CAPÍTULO IV: ANÁLISIS Y EVALUACIÓN ADMINISTRATIVA Y FINANCIERO
4.2 ANÁLISIS Y EVALUACION FINANCIERO
4.2.2 Análisis Financiero
4.2.2.1 Introducción
Let A be a f in it e o r countable s e t, and define the doubly In f in it e product space
m
X : ■ n An , where AR ■ A fo r a ll n « TL . n— »
Upon the space X we define the ( l e f t - ) s h if t transformation S by s e ttin g
(S x)n: ■ xn+1 , fo r a ll n ( I and x ■ (x fl) « X .
Depending upon whether A has f in it e or In f in it e c a r d in a lity , we sh all Id e n tify 1t w ith (as appropriate) e ith e r {0,1. . . . , k - l > fo r k ■ card(A) , o r { 0 , 1 , 2 , . . . } . The set A w ill be re fe rre d to as
the alphabet, and sets o f the form ,
{10. . . 1 t ] J : « { x - (x n) c X; X j - 10, . . . . x j + l • 1t ) ,
where l Q , . . . , 1 t < A , are calle d o ylin d e ra . By 8 we denote the o-algebra generated by a ll c y lin d e r sets.
I f P ■ ( P ( 1 ,J ) ;1 ,J < A) Is a m atrix on A , then 1t 1s called
atoohaatio 1f fo r a l l 1 « A , P(1»•) 1s a p ro b a b ility on A . The m atrix P 1s c a lle d irre d u c ib le when fo r each 1 ,J < A there e x is ts an Integer n such that Pn(1*J) > 0 .
I t 1s w ell known (see, f o r example [ F ] ) that f o r any kxk Irre d u c ib le stochastic m atrix P , there e x is ts a unique s t r i c t l y p o s itive p ro b a b ility vector p « ( p ( 0 ) , . . . , p ( k - l ) ) such th at p.P ■ p . Define fo r each c y lin d e r
«(C10 . . . 1 Jt3J)s - P(10 ).P (10 .11) . — .P(11. 1 .1 l ) .
(2 .1 )
then m extends to a unique p ro b a b ility on X , which we sh all also denote by m .
When P 1s a stochastic m atrix over a countable alphabet A , we sh all assume that we have been provided w ith a le f t -in v a r ia n t s t r i c t l y p o s itiv e p ro b a b ility vector p , and using th is we construct the p ro b a b ility (measure) m as before. Note that 1n general fo r such matrices P , n either the existence nor the uniqueness o f a su ita b le ve ctor p 1s guaranteed (see [ F ] ) .
From equation (2.1 ) one sees that the p ro b a b ility m 1s s h if t In v a ria n t, 1.e. m iS '^ B )) ■ m(B) , fo r a l l sets B < S . The process (X,8,m ,S) o r (sim ply) S , 1s ca lle d the (2-s1ded) Markov s h if t defined by the m atrix P . We sh all a ls o re fe r to S as a Markov s h if t over f i n i t e l y o r countably many sta te s (as a p p ro p ria te ), o r more p re c is e ly , over the ( f i n i t e or countable) s ta te space A .
2.2. Remarks.
The above d e fin itio n s are standard, and we re fe r the reader to [ F ] , [D ], o r [Se2] fo r fu rth e r d e t a ils . The fo llo w in g d e fin itio n s are based on those given by A. del Junco and M. Rahe 1n CJ&R], and by W. Parry 1n [P2].
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I t 1s w ell known (see, fo r example [ F ] ) th at fo r any k*k Irre d u c ib le stochastic m atrix P , there e xis ts a unique s t r i c t l y p o s itiv e p ro b a b ility vector p ■ ( p ( 0 ) , . . . , p ( k - l ) ) such that p.P ■ p . Define f o r each c y lin d e r
- P(10).P(10.1l ) . - ” .P(1t_l .1t). (2-1)
then m extends to a unique p ro b a b ility on X , which we shall also denote by m .
When P 1s a stochastic m atrix over a countable alphabet A , we shall assume that we have been provided w ith a le f t -in v a r ia n t s t r i c t l y p o sitive p ro b a b ility vector p , and using th is we construct the p ro b a b ility (measure) m as before. Note that 1n general fo r such matrices P , n eith e r the existence nor the uniqueness o f a su ita b le vector p 1s guaranteed (see [ F ] ) .
From equation (2 .1 ) one sees that the p ro b a b ility m 1s s h if t In v a ria n t, l . e . m iS '^ B )) ■ "(B ) . fo r a ll sets B < B . The process ( X ,8,m,S) o r (sim p ly) S , 1s calle d the (2-s1ded) Markov » k i f t defined by the m atrix P . We shall also re fe r to S as a Markov s h if t over f in it e l y o r countably many states (as a p p ro p ria te ), o r more p re c is e ly , over the ( f i n i t e or countable) state space A .
2.2. Remarks.
The above d e fin itio n s are standard, and we re fe r the reader to [ F ] , [D ], o r [Se2] fo r fu rth e r d e ta ils . The fo llo w in g d e fin itio n s are based on those given by A. del Junco and M. Rahe 1n [J4 R ], and by W. Parry 1n [P2 ],
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2.3. DEFINITIONS.
I f ( X j t B j , S j ) » (X2,8 2,m2,S2) are 2-s1ded Markov s h ift s , then a measure preserving Isomorphism 4 between and S2 1s calle d f in it a r y
1f fo r m^ - a.e . x « ( x n) « X^ , there corresponds a p a ir o f Integers ( s . t ) , s < 0 < t , such that f o r m1 - a .e . y t Cxsxs+i " * x t ] S » (♦(x ) ) 0 ■ (4 (y ))o •
For a c y lin d e r C ■ £xsxs+i •••X^ S , s < 0 < t , x^ t A , s < 1 < t , we define the length o f C to be t-s+1 . We may then define a 81-measurable function i : X j K by s e ttin g t ( x ) to be the minimum value of t - s+1
a ris in g from pairs of Integers ( s , t ) associated to the point x « X^ 1n the above d e fin itio n o f f in it a r y . The function t 1s defined m^ - a .e ., and 1s called the oode-lengih of the f in it a r y Isomorphism 4 .
The f in it a r y Isomorphism 4 1s said to have f i n i t e expected, code length
when
The in v e rs e code length o f 4 1s defined as above w ith 4 replaced by 4 ’ 1 . C o lle c tiv e ly , the code length and Inverse code length of a f in it a r y map 4 1 are re fe rre d to as the code lengths of 4 •
2.4. Remark.
The present d e fin itio n o f f in it a r y 1s stra ig h tfo rw a rd ly equivalent to that given by W. Parry 1n CP23.
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2.5. DEFINITION.
An In v e r tib le measure preserving (resp. n on -sin gu lar) transformation T from a measure space (X,B,m) to I t s e lf 1s c a lle d re ve re ib le i f there e x is ts a measure preserving (re sp . non-singular) Isomorphism +:X X between T and T"^ , l . e , $T * T’ ^ m -a.e..
2.6. Example.
One straightforw ard example of a re v e rs ib le transformation 1s provided by a tra n s la tio n Tg on a compact group G , by a fixe d element g < G . A reversing map + f o r 1s given by *(h) - h’ \ h < G . In th is example both Tg and + preserve the Haar p ro b a b ility measure on G .
The fo llo w in g re s u lt , together with an Idea f o r Its p ro o f, was suggested to me by W. P a rry.
2.7. THEOREM.
Let S be the Markov s h if t defined by an Irre d u c ib le , f in it e stochastic m a trix, then S 1s re ve rs ib le by a reversing map + that Is measure
preserving, f l n l t a r y , and has f in it e expected code lengths.
. P roo f.
The proof w i l l be divided In to fiv e p arts. We f i r s t need to define the reversing map + we sh all be considering.
(1) D e fin itio n o f the reversing map ♦ .
Let Xg denote the doubly In f in it e product space on which the s h if t S Is defined as the l e f t - s h i f t , 1.e .
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X . - n {0 ,1 ...k-1 > ,
* 1—
fo r some fixe d In te g e r k > 2 , and (S x)n ■ xn+^ , fo r n c Z , and x * ( x n) e X$ . (N o tice that 1f k ■ 1 , then X$ Is a sin g le point, and the theorem 1s t r i v i a l . For th is reason, we assume that k > 2 . )
We sh all denote the defining m atrix f o r S by P$ , and the o-algebra and p ro b a b ility measure associated with P$ 1n D e fin itio n s 2.1 by 8$ and m^ re s p e c tiv e ly . Let us also make s im ila r d e fin itio n s with respect to the s h if t S-1 , noting that S"^ 1s to be the l e f t - s h i f t on
(re sp. X , ) o f the form e" ■
where x^,X2, . . . , x r « { l , . . . , k - l > , and j < Z . We define a map *:CS + C by s e ttin g
The map + can be extended to countable unions and Intersections of c ylin d e rs from C j , by d e fin in g
S V (2.2) ♦ ( H C . ) : - n f t C J 4 .i 1*1 * 1 i . r Ic I > (2.3) and
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♦ ( U c . ) : « U * ( C f ) ,
(2.4)
1cl 1 1 eI 1
where I 1s a countable Index s e t, and Cj « C$ f o r a l l 1 c I .
Since the s e t, Y $ , o f points x e whose associated sequence (x n) contains I n f in i t e ly many 0 's has f u l l measure, l . e .
m^ix * (xn) £ xs » cardin e Z;(x)n ■ 0> ■ •)) ■ 1 ,
the above map 4 may be considered as a map from 7C to 7 , , or
* S ' 1
from X . to X <
* S
(Note that fo r conceptual c la r it y we c a re fu lly 1
d is tin g u is h between the spaces on which S and S are defined.)
Henceforth, we consider * as a map from X^ to X ; by It s d e fin itio n , S _ i
4 1s seen to be 8^-measurable, and to have an Inverse 4 which 1s B .-m easurable, thus 4 1s an Isomorphism.
S " 1
(11) The map 4 s a tis fie s 4S ■ S " ^ . m r-a .e ..
By th e ir d e fin itio n s , the s h ifts S and S~^ s a tis fy
S ' ( H C .) - n S '( C J ,
1c I 1 1cl 1
(2.5)
s * ( U
C .) -U
S '( C .) ,I d 1 1 d 1 (2.6)
where I 1s a countable Index set such th a t Cj c Cs , fo r a ll 1 < I , and S' - S , S-1 . Equations (2 .3 ), (2 .4 ), (2 .5 ), and (2.6) together Im ply th a t to prove 4S ■ S_14 m j-a .e . on Y $ , 1t Is s u ffic ie n t to prove 4S(C) ■ S-14(C) f o r a ll c ylin d e rs C < Cs .
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Let C be an a r b itra r y c y lin d e r chosen from Cj , then I t 1s o f the form
C - C0x1x2. . . x r _1xr 0]J
fo r some j « 1 , and x ^ ,x2, . . . , x r e { 1 , 2 , . . . , k - l } . We also have
♦S(C) « ♦(C0x1x2. . . x r _1xr 0]J ' 1)
■ [Oxpxr _^...XgX^O] ^ ” 1 ; a c y lin d e r set 1n X^_ 1
■ S "1([0 xr xr _ ^ . . . x2x10]’^) ; since S’ 1 Is the l e f t - s h i f t on X ,
S" 1 - S‘1*(C0x1x2. . . x r _1xr 0 ]j )
» S '% (C ) .
Thus +S ■ S "1# ms -a .e . on , and since m$(Xs ) ■ 1 , *S ■ S_1* ms -a .e .
Parts (1) and (11) together prove th at + 1s an Isomorphism between S and S" 1 .
(111) The map ♦ 1s measure preservin g.
From the general theory o f Harkov s h if t s , the unique s t r i c t l y p o s itiv e p ro b a b ility vector ps associated to P$ In D e fin itio n s 2.1 1s the same as th a t fo r P , , (see [ F ; V o l . I , X V .12]) and w ill be denoted
S "'
P ( 0 > ^ ( U ) - P U i . P s U . 1) (2.7)
fo r a ll 1,J « { 0 , l , . . . , k - l } . By v ir t u e of equations (2.3) and (2 .4 ), to prove th at 4 1s measure p re servin g, 1t 1s s u f f ic ie n t to prove that 4 preserves the measure o f a ll c y lin d e rs C c C$ .
Let C - C O x .x ,...x .|X 0]^ e C$ , then by equations (2.1) and (2 .2 ), the fo llo w in g ca lc u la tio n 1s v a lid :
"> _ ,(4 (C )) S 1 - p(0 ).P , (0 ,x ) ...P _•« (x , ,0) S 1 r S 1 ' P ( X ) - M l ; using equation (2.7) - p (0 ).P s (0 ,X l ) . . . Ps (x r ,0 ) - m$(C) .
The map 4 1s therefore measure p re se rvin g.
(1v) The imps 4 and a" 1 are f1 n it a r y .
Every po in t x « 7 s lie s 1n a unique c y lin d e r C (x) < C$ of the form
C (x ) - C01l 12. . . 1 M 1r 0]J
fo r some J i Z , 2 -1 e n g th (C (x))< J < 0 , and ^<|»l2*****^r * •
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Since ms (^ s ) ■ 1 , the map ♦ 1s defined n ^ -a .e ., and hence by equation (2 .2 ), fo r m$-a .e . y « C ( x ) , ( * ( y ) ) Q ■ (♦ (x ))Q . By D e fin itio n s 2.3, ♦ 1s therefore f ln lt a r y . Proceeding In a s im ila r manner, 1s also shown to be f ln lt a r y .
(v ) The maps ♦ and ♦~1 have f in it e expected code le n gth .
To provide estimates fo r ce rta in recurrence p ro b a b ilitie s , the fo llo w in g well-known re s u lt w i l l be u s e fu l.
Claim.
There e xis ts a constant 0 < X < 1 such that fo r a ll n » card(A) ■ k ,
n>s(tx»(xn)«X s ;(x)Q «0 and (x)^jK) fo r a ll 1«1, 2 , . . .,n>) < p (0 ).X n .
Proof o f claim .
By assumption the d e fin in g m atrix fo r S , denoted P$ , 1s Irre d u c ib le , and hence fo r a l l 1, j < <0,1. . . . , k - l } , there e xis ts a sm allest p o sitive In te ge r m ■ m (1,J) such that P™(1,J) ► 0 . However, since t h is Implies th a t 1t 1s possible to get from state 1 to state j , we can do so In (k -1 ) steps o r le s s , and therefore m(1,J) < (k-1 ) fo r a ll sta te s 1, J c {0,1, . . . , k -1> .
L e t us define
r : - m1n {P7(1, J ) (1,J)> , (2.8)
1,JcA 5
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t y * <x - (x n) c X$ ; ( x ) Q « 0}
and In d u c tiv e ly define the sets Au , u * 1,2, . . . , by s e ttin g
Au+1: " {x “ ( xn) £ Au; ( x ) i i 0 f o r a ll 1 ■ uk+1. . . (u+1)k> .
The Markov property (see, fo r example [ F ;V o l. I, X V . 13]), together with the d e fin itio n o f r 1n equation (2.8) , Implies that
ins (Au+1) < (1 - r ) . ^ ) , f o r a l l u ■ 0 , 1... and hence n»s(Au) < (1 - r ) % s (A0) (2.9) ■ P ( 0 ) . ( l - r ) u , fo r a l l u ■ 1,2, . . . .
Choose and f i x a constant X , 0 < X < 1 such that
Xn > (1 - r ) Cn^k] , fo r a l l n ■ k , k + l , . . . ,
where [ z ] denotes the Integer p art o f z , then In e q u a lity (2.9 ) Implies th a t fo r a l l n > k ,
ms((x-(xn)«xs»(x)o" 0 and Wi A 0 ior
- p(0)**n •
We now proceed w ith the proof that * and have f in it e expected code length.
Using the same notation as 1n part (1 v ), we define a function L:Xg + F by s e ttin g
L ( x ) : ■ le n g th (C (x)) fo r x « .
(In fa c t we have only defined L on , but has f u l l measure 1n Xs . ) The function L 1s 8$-measurable and s a tis fie s L ( x ) > t ( x ) n ^ -a .e . where t denotes the code length o f * , as 1n D e fin itio n s 2.3. To prove that + has f in it e expected code length, I t 1s therefore s u ffic ie n t to show that L « L1(Xs ,8s ,ms ) , th is we do as follo w s.
m ^ tx c Xgt L (x ) ■ n })
■ ( n - l J . m jiix » (x n) < Xs ; ( x) 0 - 0 ■ ( x)n-1 and
( x) 1 i 0 fo r a ll t ■ l ,2, . . . , n -2>)
; because the number o f c ylin d e rs o f the co rre ct form and length n , Is (n -1)
< ( n - l j . m ^ i x - (x n) < X$ ; ( x ) Q ■ 0 and ( x ) i i 0 fo r a l l 1 - 1 ,2... n -2})
< ( n - l ) . p (0) . xn -2 , fo r a ll n > k+2 ,
by the above Claim. Therefore,