TITULO SEPTIMO Delitos Contra la Salud

concentrated on the closed subset of [0, 00)

We shall prove that these two conditions completely characterise the fidis of a random measure. The idea behind the proof is simple, but matters are slightly complicated by a problem of countability. To overcome this

difficulty we need to make use of a separability property of

C .

Since the topology on S has a countable base, there exists a countable subset of

C

which is dense in the topology of uniform convergence (Bauer (1972, p. 224)). Without loss of generality it can be supposed that this subset forms a vector lattice over the field of rational numbers. Let

V

be its positive cone, and

V

be the set of [0, °°)-valued functions on

V

which satisfy the condition:

y

(<7 - yildfj) + y{<3^) for every pair

g 9 g^ € V

. Equip

V

with the weakest topology making each of the maps

y 1—

¥ y(y} continuous, for

g € V .

Clearly each Radon measure on S can be used to define a member of

V

. By means of standard Riesz space techniques (Bourbaki (1952, Chap. II,

§2)), this correspondence can be shown to be a one-to-one map of X onto

V

which establishes a homeomorphism between the two spaces. That is, X and

V

can be regarded as the same topological space for most purposes. In particular, there is a one-to-one correspondence between the Borel

probabilities on X and

V

; so to produce a random measure it suffices to construct a Borel probability on V . Since the topology of

V

is

g e n e r a t e d b y t h e c y l i n d e r s e t s o f t h e fo rm [y * y

:

[ y

G^) » •••>

y(Qk) ) * H)

,

w h e r e T = \ g ^ , . . . , c V a n d

H

i s an open s u b s e t o f [ 0 , 00) , a r o u t i n e a r g u m e n t c an b e u s e d t o p r o v e t h a t B o r e l p r o b a b i l i t i e s on

V

a r e u n i q u e l y s p e c i f i e d b y t h e m e a s u r e s o f s u c h c y l i n d e r s e t s . I t f o l l o w s t h a t a random m e a s u r e (= a B o r e l p r o b a b i l i t y on X ) i s u n i q u e l y d e t e r m i n e d by

r

i t s f i d i s P , w h e re V r a n g e s o v e r t h e f i n i t e s u b s e t s o f V .

r

r

THEOREM 1 9 „ 1 „

Suppose a Borel p ro b a b ility P

on

[ 0 , °° )

i s given

fo r each f i n i t e su b set V o f

C+ .

These are the f i d i s o f a uniquely

determined random measure i f f conditions

( i )

and

( i i )

above are s a t i s f i e d

.

P r o o f . O n ly t h e s u f f i c i e n c y n e e d s t o b e c o n s i d e r e d . A p p l y i n g a v e r s i o n o f t h e K olm ogorov e x t e n s i o n t h e o r e m (N eveu ( 1 9 6 5 , p . 8 2 ) ) , we d e d u c e fro m c o n d i t i o n ( i ) ( r e s t r i c t e d t o t h o s e T c V ) t h a t t h e r e i s a V p r o b a b i l i t y m e a s u r e P ^ on [ 0 , °°) h a v i n g t h e p r e s c r i b e d f i n i t e p d i m e n s i o n a l d i s t r i b u t i o n s

P

, f o r T c V . T h i s i s d e f i n e d on t h e c y l i n d e r G - a l g e b r a , w h i c h c o i n c i d e s w i t h t h e B o r e l ö - a l g e b r a o f [ 0 , °°)^ s i n c e V i s c o u n t a b l e . Now n o t i c e t h a t

V

i s a t o p o l o g i c a l s u b s p a c e o f [ 0 , °°)^ . I n d e e d , i t i s a c l o s e d s u b s e t o f t h a t s p a c e , b e c a u s e i t c a n b e e x p r e s s e d a s t h e ( c o u n t a b l e ) i n t e r s e c t i o n o f c l o s e d c y l i n d e r s u b s e t s o f t h e form I ' M C o ,

°°)V

:

'P{g1+g2) =

w h e re g ^ ] r a n g e s o v e r a l l p a i r s o f V f u n c t i o n s . C o n d i t i o n ( i i ) shows t h a t e a c h o f t h e s e c y l i n d e r s e t s h a s P Q m e a s u r e o n e ; h e n c e

V

a l s o h a s P^ m e a s u r e o n e . The r e q u i r e d rand om m e a s u r e P i s o b t a i n e d by t r a n s f e r r i n g P^ fro m

V

t o t h e h o m eo m o rp h ic s p a c e X .

r

T h i s P h a s t h e d e s i r e d f i d i s P , f o r T <z V ; b u t i t r e m a i n s t o p r o v e t h a t t h i s a l s o h o l d s f o r any T c C+ . S u p p o s e t h e n t h a t c C+ . C a r r y o u t t h e p r e c e d i n g a r g u m e n t a g a i n , b u t t h i s t i m e u s i n g t h e c o u n t a b l e d e n s e s u b s e t V '

o f

C+ w h ic h i s o b t a i n e d fro m t h e a u g m e n te d s e t V u Tq . T h i s p r o c e d u r e g e n e r a t e s a n o t h e r random m e a s u r e

P'

h a v i n g t h e

desired fidis for every Y c V . In particular, P and P f have the

r

same fidis for every r c D ; so P = P' , and hence P,tJ^ = P'.TT^ = P ^

!o

10

as required. □

Similar proofs for the existence of random measures on general

compact, Hausdorff spaces and a-compact, locally compact spaces were given by Prohorov (1960, 1961) and Le Cam (1961) respectively.

Random measures can also be constructed by starting with a different type of fidi. Recall that the maps x •— *- a?(4) , where A runs through the class , generate the Borel a-algebra on

X

. It is thus easy to prove that a random measure P is uniquely determined by its setwise fidis

PA . (') = €

X

:

(x(/4 ) ,

- • • • • 3-rl J-

1 n

« •} *

where {>4 , . ..9 A^\ is any finite subset of B^ . Jagers (1974, p. 193) has stated consistency conditions on the setwise fidis which ensure the existence of the corresponding random measure. Other authors who have adopted this approach include Harris (1963, 1968) and Jirina (1964, 1966, 1972). The main feature of their proofs is the use of inner regularity wrt a semi-compact paving to convert random finitely additive measures into random countably additive measures {of. the comments in Section 1).

Kallenberg (1974) has given a completely different type of existence proof, based on some preliminary weak convergence results. We comment on his method further in the next section. See also Nawrotzki (1962) and Moyal

(1962).

The setwise version of the existence theorem has the advantage that it can easily be converted into an existence theorem for point processes - see Jagers (1974, p. 201). Our Theorem 19.1 could also be adapted to this purpose, but the necessary modifications would involve some extra

complications. By definition, a random measure P is a point process iff it is concentrated on the closed subset

X^.

of

X

. As

X

can be

expressed as a countable intersection of closed cylinder subsets of

X ,

the conditions to ensure P(X^.) = 1 could be given in terms of the fidis P ; but this becomes quite messy in practice. Theorem 19.1 is not entirely useless for working with point processes though.

EXAMPLE 19ol.

Let X be an arbitrary but fixed Radon measure on

S

.

property: if ..., A^ are pairwise disjoint sets then the numbers of points falling into these sets are independent Poisson variates with means X(i4^), X [A^] . We prove that such a process exists.

If P were a process with these fidis then it would follow that, for any set of simple functions f^9 ..., / of the form

n

fj = I a-J/i with a11

0 k=l k

the joint characteristic function (C.F.) of the variates x(/*^) , x(/ ) would be given by:

I exp [it±x[f^) + ... t (dx) =

= exp I [_-l+exp [it 1f1 + ... + . (19.1) It is easy to check that this is a genuine C.F. of a distribution on

[0, 00)m . Now approximate members of C+ by such simple functions to deduce

that the joint C.F. of the variates x , ..., x(g ) , where F = {g , ..., g } c C+ , would be

$ fg, , ..., a ; t. , ..., t ) =

Völ ’ ’ vm 1 mJ

exp [-1+exp t ... + . (19.2)

Again these represent genuine C.F.’s of distributions on [0, 00) Since ®{d-\ » • • • » 5 • • • » ^ m - 1 5 £7m _-| » _{m - 1 ’ 1}» • • • 5 _m-1-1 ) and

$(pl5 9

2 >

_t) E 1 »

the corresponding measures on [0, 00) satisfy the consistency conditions of Theorem 19.1; thus there exists a random measure with fidis determined by (19*2). Working backwards now, we can verify that (19.1) is satisfied

for any simple functions f' , ..., f , and hence the random measure does

1 m

indeed constitute the required Poisson process. //

A similar procedure could be carried out for general point processes. This would involve showing that the consistency conditions for setwise fidis imply the conditions (i) and (ii) of Theorem 19.1. But it is simpler to use

t h e s e t w i s e f i d i fo rm o f t h e e x i s t e n c e t h e o r e m d i r e c t l y . See a l s o Exam ple 2 0 . 1 . The form o f t h e t h e o r e m w h ic h we h a v e s t a t e d d o e s h a v e i t s u s e s t h o u g h , a s w i l l b e shown i n t h e n e x t s e c t i o n .

In document PASCUAL ORTIZ RUBIO, Presidente Constitucional de los Estados Unidos Mexicanos, a sus (página 42-45)