By C. Preston
ISBN-10: 3540078525
ISBN-13: 9783540078524
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Additional info for Random Fields
Sample text
Like this: if E denotes the H-ergodic an affine bijection T : P(E,E) subsets of P(E,~) E , and denotes the point mass at exists; let E The representation ~P (F) denotes ~ elements of where will usually be something Po~_F) then there will be E is a suitable o-field the probability measures for any a g E , then = { ~ g E : v(I(H)) = 1 } on E (E,~) and suppose that E of . If T(6 ) = ~ . Suppose O that T g E . T : P(~o,E) It is not hard to show that extreme points. E is affine, onto, and sends extreme points to From this it is a simple matter to obtain a representation in terms of (Eo,~o) ~C=o ~)- " P(Eo,E=o) , where under the equivalence (Eo,~) relation We will now give sufficient is the quotient , where conditions ~ ~ 6 if to ensure that of space got from E(~ ) = E(6~) _Go~_V) .
H-invariant; hence the equal to a constant. x g X , thus has trivial can only take the values 0 or But tail- i . e. i . 2 is now complete. = ~ that we can consider if and only if ~ ~ g ~C~) as an affine mapping Recall ~(F) G=~V) from . 20) . 12 ; this mapping fixes the points of sends extreme points G ~) , and is thus onto, and it to extreme points. In all reasonable situations there will exist a representation of P (F) O in terms of the H-ergodic measures. like this: if E denotes the H-ergodic an affine bijection T : P(E,E) subsets of P(E,~) E , and denotes the point mass at exists; let E The representation ~P (F) denotes ~ elements of where will usually be something Po~_F) then there will be E is a suitable o-field the probability measures for any a g E , then = { ~ g E : v(I(H)) = 1 } on E (E,~) and suppose that E of .
82 < I I ~m (A) - ~n (A) lim Vn(A ) = p(A) = Dm(A) , and hence I + 2y . 2 for all There exists a unique F g A . ) are consistent. Clearly we can ex- tend this sequence to get a consistent family Be P B(F) = lim ~A (x,F) n {p0}6g~ with p o g P(B_~) i . 5) there exists a (unique) and B e P F(~_ m such that B(F) = Pm(F) for all , m~ F c ~ i . 1 we get m B(F) = lira ~ A (x,F) . 3 B(RA) = i for all A ~ C 9 Proof Using the notation of Section 2 ~) g T A then Now v(RA) = 1 . ) e_T A II that if B(RA) = n => no i TT[ ~ ~ g T% then and thus, given any _TA whenever ~)(RA) = 1 A e =C A < % ; also if for all there exists A < 7[ .
Random Fields by C. Preston
by Charles
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