Distributions, convergence of

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2010 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