Distributions, convergence of
2020 Mathematics Subject Classification: Primary: 60B10 [MSN][ZBL]
Weak convergence or convergence in variation, and defined as follows. A sequence of distributions (probability measures) on the Borel sets of a metric space
is called weakly convergent to a distribution
if
![]() | (*) |
for any real-valued bounded continuous function on
. Weak convergence is a basic type of convergence considered in probability theory. It is usually denoted by the sign
. The following conditions are equivalent to weak convergence:
1) (*) holds for any bounded uniformly-continuous real-valued function ;
2) (*) holds for any bounded -almost-everywhere continuous real-valued function
;
3) for any closed set
;
4) for any open set
;
5) for any Borel set
with
, where
is the boundary of
;
6) , where
is the Lévy–Prokhorov metric.
Let be a class of subsets of
, closed under intersection and such that every open set in
is a finite or countable union of sets in
. Then if
for all
, it follows that
. If
and
,
are the distribution functions corresponding to
,
respectively, then
if and only if
at every point
where
is continuous.
Let be a separable space and let
be the class of real-valued bounded Borel functions on
. To have
uniformly over
for every sequence
such that
, it is necessary and sufficient that:
a)
![]() |
b)
![]() |
where
![]() |
and is the open ball of radius
with centre
. If the class
is generated by the indicator functions of sets from some class
, then conditions a) and b) lead to the condition
![]() |
where
![]() |
(when each open ball in is connected,
). If
and the distribution
is absolutely continuous with respect to Lebesgue measure, then
if and only if
uniformly over all convex Borel sets
.
Let ,
be distributions on a metric space
such that
and let
be a continuous
-almost-everywhere measurable mapping of
into a metric space
. Then
, where for any distribution
on
, the distribution
is its
-image on
:
![]() |
for any Borel set .
A family of distributions on
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
is called tight if, for any
, there is a compact set
such that
, for all
. Prokhorov's theorem now states: If
is tight, then it is relatively compact; if, moreover,
is separable and complete, then weak relative compactness of
implies its tightness. In the case when
, a family
of distributions is weakly relatively compact if and only if the family of characteristic functions corresponding to
is equicontinuous at zero.
Now let ,
be distributions on a measure space
, where
is a
-algebra. Convergence in variation of
to
means uniform convergence on all sets in
or, equivalently, convergence on all sets in
or, equivalently, convergence of the variation
![]() |
to zero. Here, and
are the components in the Jordan–Hahn decomposition of the signed measure
.
References
[1] | P. Billingsley, "Convergence of probability measures" , Wiley (1968) |
[2] | M. Loève, "Probability theory" , Princeton Univ. Press (1963) |
[3] | R.N. Bhattacharya, R. Ranga Rao, "Normal approximations and asymptotic expansions" , Wiley (1976) |
Comments
For more information on weak convergence see Weak convergence of probability measures; Convergence of measures.
Distributions, convergence of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distributions,_convergence_of&oldid=11307