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) $ \{ 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 $.

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