The theory of weak convergence, as it is currently applied in
probability theory, has come to be regarded as a study of convergence of Borel probability measures on Polish spaces, of which the most popular representatives are the function spaces like C[0, 1] and £>[0, 1] . The latter occur quite naturally when working with stochastic processes via their sample path behaviour e.g. a process with a.s. continuous sample paths is usually realisable as a probability measure on a space of
continuous functions (Billingsley (1968, p. 66)). The obvious success of this Polish space theory is in large part due to the fundamental work of Prohorov (1956), who not only pointed out the significance of this
particular approach, but also developed much of the necessary technical apparatus.
However even as early as in the work of Alexandroff (1943), there have been efforts to treat weak convergence in a more general setting. One paper that stands out in this respect is that of Le Cam (1957). He used functional analytic techniques in a study of the weak convergence behaviour of measures on uniform spaces. In particular he proved a number of results on compactness for sets of tight measures, including what may be identified as a generalised form of "Prohorov’s theorem" (Billingsley (1968, p. 35)). We believe that this more general weak convergence theory could be used to greater advantage in probability theory (see Chapter 6 for example).
methods for obtaining criteria for relative compactness in various spaces of measures. We have three reasons for proceeding in this way. Firstly, problems of relative compactness provide a non-trivial illustration of the techniques available in the general theory. Secondly, this is a topic of some theoretical importance, as shown by the substantial amount of attention it has received in the literature (see Section 11). Lastly, our necessary and sufficient conditions generalise Prohorov’s theorem; and it is that result which dominates the Polish space theory in its applications to probability. Let us see why this is so.
As expounded by Billingsley (1968), the standard form of weak
convergence theory can be developed for an arbitrary metric space X . The separability and completeness assumptions are not needed initially, although they later prove desirable. Let C(X) denote the space of all bounded, continuous real functions on X . Then the sequence {P^} Borel probabilities on
(written P^ =* P )
X is said to converge weakly to the target measure
if:
P
P^(/) -*■ P(/) for every / € P(X) . (9.1) From this weak convergence it can be deduced that (9.1) also holds if f
is only bounded and continuous P a.s.0 More generally, if T is a P a.s. continuous map into another metric space, then it can be proved that the induced measures P^.T ^ also converge weakly, to P.T ^ . Here lies the great usefulness of the whole method.
The mechanics of proving weak convergence are usually handled in two stages. First it is verified that the convergence in (9.1) is valid for all / ’s in some separating class of functions F (i.e. knowledge of
P(f) for all / ( F uniquely determines P amongst the class of all Borel probabilities on X ). Then the uniform tightness of the sequence
{P^} must be proved: for every £ > 0 there exists a compact set
such that liminf P > 1 - £ . From uniform tightness it follows that {P^} has a vital sequential compactness type of property: for every subsequence of } there is a further sub-subsequence weakly convergent to some probability measure on X . The weak convergence P^ => P is then easy to deduce.
p la y e d by u n if o r m t i g h t n e s s . In t h i s s e n s e modern weak co n v e rg e n c e th e o r y can be d a te d from P r o h o r o v ’ s 1956 p a p e r , s i n c e he i s th e one u s u a l l y
c r e d i t e d w ith d e v e lo p in g th e u n ifo rm t i g h t n e s s id e a ( of . o u r rem a rk s
e a r l i e r on t h e work o f Le Cam). In a d d i t i o n , P ro h o ro v showed t h a t u n ifo rm t i g h t n e s s i s a c t u a l l y a n e c e s s a r y and s u f f i c i e n t c o n d i tio n f o r " s e q u e n t i a l c o m p a c tn e s s " , when X i s a P o l i s h s p a c e ; so i t i s a l s o a n e c e s s a r y
c o n d i tio n f o r weak c o n v e rg e n c e ( s e e S e c tio n 16 f o r c a s e s w here i t i s
p o s s i b l e t o a v o id c h e c k in g u n ifo rm t i g h t n e s s d i r e c t l y ) . F o r g e n e r a l s p a c e s , th e s i t u a t i o n i s n o t so c l e a r c u t .
As w i l l b e e x p la in e d in th e n e x t s e c t i o n , i t i s n a t u r a l t o r e p l a c e th e " s e q u e n t i a l c o m p a c tn e ss" c o n d i t i o n by th e p r o p e r ty o f r e l a t i v e c o m p a c tn e ss . F o r P o l i s h s p a c e s th e two c o - i n c i d e , so P r o h o r o v ’ s r e s u l t may b e s t a t e d a s : a fa m ily o f p r o b a b i l i t y measures i s r e l a t i v e l y oompaot i n the space o f a l l t i g h t p r o b a b i l i t y measures i f f i t i s uniform ly t i g h t . [ R e c a l l t h a t any B o re l p r o b a b i l i t y m easu re on a P o l i s h s p a c e i s a u t o m a t i c a l l y t i g h t ( B i l l i n g s l e y (1 9 6 8 , p . 1 0 ) ) , s o t h e r e s t r i c t i o n t o th e s p a c e o f t i g h t
p r o b a b i l i t i e s i s i n o p e r a t i v e i n t h i s c a s e . ] In g e n e r a l t h i s e q u iv a le n c e i s no lo n g e r v a l i d ; u n if o r m ly t i g h t f a m i l i e s a r e r e l a t i v e l y co m p act, b u t n o t
c o n v e r s e ly . Those s p a c e s f o r w hich r e l a t i v e l y com pact f a m i l i e s a r e n e c e s s a r i l y u n if o r m ly t i g h t a r e so m etim es c a l l e d Prohorov s p a c e s . As Tops^e (1 9 7 4 b ) h a s s u rv e y e d t h i s a r e a we s h a l l n o t d w e ll on i t f u r t h e r , sa v e t o r e m a r k .t h a t P r e i s s (1 9 7 3 ) h a s shown t h a t th e r a t i o n a l num bers do n o t c o n s t i t u t e a P ro h o ro v s p a c e . We f e e l t h a t r e l a t i v e co m p actn e ss i s th e more im p o r ta n t c o n c e p t i n g e n e r a l , and t h a t u n ifo rm t i g h t n e s s s h o u ld be r e g a r d e d as j u s t a u s e f u l s u f f i c i e n t c o n d i t i o n . Hence o u r aim w i l l be t o
f i n d n e c e s s a r y and s u f f i c i e n t c o n d i tio n s f o r r e l a t i v e c o m p a c tn e ss , r a t h e r th a n t o e x p l o r e u n ifo rm t i g h t n e s s f u r t h e r .