Strongly continuous chains

H e n c e f o r th i t w i l l be assumed t h a t X i s a t o p o l o g i c a l s p a c e and t h a t 8 i s i t s B o r e l o - a l g e b r a . To b e g i n w i t h we employ q u i t e a s t r o n g

c o n t i n u i t y c o n d i t i o n f o r t h e t r a n s i t i o n p r o b a b i l i t i e s . Even th o u g h t h i s w i l l p r o v e enough t o g u a r a n t e e t h a t r e l a t i v e l y compact s e t s a r e R- s e t s and E - s e t s , we f e e l t h a t i t i s g e n e r a l l y t o o s t r o n g a c o n s t r a i n t t o p l a c e upon a c h a i n . But i t i s i n s t r u c t i v e t o f o l l o w th r o u g h w ith t h e a rg u m e n ts i n t h i s c a s e , i f o n ly t o a p p r e c i a t e t h e e x t r a a s s u m p tio n s r e q u i r e d when t h e s t r o n g c o n t i n u i t y c o n d i t i o n i s r e l a x e d . We s a y t h a t a Markov c h a i n i s s t r o n g l y c o n t i n u o u s i f , f o r e a c h A € 8 , P ( x , A) i s a c o n t i n u o u s f u n c t i o n o f x . I t i s c o n v e n i e n t t o r e - e x p r e s s t h i s i n te rm s o f t h e o p e r a t o r P in d u c e d by t h e t r a n s i t i o n p r o b a b i l i t i e s on t h e s p a c e B(X) o f b o u n d e d , m e a s u r a b le r e a l f u n c t i o n s on X . T h is o p e r a t o r maps g £ B(X) i n t o t h e f u n c t i o n P ( • , g) (= P ( • , d y ) g ( y ) ) . S tr o n g c o n t i n u i t y i s t h u s e q u i v a l e n t t o : Pg € C(X) f o r e v e r y g Z B( X) . Yl S in c e C(X) c B(X) , t h e i t e r a t e s P a r e t h e r e f o r e a l s o s t r o n g l y c o n t i n u o u s , and h en ce G j i x , g) must be c o n t i n u o u s i n x f o r e v e r y g € B(X) and

0 < r < 1 .

THEOREM 2 3 01 „ For any ( M - i r r e d u c i h l e ) s t r o n g l y c o n t i n u o u s Markov c h a i n } e v e r y r e l a t i v e l y com pact A € 8 + i s an R - s e t .

P r o o f . L e t A be a r e l a t i v e l y compact member o f 8 + . I t s u f f i c e s t o show t h a t i f , f o r some r > 1 , £ € X and B € 8 + , G ^ (£ , 8 ) i s c o n v e r g e n t t h e n G^(C, A) c o n v e r g e s t o o .

D e fin e = [ij : G ^ ( y, B) > n f o r

n

= 1 , 2, . . . . The c o n t i n u i t y o f £ , ( • , B) i m p l i e s t h a t each U i s o p en . A ls o , s i n c e M(B) > 0 i m p l i e s

* 2 n

B = X , G A y, B) > 0 f o r e v e r y y € X . Thus U t X , and s o t h e

■'i? n

r e l a t i v e l y compact s e t A must be c o v e r e d by one o f t h e > s a Y

A

s

uu

•

We show t h a t G ( c , Um 1 < 00 , from which i t w i l l f o l l o w t h a t A € 8„ . To p r o v e t h i s we u se t h e Chapman-Kolmogorov r e l a t i o n

m n+mn+m

valid for m, n > 1 . Choose 0 = (2r) ^ and sum over first n then m to give e d - e ) ' 1^ (c, B) 2 V dy)G^(y, Z?) ,-l

4/).fl

a ,-i

By assumption G (C, ß) < 00 ; thus (7 (c, Z/,7) < 00 too.

The argument to show that every relatively compact member of

B

is an L-set can be carried out in a similar fashion by using the criterion of Theorem 22.1.

T H E O R E M 2 3 o 2 0

If an (M-irreducible) Markov chain is strongly

continuous and R-recurrent then every relatively compact

B +

set is an L-set.

Proof.

Let A €

B +

be relatively compact. Since M and Q are

equivalent, Q(A) > 0 . We construct a sequence of open sets U f

X

with

Q (c/ ) < 00 and f > n \(2i?-l) a.s. on for every n . As before,

A c: for some N , hence A will satisfy the sufficient conditions of Theorem 22.1 for it to be an L-set.

Iterate the defining equation (22.7) and sum to obtain r 00

£ (2Z?) nf(x) =

n -1 J

Thus the extended real function

£ (%)"*” («, dy)f(y) M a.s. .

n-1

/ * ( # ) = dy)f(y)

-1

equals (2Z?-1) /(a;) for almost all x 6

X

. Notice that f*(x) > 0 for every x €

X

(since f has the same property) and that /* is l.s.c., being the pointwise supremum of the continuous functions

G^(x, dy)min(n, f) , n = 1, 2,

I t e r a t i n g ( 2 2 .8 ) i n th e same f a s h io n we o b ta in

Q(B) = (2 i? -l) Q( d y ) G^ ( y, B) ( 2 3 .1 )

S in c e Q i s a - f i n i t e t h e r e e x i s t s a se q u e n c e 0^ t X w ith 0 < Q [b^] < 00 f o r each n . D e fin e V^ = jy : G p { y, B^) > n . T hese form an

i n c r e a s i n g se q u e n c e o f open s e t s w hich c o v e r X , s i n c e { £ , ( • , B ) } i s an i n c r e a s i n g se q u e n c e o f s t r i c t l y p o s i t i v e , c o n tin u o u s f u n c t i o n s . A ls o , from ( 2 3 . 1 ) ,

“ > « ( s j 5 ( 2 i ? - l ) . e p J . : - 1

The r e q u i r e d se q u e n c e {f/^} can t>e d e f in e d by U - n > n .□ H aving p ro v e d t h a t s t r o n g l y c o n tin u o u s c h a in s h av e t h e s e n i c e p r o p e r t i e s , we s h o u ld a l s o show t h a t t h e r e e x i s t n o n - t r i v i a l ex am p les o f su c h c h a i n s . C e r t a in random w alk s f a l l i n t o t h i s c l a s s .

EXAMPLE

2 3

o

1

. C o n s id e r a random w alk on th e r e a l l i n e

R

, i . e . a Markov c h a in w ith t r a n s i t i o n p r o b a b i l i t i e s

P ( x , A) = p(A-a;)

f o r some B o re l p r o b a b i l i t y

y

on

R

. I f

y

i s a b s o l u t e l y c o n tin u o u s w r t Lebesgue m easure th e n th e c h a in i s s t r o n g l y c o n tin u o u s . F o r i f

y

h a s d e n s i t y f u n c t i o n h th e n f o r g £

B ( R ) ,

P ( x , g) = h ( y - x ) g ( y ) d y

J R

- h * g ( x ) w here h( x ) = h ( - x ) . T h is l a s t e x p r e s s io n i s th e c o n v o lu tio n o f an L L CO f u n c t i o n w ith an L f u n c t i o n w h ic h , by a th e o re m o f c l a s s i c a l a n a l y s i s (H e w itt 6 S tro m b e rg (1 9 6 5 , p . 3 9 8 ) ) , i s a c t u a l l y u n ifo r m ly c o n tin u o u s on

R

. Thus th e c h a in p o s s e s s e s t h e a s s e r t e d s t r o n g c o n t i n u i t y p r o p e r t y .

The q u e s ti o n o f i r r e d u c i b i l i t y p o s e s more o f a p ro b le m . The o b v io u s c a n d id a te f o r th e i r r e d u c i b i l i t y m easure cp i s Lebesgue m e a su re . I t w ould th e n s u f f i c e t o have h s t r i c t l y p o s i t i v e in some n e ig h b o u rh o o d o f th e o r i g i n . A n o th er p o s s i b i l i t y w ould be M( •) = G^(0 , • ) ; i f th e o r i g i n w ere an i n t e r i o r p o i n t o f th e s u p p o r t o f M th e n t h i s to o w ould be a s u i t a b l e c h o ic e f o r th e i r r e d u c i b i l i t y m e a su re . / /

In document Topological measure theory, with applications to probability (Page 119-122)