CAPÍTULO IV: MARCO PROPOSITIVO
4.2 CONTENIDO DE LA PROPUESTA
4.2.2 FASE II: PLANIFICACIÓN
Choose and f i x sequences ( « t ) t i j • ( P ^ t a l * and ^‘ t^ ta l such that •
V
4-» •O ,m 3 (6.45) P-j > P2 ^ . . . > 0 , pt 0 as t ■* • , (6.46) m E Pt ■ - . t-1 '- 155 - ci > c2 > . . . > 0 , “ E E l < • • t-1 1 and define (6.47) \ : " u -t cu (6.48)
Let x^ be a p o s itiv e number greater than one, and define X® : - X^ • fo r a ll a c [0,1] . Now choose n] , an Integer greater than one, and
, an Integer greater than 1 + n^ . Using equations (6 .4 2 ), (6.43) and (6 .4 4 ), we now define HQ , N^ , and Mj . Let : ■ [0 ,1 ] .
Assume th a t (* y inu»mu) » mu > ^u* ^u* f°r u * 1 , 2 , . . . , t-1 , and x1 > x2 > . . . > xt-1 > 1 , are a ll chosen.
Choose xt such th a t both
1 < x^ < |(1 ♦ x ^ _ j) , (6.49)
and,defining x® : - * ( ! - » ■ xt ) + f ( x t - 1) , fo r a ll a c [0 ,1 ] ,
(6.50)
fo r «11 a < [0 ,1 ] .
Now take pt > 0 s a tis fy in g 2M
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fo r a ll a e [0 ,1] .
Let ct > 0 be such that
ct
- L f e xp (-s 2/2)ds ■ p* . (6-5J
* * J - c t 1
For any fix e d a « [0 ,1 ] , we may use the normal d is trib u tio n 's approximation to the binomial d is trib u tio n to Imply that fo r a ll large enough Integers n^ > , both
U“ - 2pt < |k - n jx j| < (6.53) and (6.54) where fo r k ■ 0 , l , . t . , n ^ , and
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U“ - ( n j ) * ( x j ) 1 c t /(1 + x\) .
Since polntwlse convergence Im plies convergence 1n p ro b a b ility , we can fin d an In te ge r nfc > Mt _^ , and a set c [0 ,1 ] w ith Lebesgue measure greater than 1 - 6^/2 , such that
and equations (6.53) and (6.54) are s a tis fie d fo r a ll a « At , with n? replaced by n*. . Equations (6.53) and (6.54) Imply that
t »
Ef^(lc) > Pt /4 , (6.56)
fo r a l l o < A^ .
For x < X , we define
fo r a c [0 ,1 ] , (6.57)
fo r u ■ 1 ,2 ,., . * t , and fo r R < S (to be sp ecified la te r)
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We now use the same arguments as 1n T. HamacM's paper [Ha] to show that 1f q 1s the measure defined by
q : ■ n u , 1 —
(6.59)
and 1 f Eg denotes In te g ra tio n with respect to q , then the follow ing equations hold fo r a l l a e : 11m
îr t flW x )
E (F°t H) l F^(Tj x) J-0 t Eq(F°t) B l ‘ E (F?H) - l f > ) , * 1 RsksS 1 V W - 2" t • (6.60) (6.61) (6.62) (6.63) • where 1s defined fo r x c X by ^ ( x ) ! - xD (t, 0 ) ( xi ♦ *2 ♦ ••• 4 xnt ) ' (6.64)and D (t,a ) 1s the set given by
D (t,a ):
r n ,x0 nt x.
J _ t . U“ , - 4 + u?
H X
l+x:
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I f L denotes Lebesgue measure on the In te rv a l [0 ,1 ] , then since polntwlse convergence Implies convergence 1n p ro b a b ility , equations (6.60), (6.61), and (6.63) Imply th a t there e x is ts an Integer J > J ( t ) such th at fo r a ll m i J
l" i 1F j(T Jx)H“ (TJx) a
L ({a c [0 ,l];q ({x < X ; ^ --- < 2pt >) > 1 - ct >) > 1 - E F“ (TJx)
J - o 1
Choose an Integer m^ > J ( t ) such th a t both
nt - Nt > J (T ) and
3Nt Nt e x p ^ n ^ M x , )
---'--- < t j 2
For a < [0 ,1 ] , define the set
m
£ F^(Tj x )H j(T J x ) • ? = ■ < « * > S ^ T - --- ‘ * t> I J-0 1 and le t A^ : ■ ( « « A j I q(B^) * 1 - *t ) . (6.6 6) (6.67) (6.68)- 160 -
L(AJ.) > l - at .
We have now chosen X^ , x® f o r i < [0 ,1 ] , n( and , and thus completed step t o f the In d uctive construction.
Let us d e fin e the set A by
A : - H A i , t-1 1
then by the choice o f the sequence • the set A has po sitive Lebesgue measure« and 1s thus uncountable. For each a < A , we have an associated sequence (x“ ,n t ,mt ) t l j • that de^ nes 8 measure
p° : ■ n p® , k— K
via the equations (6.40), (6 .4 1 ), (6.42), (6.4 3), and (6.4 4). This
I measure pa possesses a ll the properties used 1n T . Hamachl's calcu lations
(see CHa]).
The con stru ction 1s now complete.
□
6.10. THEOREM. (Ref. [Ha;Theorem 2])
Let T be the s h if t on the product space X , then w ith respect to any o f the uncountable I n f in i t y o f measures pa , fo r a < A , T Is n on -sin gu lar, ergodlc, and admits no o-f1n1te In va ria n t measure equivalent to p® . F o r a,9 < A , a j* 0 , the measures p® and p* are mutually sin g u la r.
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Proof.
The f i r s t p a rt o f the theorem 1s proved by noting that by construction, 1f a c A then we can Immediately use T . Hamachi's calculations fo r the measure pa . To prove that the measures are a ll mutually sin g u la r, suppose a,0 c A and a j* e . By applying Remark 6 .5 , the follo w in g calcu latio n
6.11. Remark.
Let p® , a « A c [0 ,1 ] be a p ro b a b ility measure obtained from Construction 6.9, and le t A denote the past o-algebra corresponding
to the s h if t T , that 1s the product o-algebra generated by the co-ordinate maps X1 , fo r 1 a 0 . I f C denotes the o-algebra generated by the co-ordinate maps , fo r t < 0 , then we can calculate the Information cocycle w ith respect to both A and C (using the measure p®) as In Section 5. By doing th is one obtains:
n.
A Im plies that p i p .•
i
i
2
a z (— — - — U r) .n«. ; by equations (6.41) and (6.40) i* ia 1 ■ • ; using In e q u a lity (6.55).The proof o f Theorem 6.10 1s complete.
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