functions corresponding to $  P _ {n} $,  
 +
$  P $
 +
respectively, then $  P _ {n} \Rightarrow P $
 +
if and only if $  F _ {n} ( x) \rightarrow F ( x) $
 +
at every point $  x $
 +
where $  F $
 +
is continuous.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356038.png" /> be a [[Separable space|separable space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356039.png" /> be the class of real-valued bounded Borel functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356040.png" />. To have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356041.png" /> uniformly over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356042.png" /> for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356044.png" />, it is necessary and sufficient that:
+
Let $  S $
 +
be a [[Separable space|separable space]] and let $  {\mathcal F} $
 +
be the class of real-valued bounded Borel functions on $  S $.  
 +
To have $  \int _ {S} f  dP _ {n} \rightarrow \int _ {S} f  dP $
 +
uniformly over $  f \in {\mathcal F} $
 +
for every sequence $  \{ P _ {n} \} $
 +
such that $  P _ {n} \Rightarrow P $,  
 +
it is necessary and sufficient that:
  
 
a)
 
a)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356045.png" /></td> </tr></table>
+
$$
 +
\sup _ {f \in F }  \omega _ {f} ( S)  < \infty ,
 +
$$
  
 
b)
 
b)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356046.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\epsilon \downarrow 0 }  \sup _ {f \in {\mathcal F} } \
 +
P ( \{ {x } : {\omega _ {f} ( S _ {x, \epsilon }  ) > \delta } \}
 +
= 0 \ \
 +
\textrm{ for }  \textrm{ all }  \delta > 0,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356047.png" /></td> </tr></table>
+
$$
 +
\omega _ {f} ( A)  = \
 +
\sup \
 +
\{ {| f ( x) - f ( y) | } : {x, y \in A } \}
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356048.png" /> is the open ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356049.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356050.png" />. If the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356051.png" /> is generated by the indicator functions of sets from some class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356052.png" />, then conditions a) and b) lead to the condition
+
and $  S _ {x, \epsilon }  $
 +
is the open ball of radius $  \epsilon $
 +
with centre $  x $.  
 +
If the class $  {\mathcal F} $
 +
is generated by the indicator functions of sets from some class $  E $,  
 +
then conditions a) and b) lead to the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356053.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\epsilon \downarrow 0 } \
 +
\sup _ {A \in E } \
 +
P ( A  ^  \epsilon  \setminus
 +
A ^ {- \epsilon } )  = 0,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356054.png" /></td> </tr></table>
+
$$
 +
A  ^  \epsilon  = \
 +
\cup _ {x \in A }
 +
S _ {x, \epsilon }  ,\ \
 +
A ^ {- \epsilon }  = \
 +
S \setminus  ( S \setminus  A)  ^  \epsilon
 +
$$
  
(when each open ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356055.png" /> is connected, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356056.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356057.png" /> and the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356058.png" /> is absolutely continuous with respect to Lebesgue measure, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356059.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356060.png" /> uniformly over all convex Borel sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356061.png" />.
+
(when each open ball in $  S $
 +
is connected, $  A  ^  \epsilon  \setminus  A ^ {- \epsilon } = ( \partial  A)  ^  \epsilon  $).  
 +
If $  S = \mathbf R  ^ {k} $
 +
and the distribution $  P $
 +
is absolutely continuous with respect to Lebesgue measure, then $  P _ {n} \Rightarrow P $
 +
if and only if $  P _ {n} ( A) \rightarrow P ( A) $
 +
uniformly over all convex Borel sets $  A $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356063.png" /> be distributions on a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356064.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356065.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356066.png" /> be a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356067.png" />-almost-everywhere measurable mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356068.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356069.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356070.png" />, where for any distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356071.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356072.png" />, the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356073.png" /> is its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356074.png" />-image on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356075.png" />:
+
Let $  P _ {n} $,  
 +
$  P $
 +
be distributions on a metric space $  S $
 +
such that $  P _ {n} \Rightarrow P $
 +
and let $  h $
 +
be a continuous $  P $-
 +
almost-everywhere measurable mapping of $  S $
 +
into a metric space $  S  ^  \prime  $.  
 +
Then $  P _ {n} h  ^ {-} 1 \Rightarrow Ph  ^ {-} 1 $,  
 +
where for any distribution $  Q $
 +
on $  S $,  
 +
the distribution $  Qh  ^ {-} 1 $
 +
is its $  h $-
 +
image on $  S  ^  \prime  $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356076.png" /></td> </tr></table>
+
$$
 +
Qh  ^ {-} 1 ( A)  = Q ( h  ^ {-} 1 ( A))
 +
$$
  
for any Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356077.png" />.
+
for any Borel set $  A \in S  ^  \prime  $.
  
A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356078.png" /> of distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356079.png" /> is said to be weakly relatively compact if every sequence of elements of it contains a weakly convergent subsequence. A condition for weak relative compactness is given by Prokhorov's theorem. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356080.png" /> is called tight if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356081.png" />, there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356082.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356083.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356084.png" />. Prokhorov's theorem now states: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356085.png" /> is tight, then it is relatively compact; if, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356086.png" /> is separable and complete, then weak relative compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356087.png" /> implies its tightness. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356088.png" />, a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356089.png" /> of distributions is weakly relatively compact if and only if the family of characteristic functions corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356090.png" /> is equicontinuous at zero.
+
A family $  {\mathcal P} $
 +
of distributions on $  S $
 +
is said to be weakly relatively compact if every sequence of elements of it contains a weakly convergent subsequence. A condition for weak relative compactness is given by Prokhorov's theorem. A family $  {\mathcal P} $
 +
is called tight if, for any $  \epsilon > 0 $,  
 +
there is a compact set $  K \subset  S $
 +
such that $  P ( K) > 1 - \epsilon $,  
 +
for all $  P \in {\mathcal P} $.  
 +
Prokhorov's theorem now states: If $  {\mathcal P} $
 +
is tight, then it is relatively compact; if, moreover, $  S $
 +
is separable and complete, then weak relative compactness of $  {\mathcal P} $
 +
implies its tightness. In the case when $  S = \mathbf R  ^ {k} $,  
 +
a family $  {\mathcal P} $
 +
of distributions is weakly relatively compact if and only if the family of characteristic functions corresponding to $  {\mathcal P} $
 +
is equicontinuous at zero.
  
Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356092.png" /> be distributions on a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356094.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356095.png" />-algebra. Convergence in variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356096.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356097.png" /> means uniform convergence on all sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356098.png" /> or, equivalently, convergence on all sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d03356099.png" /> or, equivalently, convergence of the variation
+
Now let $  P _ {n} $,  
 +
$  P $
 +
be distributions on a measure space $  ( X, A) $,  
 +
where $  A $
 +
is a $  \sigma $-
 +
algebra. Convergence in variation of $  P _ {n} $
 +
to $  P $
 +
means uniform convergence on all sets in $  A $
 +
or, equivalently, convergence on all sets in $  A $
 +
or, equivalently, convergence of the variation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d033560100.png" /></td> </tr></table>
+
$$
 +
| P _ {n} - P |  = \
 +
( P _ {n} - P)  ^ {+} +
 +
( P _ {n} - P)  ^ {-}
 +
$$
  
to zero. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d033560101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d033560102.png" /> are the components in the Jordan–Hahn decomposition of the signed measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033560/d033560103.png" />.
+
to zero. Here, $  ( P _ {n} - P)  ^ {+} $
 +
and $  ( P _ {n} - P)  ^ {-} $
 +
are the components in the Jordan–Hahn decomposition of the signed measure $  P _ {n} - P $.
  
 
====References====
 
====References====

Latest revision as of 19:36, 5 June 2020


2020 Mathematics Subject Classification: Primary: 60B10 [MSN][ZBL]

Weak convergence or convergence in variation, and defined as follows. A sequence of distributions (probability measures) $ \{ P _ {n} \} $ on the Borel sets of a metric space $ S $ is called weakly convergent to a distribution $ P $ if

$$ \tag{* } \lim\limits _ { n } \int\limits _ { S } f dP _ {n} = \int\limits _ { S } f dP $$

for any real-valued bounded continuous function $ f $ on $ S $. Weak convergence is a basic type of convergence considered in probability theory. It is usually denoted by the sign $ \Rightarrow $. The following conditions are equivalent to weak convergence:

1) (*) holds for any bounded uniformly-continuous real-valued function $ f $;

2) (*) holds for any bounded $ P $- almost-everywhere continuous real-valued function $ f $;

3) $ \lim\limits _ {n} \sup P _ {n} ( F) \leq P ( F) $ for any closed set $ F \subset S $;

4) $ \lim\limits _ {n} \inf P _ {n} ( G) \geq P ( G) $ for any open set $ G \subset S $;

5) $ \lim\limits _ {n} P _ {n} ( A) = P ( A) $ for any Borel set $ A \subset S $ with $ P ( \partial A) = 0 $, where $ \partial A $ is the boundary of $ A $;

6) $ \lim\limits _ {n} p ( P _ {n} , P) = 0 $, where $ p $ is the Lévy–Prokhorov metric.

Let $ U $ be a class of subsets of $ S $, closed under intersection and such that every open set in $ S $ is a finite or countable union of sets in $ U $. Then if $ \lim\limits _ {n} P _ {n} ( A) = P ( A) $ for all $ A \in U $, it follows that $ P _ {n} \Rightarrow P $. If $ S = \mathbf R ^ {k} $ and $ F _ {n} $, $ F $ are the distribution functions corresponding to $ P _ {n} $, $ P $ respectively, then $ P _ {n} \Rightarrow P $ if and only if $ F _ {n} ( x) \rightarrow F ( x) $ at every point $ x $ where $ F $ is continuous.

Let $ S $ be a separable space and let $ {\mathcal F} $ be the class of real-valued bounded Borel functions on $ S $. To have $ \int _ {S} f dP _ {n} \rightarrow \int _ {S} f dP $ uniformly over $ f \in {\mathcal F} $ for every sequence $ \{ P _ {n} \} $ such that $ P _ {n} \Rightarrow P $, it is necessary and sufficient that:

a)

$$ \sup _ {f \in F } \omega _ {f} ( S) < \infty , $$

b)

$$ \lim\limits _ {\epsilon \downarrow 0 } \sup _ {f \in {\mathcal F} } \ P ( \{ {x } : {\omega _ {f} ( S _ {x, \epsilon } ) > \delta } \} ) = 0 \ \ \textrm{ for } \textrm{ all } \delta > 0, $$

where

$$ \omega _ {f} ( A) = \ \sup \ \{ {| f ( x) - f ( y) | } : {x, y \in A } \} $$

and $ S _ {x, \epsilon } $ is the open ball of radius $ \epsilon $ with centre $ x $. If the class $ {\mathcal F} $ is generated by the indicator functions of sets from some class $ E $, then conditions a) and b) lead to the condition

$$ \lim\limits _ {\epsilon \downarrow 0 } \ \sup _ {A \in E } \ P ( A ^ \epsilon \setminus A ^ {- \epsilon } ) = 0, $$

where

$$ A ^ \epsilon = \ \cup _ {x \in A } S _ {x, \epsilon } ,\ \ A ^ {- \epsilon } = \ S \setminus ( S \setminus A) ^ \epsilon $$

(when each open ball in $ S $ is connected, $ A ^ \epsilon \setminus A ^ {- \epsilon } = ( \partial A) ^ \epsilon $). If $ S = \mathbf R ^ {k} $ and the distribution $ P $ is absolutely continuous with respect to Lebesgue measure, then $ P _ {n} \Rightarrow P $ if and only if $ P _ {n} ( A) \rightarrow P ( A) $ uniformly over all convex Borel sets $ A $.

Let $ P _ {n} $, $ P $ be distributions on a metric space $ S $ such that $ P _ {n} \Rightarrow P $ and let $ h $ be a continuous $ P $- almost-everywhere measurable mapping of $ S $ into a metric space $ S ^ \prime $. Then $ P _ {n} h ^ {-} 1 \Rightarrow Ph ^ {-} 1 $, where for any distribution $ Q $ on $ S $, the distribution $ Qh ^ {-} 1 $ is its $ h $- image on $ S ^ \prime $:

$$ Qh ^ {-} 1 ( A) = Q ( h ^ {-} 1 ( A)) $$

for any Borel set $ A \in S ^ \prime $.

A family $ {\mathcal P} $ of distributions on $ S $ is said to be weakly relatively compact if every sequence of elements of it contains a weakly convergent subsequence. A condition for weak relative compactness is given by Prokhorov's theorem. A family $ {\mathcal P} $ is called tight if, for any $ \epsilon > 0 $, there is a compact set $ K \subset S $ such that $ P ( K) > 1 - \epsilon $, for all $ P \in {\mathcal P} $. Prokhorov's theorem now states: If $ {\mathcal P} $ is tight, then it is relatively compact; if, moreover, $ S $ is separable and complete, then weak relative compactness of $ {\mathcal P} $ implies its tightness. In the case when $ S = \mathbf R ^ {k} $, a family $ {\mathcal P} $ of distributions is weakly relatively compact if and only if the family of characteristic functions corresponding to $ {\mathcal P} $ is equicontinuous at zero.

Now let $ P _ {n} $, $ P $ be distributions on a measure space $ ( X, A) $, where $ A $ is a $ \sigma $- algebra. Convergence in variation of $ P _ {n} $ to $ P $ means uniform convergence on all sets in $ A $ or, equivalently, convergence on all sets in $ A $ or, equivalently, convergence of the variation

$$ | P _ {n} - P | = \ ( P _ {n} - P) ^ {+} + ( P _ {n} - P) ^ {-} $$

to zero. Here, $ ( P _ {n} - P) ^ {+} $ and $ ( P _ {n} - P) ^ {-} $ are the components in the Jordan–Hahn decomposition of the signed measure $ P _ {n} - P $.

References

[B] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[L] M. Loève, "Probability theory" , Princeton Univ. Press (1963) MR0203748 Zbl 0108.14202
[BR] R.N. Bhattacharya, R. Ranga Rao, "Normal approximations and asymptotic expansions" , Wiley (1976) MR0436272

Comments

For more information on weak convergence see Weak convergence of probability measures; Convergence of measures.

How to Cite This Entry:
Distributions, convergence of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distributions,_convergence_of&oldid=46754
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article