STOCHASTIC ANALYSIS
OF
MULTIVARIATE POINT PROCESSES
by
R.K. KEINE
A thesis submitted to the Australian National University
for the degree of Doctor of Philosophy
in the
Department of Statistics Research School of Social Sciences
CONTENTS
A c k n o w le d g e m e n ts v
S u m m ary v i
S y m b o ls a n d A b b r e v i a t i o n s x
1 . INTRODUCTION TO MULTIVARIATE PO IN T PROCESSES
1 . 1 I n t r o d u c t i o n 1
1 . 2 D e f i n i t i o n o f a M u l t i v a r i a t e P o i n t P r o c e s s 2 1 . 3 N o t a t i o n , D e f i n i t i o n s , a n d P r o p e r t i e s 7 1 . 4 Som e E x a m p le s o f B i v a r i a t e P o i n t P r o c e s s e s 12
2 . SIMPLE PROOFS OF SOME THEOREMS ON MULTIVARIATE PO IN T PROCESSES
2 . 1 I n t r o d u c t i o n 16
2 . 2 I n t e n s i t i e s , P a r a m e t e r s , a n d C-roup S i z e s 17
2 . 3 H i g h e r - o r d e r M o m en ts 2 2
2 . 4 S t a t i o n a r i t y 2 3
3 . PALM FUNCTIONS FOR MULTIVARIATE POINT PROCESSES
3 . 1 I n t r o d u c t i o n 2 6
3 . 2 D e f i n i t i o n a n d E x i s t e n c e : P a lm F u n c t i o n s 2 7 3 . 3 D e f i n i t i o n a n d P r o p e r t i e s : P a lm M e a s u r e s 32
3 . 4 P a l m - K h i n c h i n F o r m u l a e 3 8
3 . 5 P t i r t h e r P a l m - K h i n c h i n R e l a t i o n s a n d a n E x a m p le 42 3 . 6 C o n n e c t i o n w i t h P a lm F u n c t i o n s f o r N o n - o r d e r l y U n i v a r i a t e
P o i n t P r o c e s s e s 45
4 . THE PROBABILITY GENERATING FUNCTIONAL FOR MULTIVARIATE
PO IN T PROCESSES
4 . 1 I n t r o d u c t i o n 4 8
4 . 2 D e f i n i t i o n a n d P r e l i m i n a r y R e s u l t s 4 8
4 . 3 A C h a r a c t e r i z a t i o n T he orem 52
4 . 4 C o n v e r g e n c e T h e o r e m s 55
4 . 5 I n d e p e n d e n c e 58
5 . IN F IN IT E L Y D IV IS IB L E MULTIVARIATE PO IN T PROCESSES
5 . 1 I n t r o d u c t i o n 6 l
5 . 2 D e f i n i t i o n s a n d C a n o n i c a l F o rm s 6 2
5 . 3 Som e P r o p e r t i e s a n d E x a m p le s 7 0
5 . 4 Two B i v a r i a t e P o i s s o n C l u s t e r P r o c e s s e s 7 4 5 . 5 I n f . D iv . M u l t i v a r i a t e P o i s s o n P r o c e s s e s 78 5 . 6 Som e P r o p e r t i e s o f I n f . D i v . B i v a r i a t e P o i s s o n P r o c e s s e s 82 5 . 7 C o n v e r g e n c e o f I n f . D i v . M u l t i v a r i a t e P o i n t P r o c e s s e s 9 0
6 . SUPERPOSITIONS OF MULTIVARIATE POINT PROCESSES
6 . 1 I n t r o d u c t i o n 9 4
6 . 2 L i m i t T h e o r e m s f o r S u p e r p o s i t i o n s 95
6 . 3 F u r t h e r S u p e r p o s i t i o n R e s u l t s 99
6 . 4 E x t e n s i o n s t o P o i n t P r o c e s s e s i n R 1 0 3
iv
7 . I DEN T I F I ABILITY FOR RANDOM TRANSLATIONS OF POISSON PROCESSES
7 . 1 I n t r o d u c t i o n 107
7 . 2 The J o i n t D i s t r i b u t i o n s 109
7 . 3 I d e n t i f i a b i l i t y 1 1 1
A p p e n d ix 117
ACKNOWLEDGEMENTS
I w ish t o th a n k t h e A u s t r a l i a n Government f o r t h e i r g e n e ro u s s u p p o r t
u n d e r th e Commonwealth S c h o la r s h i p and F e llo w s h ip P la n and th e s t a f f and s t u d e n t s o f b o th d e p a rtm e n ts o f s t a t i s t i c s i n t h e A u s t r a l i a n N a ti o n a l U n iv e r s it y f o r h a v in g c o n t r i b u t e d t o m aking niy s t a y i n C a n b e rra so p l e a s a n t and re w a r d in g .
S p e c i a l th a n k s a r e due t o my s u p e r v i s o r P r o f e s s o r P .A .P . H oran f o r h i s g e n t l e h e lp and en co u ra g em en t a t a l l s ta g e s o f my work and f o r h i s
s u g g e s tin g th e p ro b le m o f i d e n t i f i a b i l i t y f o r th e s e r v i c e - t i m e d i s t r i b u t i o n o f M/G/«>. I am v e ry g r a t e f u l t o P r o f e s s o r D. V e re -J o n e s f o r s t i m u l a t i n g my i n t e r e s t in p o i n t p r o c e s s e s and in p a r t i c u l a r f o r h i s h e lp w ith some
o f th e m a t e r i a l o f C h a p te r 2 . To Dr M. W e s tc o tt, f o r h i s w i l l i n g n e s s
t o l i s t e n , d is c u s s and c r i t i c i z e , an d f o r h i s c o o p e r a tio n w ith th e m a t e r i a l o f t h e A ppendix and o u r j o i n t p a p e r on G a u s s -P o is s o n p r o c e s s e s , I e x p r e s s my s i n c e r e g r a t i t u d e .
I am in d e b te d t o P r o f e s s o r H.O. L a n c a s te r and Dr G.K. E a g le s o n o f th e U n iv e r s it y o f Sydney f o r an e n jo y a b le and f r u i t f u l t e r m 's s t a y w hich c o n t r i b u t e d much t o my u n d e r s ta n d in g o f b i v a r i a t e d i s t r i b u t i o n s .
To Dr P . M andl, I s h a l l alw ay s be g r a t e f u l f o r a h e l p f u l i n t r o d u c t i o n t o th e b e a u t i e s o f th e German la n g u a g e .
F i n a l l y , I w ould l i k e t o th a n k Mrs B. C ra n s to n f o r h e r c h e e r f u l and e x c e l l e n t ty p in g o f th e f i n a l v e r s i o n o f t h i s t h e s i s .
Sm.ftfi.ARY
This thesis is concerned with multivariate point processes in R
1
•
For the purposes of this general survey a multivariate point process may
be thought of as a series of events of finitely many distinguishable
types happening in time.
Chapter One defines a multivariate point process and shows that such
a process is uniquely specified once a consistent set of finite-dimensional
distributions is given. This result is essentially known. Notions such
as those of independence, superposition, moment measures, stationari ty,
intensities, parameters, orderliness, fixed atoms, convergence in distribution,
triangular array, and complete randomness are then defined. Most of these
are fairly straightforward extensions of the definitions for univariate
point processes.
Finally, we present some examples of multivariate point
processes and define, in particular, what we mean by
aPoisson process.
The next chapter is based on ideas in Milne (
1971)
but the results
are presented here for bivariate point processes in
R
1
instead of for
univariate point processes in Rn
asin that paper. The basic result,
well-known in the univariate case, is an extension to non-orderly processes
of Korolyuk's theorem connecting the intensity and the parameter. We
give related results for higher-order moments and some stationarity results
which are used later. Our methods are extensions of techniQues of
Leadbetter
(1968)
and are capable of further extension e.g. to processes
in Rn
(Milne,
1971).
In Chapter Three we study extensions to multivariate point processes
of the Palm functions introduced for univariate point processes by Palm
(1943)
and Khinchin
(1955).
These functions are of interest in their own
right
aswell as being useful in later discussion of superposition results.
-It is shown that the usual subaddi ti
vity and convexity methods appear to be
vii
inadequate for a full treatment of Palm functions in the multivariate case but that we can proceed using extensions of recent techniques of Belyaev
(1968, 1970) and Leadbetter (1970). Next, we derive some generalizations
of the sc-called Palm-Khinchin formulae for univariate point processes. Examples of bivariate Palm functions are exhibited for the randomly
translated Poisson process and given an intuitive interpretation. Finally,
the representation due to Fieger (196h) for the probabilities of a general
non-orderly, stationary, univariate point process is derived from an extension of our representation in terms of multivariate Palm functions
for the probabilities of a stationary, strongly orderly, multivariate point process.
The fourth chapter introduces probability generating functionals for
multivariate point processes. These are our main tool in later discussion
of infinite divisibility and superpositions. Most of the results are
extensions of previous work for univariate point processes (Moyal, 1962; Vere-Jones, 1968, 1970; Westcott, 19715) but we pay special attention to
the complications arising from fixed atoms. An example is given to
illustrate the complications which arise from such fixed atoms when
convergence in distribution is discussed. The concept of independence
for multivariate point processes is considered in relation to the probability generating functional.
Infinitely divisible multivariate point processes are introduced in Chapter Five which outlines some results about their finite-dimensional distributions and gives a constructive derivation of the canonical form
of the probability generating functional of such a process. Multivariate
Poisson cluster processes are considered and the randomly translated
Poisson process looked at from this point of view. We then investigate
viii
recent work of Newman (1970), and Milne and Westcott (1972) on
Gauss-Poisson processes. Finally, some results on convergence of infinitely
divisible multivariate point processes are derived.
This last result is applied in Chapter Six in discussing the convergence
of the 'row sums’ of a triangular array of multivariate point processes
to a multivariate Poisson process. We first consider convergence to a
general infinitely divisible multivariate process and then specialize our result to the case of convergence to an infinitely divisible multivariate
Poisson process with independent marginals. Also, in this case the conditions
for convergence are rephrased in terms of multivariate Palm functions using the results of Chapter Three and connections made with the previous work
by Khinchin (1955)» Ososkov (195^), Grigelionis (1963) on univariate
point processes and by £inlar (l9oö) on multivariate point processes.
It is shown that a superposition theorem of Vere-Jones (1968) is an
interesting special case of the result of Grigelionis (1963) and hence that
the conditions of the former theorem may be made necessary as well as
sufficient. Lastly, as a diversion to illustrate a direct approach to
superposition problems, we improve slightly a theorem of Goldman (l967fc)
about convergence to a stationary univariate Poisson process in Rn .
The final chapter returns to the oft-recurring randomly translated Poisson process to discuss a special identifiability problem viz, how much information a complete input-output record contains about the
displacement distribution. The result for Poisson processes in R^ is
contained in Milne (1970) and this chapter shows how, with minor modifications to the argument, the result may be extended to Poisson processes in Rn
i.e. we consider a Poisson process in Rn randomly displaced by a bivariate
distribution. It is shown that, from a complete input-output record, the
displacement distribution is identifiable with probability one. This
ix
theorem using some results which are derived about the form of some
joint distributions. The connection of this identifiability result
with recent work of Brown (1970) is also discussed.
The interdependence among chapters is summarized in the following diagram.
4
i
2
<
--- & 2
SYMBOLS M D ABBREVIATIONS
We c o l l e c t h e r e f o r r e f e r e n c e some g e n e r a l sy m b o ls a n d a b b r e v i a t i o n s u s e d i n t h e t h e s i s . M ost n o t a t i o n , e s p e c i a l l y t h a t p e c u l i a r t o t h i s t h e s i s i s d e f i n e d a s n e e d e d .
cl • 0 • a lm o s t e v e ry w h e re
a . s • a lm o s t s u r e l y
A+t ( x + t : x£A} w h e re A i s a s u b s e t o f R A+t
% { x + t : XEA} " "
" " " " R 2
% % a,
B c f - a l g e b r a o f B o r e l s e t s o f R
B
n
II 11 SI tl 11
c a r d { *} c a r d i n a l i t y o f t h e s e t {•}
cov(X ,Y ) c o v a r i a n c e o f X a n d Y
+
Dt r i g h t - h a n d d e r i v a t i v e wo t
£ { • } e x p e c t a t i o n wo t h e a p p r o p r i a t e p r o b a b i l i t y m e a s u re e .
^ 5
u n i t v e c t o r i n Zm ( f o r a p p r o p r i a t e in) w i t h a ’ 1* i n t h e j t h p o s i t i o n
m - v a r i a t e m u l t i v a r i a t e w i t h m com p o n en ts
P p r o b a b i l i t y m e a s u re on t h e a p p r o p r i a t e s p a c e
p . g . f l " g e n e r a t i n g f u n c t i o n a l
p . g . f n " Ti f u n c t i o n
R r e a l l i n e
RS s - f o l d C a r t e s i a n p r o d u c t o f P.
W O ,
w . r . t .* w i t h r e s p e c t t o
w . & . g . w i t h o u t l o s s o f g e n e r a l i t y Z s e t o f n o n - n e g a t i v e i n t e g e r s
Z " " n e g a t i v e "
z o ” " p o s i t i v e ”
z s
s - f o l d C a r t e s i a n p r o d u c t o f Z ( s e Z )x i
Z
zs
1
a,
AxA( - )
|A |
[x ]
• / • / k e t c
*
L . H . S .
R . H . S .
u = 0 ( v )
u = o ( v )
m e a s u r a b l e
c o u n t a b l e C a r t e s i a n p r o d u c t o f Z
g
s e t o f e l e m e n t s o f Z d i f f e r e n t f r o m 0
''U
v e c t o r i n Z r ' L ( f o r a p p r o p r i a t e m) w h o s e e l e m e n t s a r e
s y m m e t r i c d i f f e r e n c e o p e r a t o r f o r s e t s
i n d i c a t o r f u n c t i o n o f t h e s e t A
L e b e s g u e m e a s u r e o f t h e s e t A
g r e a t e s t i n t e g e r l e s s t h a n x
q u e u e i n g n o t a t i o n - K e n d a l l ( 1 9 6 4 ) .
c o n v o l u t i o n
e n d o f p r o o f
l e f t - h a n d s i d e
r i g h t - h a n d >r
— r e m a i n s b o u n d e d a s t h e p a r a m e t e r t e n d s
n p m m i i M
E 8 -m e a s u ra b le u n le s s s t a t e d o th e rw is e
a l l * 1 5 s .
/
it ii iiT h e o r e m s a n d lem m as a r e n u m b e r e d c o n s e c u t i v e l y w i t h i n e a c h c h a p t e r i . e . T h e o r e m x . y i s t h e y t h t h e o r e m i n C h a p t e r x .
1. INTRODUCTION TO MULTIVARIATE POINT PROCESSES
1 .1 I n t r o d u c t i o n
A lth o u g h t h i s t h e s i s i s c o n c e rn e d w ith m u l t i v a r i a t e p o i n t p r o c e s s e s ,
i t seem s w o rth w h ile t o b e g in w ith a b r i e f h i s t o r i c a l s u rv e y o f p o i n t p r o c e s s e s . I t w o u ld , o f c o u r s e , b e im p o s s ib le t o c i t e a l l r e l e v a n t r e f e r e n c e s h e r e b u t we t r y t o c o v e r th o s e w h ich seem th e m ost im p o r ta n t and th o s e w hich a r e e i t h e r r e p r e s e n t a t i v e o r c o n t a in many more r e f e r e n c e s . Where r e l e v a n t , f u r t h e r m e n tio n i s made o f t h e c o n t e n t o f t h e s e p a p e r s i n t h e a p p r o p r i a t e p a r t ( s ) o f t h i s t h e s i s . I t i s h o p e d , a t l e a s t , t h a t c r e d i t h as b een g iv e n w h e re v e r i t i s due.
The g e n e r a l th e o r y o f p o i n t p r o c e s s e s h a d i t s b e g in n in g s i n th e s tu d y o f n u c le o n c a s c a d e s (B h ab h a, 1950; R a m a k rish n a n , 1950) and p o p u la tio n p r o c e s s e s ( K e n d a ll 19 ^9 ; B a r t l e t t , 195*+; M oyal, 1962; H a r r i s , 1 9 6 3 ). The s tu d y o f s p e c i a l p o i n t p r o c e s s e s v i z , P o is s o n p r o c e s s e s and re n e w a l p r o c e s s e s h a d begun much e a r l i e r b u t th e fo rm e r w ere s t i l l b e i n g i n t e n s i v e l y s t u d i e d
i n e a s t e r n E urope in th e e a r l y ’ f i f t i e s ( e . g . R e n y i, 1951; P r e k o p a , 1952; I ty ll- N a r d z e w s k i, 19 5 3 , 195*0. M ean w h ile, th e work o f Palm (l9*+3) was g e n e r a l i z e d and r i g o r i z e u by Wold (l9*+ 9), and K h in c h in (1 9 5 5 ). A sp e cts o f t h i s work w ere c o n tin u e d i>n K h in c h in ( 1956a,,b) and F ie g e r (196*+) ( p r o c e s s e s w ith o u t a f t e r - e f f e c t ) , and O soskov ( 1 9 5 6 ) , G r i g e l i o n i s ( 1 9 6 3 ) , and B e ly a e v
(1 9 6 3 ) ( l i m i t th e o re m s ) . I n t e r e s t i n th e g e n e r a l th e o r y o f p o i n t p r o c e s s e s becam e more w id e s p r e a d . F o r t e t ( e . g . 1 9 6 8 ), McFadden ( 1 9 6 2 ) , S liv n y a k
(1 96 2, 1 9 6 6 ), K e r s ta n and M atth es e t a l . ( e . g . 196*+a, 1 9 6 5a) fo llo w e d by
B e u t l e r and Leneman ( 1 9 6 6 ) , Cox an d Lewis ( 1 9 6 6 ) , D aley ( 1 9 7 1 ) , Gnedenko and K ovalenko ( 1 9 6 6 ) , Goldman ( 1 9 6 7 a ,b ) , L e a d b e tt e r (1966, 1 9 6 8 ), Lee ( 1 9 6 8 ) ,
Lewis (196**, 1969 ), V e re -J o n e s ( 1 96 8, 1 9 7 0 ), and W e s tc o tt ( 1 9 7 1 a ,b )
c o n c e rn e d th e m s e lv e s w ith v a r io u s a s p e c t s o f p o i n t p r o c e s s e s i n R^. C u r r e n tly
§1 . 2 2
f a s h i o n a b l e a r e g e n e r a l i z a t i o n s and e x te n s i o n s t o p o i n t p r o c e s s e s and random m e a su re s i n v e ry g e n e r a l s p a c e s e . g . Agnew ( 1 9 6 8 ) , Meeke ( 1 9 6 7 ) , K aw ro tzk i
( 1 9 6 8 ) , and T o r t r a t ( 1 9 6 9 ) .
M u l t i v a r i a t e p o i n t p r o c e s s e s h a v e , u n t i l v e ry r e c e n t l y , rem a in ed
r e l a t i v e l y u n e x p lo re d th o u g h some r e s u l t s can be dedu ced from t h e e a r l y w ork o f K e r s ta n and M atth es e t a l . ( e . g . 1 9 6 4 a, 1965a) on m arked p o i n t
p r o c e s s e s . £ i n l a r ( 1 9 6 8 ) , and G r i g e l i o n i s (1 970) h av e s t u d i e d l i m i t th e o re m s f o r s u p e r p o s i t i o n s o f m u l t i v a r i a t e p o i n t p r o c e s s e s . C in l a r and Agnew (1968) h av e c o n s id e r e d s u p e r p o s i t i o n s o f two p o i n t p r o c e s s e s , t h e i r r e s u l t s b e in g e s s e n t i a l l y a b o u t dependence b etw ee n th e two p r o c e s s e s . A more co m p reh en siv e s tu d y o f m u l t i v a r i a t e p o i n t p r o c e s s e s h a s j u s t b een p ro d u c e d by Cox and
Lewis (3.970). The s m a ll am ount o f o v e r la p b etw ee n t h e i r work and o u rs w i l l b e p o i n t e d o u t as i t o c c u r s . Our r e s u l t s , on t h e o t h e r h a n d , e n a b le us t o
an sw er s e v e r a l q u e s tio n s w h ich Cox and Lewis l e f t op en . So f a r as we a r e aw are t h e r e i s o n ly a h a n d f u l o f o t h e r p a p e r s t r e a t i n g some s p e c i a l m u l t i v a r i a t e p o i n t p r o c e s s e s and t h e s e seem m ost a p p r o p r i a t e l y r e f e r r e d t o l a t e r .
We have p u r p o s e ly a v o id e d any d i s c u s s i o n o f th e v e ry w ide f i e l d o f a p p l i c a t i o n s o r o f s t a t i s t i c a l a n a l y s i s o f m u l t i v a r i a t e p o i n t p r o c e s s e s .
We r e f e r t o Cox and Lewis (197 0 ) who t r e a t b o th o f t h e s e t o p i c s .
The re m a in d e r o f t h i s c h a p t e r c o l l e c t s th o s e b i t s o f n o t a t i o n and
th o s e d e f i n i t i o n s , p r o p e r t i e s , a n d exam ples w h ich b e c a u s e o f t h e i r a p p e a ra n c e i n s e v e r a l o f th e s u c c e e d in g c h a p te r s seem a p p r o p r i a t e l y b r o u g h t t o g e t h e r . We hope t h a t th e n o t a t i o n u s e d i s n o t t o o f o r m id a b le . T h ro u g h o u t th e
t h e s i s we h av e t r i e d t o be c o n s i s t e n t th o u g h i n some s e c t i o n s t h i s may w e l l b e a t th e e x p en se o f a s im p l e r n o t a t i o n . A g l o s s a r y o f sym bols and
a b b r e v i a t i o n s h as b een p r o v id e d f o r c o n v e n ie n c e . 1 .2 D e f i n i t i o n o f a M u l t i v a r i a t e P o i n t P r o c e s s
§1 . 2 3
s e t K = { l , 2 , . . . , m } ( f o r some f i x e d p o s i t i v e i n t e g e r m) i n d i c a t i n g t o w h ich o f t h e m component p r o c e s s e s t h a t p o i n t b e l o n g s . A f o r m a l s t r u c t u r e
can b e s e t up much as f o r u n i v a r i a t e p o i n t p r o c e s s e s .
Let 0^ b e t h e c l a s s o f a l l s e q u e n c e s { [ t ^ , k ^ ] : i e 1} (where I i s any s u b s e t o f c o n s e c u t i v e e l e m e n t s o f Z \J Z” ) o f e l e m e n t s from R x K f o r w hich t . < t . _ f o r e a c h i and f o r w hich t h e s e t {t .} h as no
l — l + l l
f i n i t e l i m i t p o i n t s . The e l e m e n t s to, o f w i l l be t h e p o s s i b l e r e a l i z a t i o n s o f o u r m - v a r i a t e p o i n t p r o c e s s . The p o i n t s t ^ w i l l be
r e f e r r e d t o as t h e l o c a t i o n s or tim e s o f o c c u r r e n c e o f p o i n t s i n o u r p r o c e s s .
, 4 * ( ? ) / \ ■A k ( p ) / \
F o r a p a r t i c u l a r w, we d e f i n e c o u n tin g m&aiuA&S N ( . ) , N ( • )
U £ K) by
( Z )
Nv (A) = c a r d {i : [t.^ jk ^ ] e A x {£}}
= t h e number o f t ^ £ A w h ich h av e l a b e l £ ,
N ^ ^ (a) = c a r d { t^ : [t_^,k^] £ A x {£}}
= t h e number o f d i s t i n c t t £ A w h ich have l a b e l jj, ,
++
f o r A £ B and w r i t e
N(A) = ( N ^ ( A ) , N ^ ( A ) , . . . ,W(m )(A)) and
N(A) = (N( 1 ) (A ), N( 2 ) (A) , . . . ,K(in)( A ) ) .
F o r o b v io u s r e a s o n s we s h a l l r e f e r t o t h e o p e r a t i o n d e n o te d h e r e by as CO-ttap^Zug. We n o t e t h a t £T ' ( A) e t c . s h o u l d r e a l l y b e w r i t t e n
N '(to;A) e t c . , b u t t h a t t h e w i s , as u s u a l , s u p p r e s s e d . The a s s u m p tio n
t h a t ( t ^ } h a s no f i n i t e l i m i t p o i n t s j u s t e n s u r e s t h a t (A) < oo f o r eac h £ £ K and a l l bo u n d ed s e t s A e 8 . I t can b e shown as an e x t e n s i o n
+
By a c o u n t i n g m easu re we mean a c o u n t a b l y a d d i t i v e s e t f u n c t i o n d e f i n e d f o r s e t s o f a c e r t a i n c l a s s and t a k i n g v a l u e s 0 , 1 , 2
• W *
§1 . 2
h
o f t h e r e s u l t o f M oyal (1962) f o r u n i v a r i a t e p o i n t p r o c e s s e s ( s e e a l s o H a r r i s , 1968) t h a t t h e r e i s a o n e-o n e c o rre s p o n d e n c e b etw een £1. and th e s e t o f a l l v e c t o r c o u n tin g m e asu re s N whose m com ponents a re a - f i n i t e
c o u n tin g m e asu re s on
B.
H e n c e fo rth , we i d e n t i f y th e two s e t s and in f u t u r e u s e th e same sym bol t o d e n o te an e le m e n t o f £1 and th e c o r r e s p o n d in gc o u n tin g m easu re e x c e p t w here i t seems more c o n v e n ie n t t o s e p a r a t e t h e tw o.
The s e n s e w i l l b e o b v io u s from th e c o n t e x t .
We d e f in e a ö - a l g e b r a F as t h a t g e n e r a te d by th e
(lijtindah.
6dtt>
K
i n
&
i . e . s e t s o f th e form K(N(Ai ) =
k± ; i = 1 , 2 , . . . ,s }
w here s i s a n o n - n e g a tiv e i n t e g e r , and f o r e a c h i = l , 2 , . . . , s k. £ Zm
and A. £ B.
1
We now d e f in e an m-vaAicutd
p a in t
ptioceAA by a t r i p l e ( £ y , F , P) w here P i s a p r o b a b i l i t y m easu re on F , A th e o re m w hich i s an e x te n s i o no f K olm ogorov’ s th e o re m f o r o r d in a r y s t o c h a s t i c p r o c e s s e s a s s e r t s t h a t su ch a P i s u n iq u e ly d e te r m in e d by a s e t o f
{^rUX.e.-clbnMib'LOncLt I fiZcU)
cUA&hibtubLonA ( p r o b a b i l i t i e s a s s ig n e d t o t h e c y l i n d e r s e t s ) w hich s a t i s f y , i n a d d i t i o n t o th e u s u a l c o n s is te n c y c o n d i t i o n s , some c o n d i tio n s w hich r e f l e c t t h e c o u n tin g m easure p r o p e r t i e s o f F o r u n i v a r i a t e p o i n t p r o c e s s e s t h i s
r e s u l t i s due t o H a r r i s (1 9 6 3 , C h a p te r 3) when th e t o t a l num ber o f p o i n t s i s f i n i t e w ith p r o b a b i l i t y one, and in d e p e n d e n tly t o Moyal (1 96 2 ) and
N a w ro tz k i (1 9 6 2 ) i n t h e C J - f in it e c a s e . H a r r i s (1 9 6 8 , Theorem 6 . 1 ; 19 7 1 , Theorem 2 .3 ) c o n s id e r e d p o i n t p r o c e s s e s i n a c o m p le te s e p a r a b l e m e tr ic s p a c e . M oyal*s f o r m u l a tio n i n f a c t c o v e re d p o i n t p r o c e s s e s i n a r b i t r a i y s p a c e s . The f o ll o w i n g th e o re m i s e s s e n t i a l l y c o n ta in e d i n t h e s e r e s u l t s .
Theorem 1 .1 W ith e a c h s e t o f f u n c tio n s
{p(A1 ,A2 , . . . ,Ag ; r l 9r 2 >. . . , r g ) : s e Z , r . e zm, A. e B, i = l , 2 , . . . , s }
§1.2 5
p(A 1 , . . . , A s ; ^ 1 # . . . *£s ) 1 0 ;
p(A 1 , . . . , A s ; = p(A. , . . . , A . ; rJ. . , « « « . )
. - 'VI - ’ 'V l
1 s 1 s
p(A1 U . . . U Ag ; r ) = E
r . + , . j: = r
a , l o-s a ,
p(A, )• • • jA } )
* 1 9 9 S 5 'Vl5 % s
I
where t h e A ^'s a r e d i s j o i n t ;
(2. 1)
f o r any p e r m u t a t i o n , . . . , i g ) o f ( l , 2 , . . . , s ) ;
P ^A1 ’ ” ‘ ,As ,A; ‘ - ’o s , p ) = P ^A1 ... As ; b V ’& j ;
(2.2)
( 2 . 3 )
(2.1)
l i m p(A^.;^) = 1 i f {A^.} i s a se q u e n c e o f b o unded s e t s from
k - K o
8 such t h a t Ak + 0 as k •+• c» ( 2 . 5 )
we c a n a s s o c i a t e a u n iq u e p r o b a b i l i t y m easu re P on t h e a - a l g e b r a Fr f o r which
P { N ( A j = k. ; i = l , 2 , . . . , s } = p( i ^ . A g , . . . fAg ;
v ^ r ^ , . . . ,
1^) .
We rem ark t h a t , s i n c e t h e c l a s s o f a l l b o u n d ed h a l f - o p e n i n t e r v a l s w i t h r a t i o n a l e n d p o i n t s g e n e r a t e s t h e a - a l g e b r a , ß , o f B o r e l s e t s o f R
(Kingman and T a y l o r , 1 9 6 6 , p . HU) , i t i s s u f f i c i e n t t o r e s t r i c t a t t e n t i o n i n Theorem 1 . 1 t o A^’ s b e l o n g i n g t o t h i s c l a s s . H a r r i s ( 1 9 6 3 , p . 53-5*0 and N a w ro ts k i ( 1962 ) have a l s o shown ( i n t h e u n i v a r i a t e c a s e th o u g h t h e e x t e n s i o n i s o b v io u s ) t h a t , w i t h a m o d i f i e d c o n d i t i o n ( 2 . U ) , we may t a k e t h e s e t s A^ t o b e d i s j o i n t .
I t s h o u l d b e m e n tio n e d t h a t t h e f o r m a l s t r u c t u r e we h av e o u t l i n e d f o r
m u l t i v a r i a t e p o i n t p r o c e s s e s h a s much i n common w i t h t h e c o n c e p t o f a
§1.2
6
restrictions which will be imposed later) any number of points with the
same location in any of the component processes. It is true that, by taking
a more complex mark space, the marked point processes of Matthes could
cope with this situation. However, we prefer our present formulation
principally because it lends easily to our using the multivariate probability generating functional (Chapter 4).
In this section we have considered multivariate point processes in
which are the main concern of this thesis. There is, however, no extra
2
difficulty involved in treating multivariate point processes in R i.e.
whose components are univariate point processes in R (cf. Goldman, 1967a).
We only venture into such processes towards the end of the thesis. Thus,
unless otherwise stated, it will be assumed that we are dealing with
processes evolving on the real line. If it is not obvious from the context
we will usually specify whether the point processes under discussion are
univariate or multivariate. In the univariate case we drop the subscripts
and talk of (^,
F)
instead of ,F
). We point out that our terminology,whilst agreeing with that of Cox and Lewis (1970), is at variance with that
adopted by Cinlar (1968) who uses the adjective ’multidimensional’ for
point processes we prefer to call ’multivariate’. We think multidimensional is better reserved to describe point processes in Rn (n > l). Our
preference is also more in keeping with the convention suggested by Bartlett (1966, p.13) for ordinary stochastic processes.
In general we follow a policy of stating and proving results for bivariate point processes unless the m-variate result involves little
extra effort or notation. In most situations where only the bivariate
§ 1 .3
7
1 .3 N o t a t i o n , D e f i n i t i o n s . a n d P r o p e r t i e s
We s h a l l u s e N o r th e c o r r e s p o n d in g m easu re P t o d en o te an. a r b i t r a r y
( l )
m - v a r i a t e p o i n t p r o c e s s . We c a l l th e u n i v a r i a t e p o i n t p r o c e s s e s N , Nv , . . . , N V ' th e
componante, mcuigincUt p/ioc&AAeA,
o r j u s t th emaAgüiat*
o f t h e m - v a r i a te p o i n t p r o c e s s N. F o r P{N(a) = we w r i t e p ( k ; A) w here k e zm and A
e B.
I f A = [ 0 , t ) we w r i t e N (a) as N [ 0 ,t )>\, 'Xj
an d p(j£; A) as p ( k ; t ) .
The b a s i c p r o b a b i l i s t i c n o tio n o f in d e p e n d e n c e c a r r i e s o v e r t o
m u l t i v a r i a t e p o i n t p r o c e s s e s i n a n a t u r a l w ay. A c o l l e c t i o n o f m u l t i v a r i a t e
p o i n t p r o c e s s e s i s s a i d t o be i.nde.pe.nde.nt i f f f o r e v e ry f i n i t e s u b c o l l e c t i o n
( s a y )
S
e F! * p2 * • • • x V = n e F .} ( 3 .1 )
1=1
w here F^ £ , th e G - f i e l d g e n e r a te d by th e c y l i n d e r s e t s i n ,
vi vi
and K. = { 1 , 2 , . . . ,m.} w ith m. < 00 f o r e a c h i .
l i i
We r e c a l l th e d e f i n i t i o n o f th e -6u.peApo4
^ööion
o f th e u n i v a r i a t e p o i n t p r o c e s s e s N ^jN ^, . . . ,Ng . T h e ir s u p e r p o s i t i o n i s t h e p o i n t p r o c e s s whoset t s
r e a l i z a t i o n i s
\J
w hen, f o r eac h i , ok i s th e r e a l i z a t i o n c o r r e s p o n d in g t o IM ( c f . M atth e s , 1963b; Goldman, 1 9 6 7 a ). F o r o b v io u s r e a s o n s we w r i t eg
t h i s s u p e r p o s i t i o n as E . _ n N. . The s u p e r p o s i t i o n o f th e m - v a r ia te p o i n t
1—1 i
p r o c e s s e s N ^ j N ^ , . . . , ^ can th e n be d e f in e d com ponentw ise by t a k i n g th e s u p e r p o s i t i o n o f t h e c o r r e s p o n d in g com ponents o f th e g iv e n p r o c e s s . We w r i t e t h i s s u p e r p o s i t i o n as £j__p When th e m - v a r i a te p o i n t p r o c e s s e s
§ 1 .3 8
A s p e c i a l s u p e r p o s i t i o n p r o c e s s a s s o c i a t e d w i t h an m - v a r i a t e p o i n t
p r o c e s s ^ i s t h a t o b t a i n e d by s u p e r p o s i n g a l l i t s com ponents. Vie c a l l
m
t h i s t h e
MipeJipoAed pA.0C£6
-6 and d e n o te i t by N. Thus we h a v e N = NMoment meaduAeA ( c f . Moyal, 1 9 6 2 ; V e r e - J o n e s , 1968; f o r u n i v a r i a t e
p o i n t p r o c e s s e s ) may be d e f i n e d i n t h e o b v io u s way. S p e c i f i c a l l y , f o r t h e f i r s t moment m e asu re s and t h e f i r s t cross-m om ent m easu res (w hich i s a l l ve s h a l l n e e d i n t h i s t h e s i s ) we w r i t e
M ^ ( A ) = E N ^ ( A ) ( 3 . 2 )
M ^ * ^ (A x B) = E 1 ^ ( A) N ^ ( B ) ( 3 . 3 )
w h ere i , j
e K
and A, B e B. We a l s o d e f i n eM(A) =
E
n(A) (= £ MU '( A ) ) (3.U )i = l
f o r A £ ß. V7e s h a l l l a t e r , e s p e c i a l l y i n C h a p te r
b, assume
M t o b e a B o r e l m easu re ( i . e . MCA) < 00 f o r compact s e t s A £ B; s e e H alm o s, 1950,( k )
p . 2 2 3 ) . T h is i s c l e a r l y e q u i v a l e n t t o r e q u i r i n g t h a t M " b e a B o r e l
y \ / v i ) / \
m easu re f o r eac h k e K and i m p l i e s t h a t N(A) = L. W (A) < 00 a . s .
l — J .
a p r o p e r t y w hich h as a l r e a d y f o l l o w e d from t h e a s s u m p tio n ( § 1 . 2 ) t h a t { t^ } h a d no f i n i t e l i m i t p o i n t s .
An m - v a r i a t e p o i n t p r o c e s s J7 w i l l b e c a l l e d
ptk ohdex AtcutionaAy
(p e ZQ) + i f
,r .i,
p{N(Ai + t ) = k ^ ; i = l , 2 , . . . , p } = P{w( A^) = k ^ ; i = 1 , 2 , . . . ,p} ( 3 . 5 )
f o r a l l r e a l t and k^ £ Z^, A^ £ B i = l , 2 , . . . , p . We s t r e s s t h e
d i f f e r e n c e b e tw e e n t h i s s t a t i o n a r i t y and t h a t d e f i n e d by Cox and Lewis ( 1 9 7 0 ) . We f e e l t h e p r e s e n t d e f i n i t i o n t o be more n a t u r a l i n t h a t f i r s t - o r d e r
s t a t i o n a r y i s a l l t h a t i s r e q u i r e d i n ( i ) , ( i i ) , and ( i i i ) b elo w and i n C h a p te r 2 w h e re a s f o l l o w i n g Cox and Lewis t h e a p p a r e n t l y s t r o n g e r a s s u m p tio n
t f t
§ 1 .3 9
o f s im p le s t a t i o n a r i t y w ould be n e e d e d . H ow ever, i t m ust b e a d m itte d t h a t
i n th e e a r l y p a r t o f C h a p te r 3, f o r d i s c u s s i n g Palm f u n c ti o n s , we r e q u i r e t h i r d - o r d e r s t a t i o n a r i t y w h ereas w ith Cox an d L ew is’s d e f i n i t i o n se c o n d - o r d e r s t a t i o n a r i t y i s s u f f i c i e n t .
An m - v a r i a te p o i n t p r o c e s s w hich i s p t h o r d e r s t a t i o n a r y f o r a l l p e w i l l be c a l l e d
^V
iL
qXZ
ljAtcutionaAy.
T h is means j u s t t h a t a l l th e f i d id i s t r i b u t i o n s o f N a r e i n v a r i a n t u n d e r t r a n s l a t i o n . Where t h e r e i s no a m b ig u ity we dro p th e a d j e c t i v e and t a l k j u s t o f s t a t i o n a r i t y . We rem ark t h a t t h i s i s a r a t h e r s t r o n g c o n d i tio n - a j o i n t s t a t i o n a r i t y c o n d i tio n on th e m a r g in a l p r o c e s s e s - and o f c o u rs e im p lie s t h a t th e m a r g in a l p r o c e s s e s a r e e a c h s t a t i o n a r y i n t h e u s u a l s e n s e .
I t i s o b v io u s t h a t p th o r d e r s t a t i o n a r i t y im p lie s q th o r d e r s t a t i o n a r i t y f o r q p b u t i t a p p e a rs t o b e an open q u e s ti o n as t o w h e th e r f o r eac h p t h e r e a r e m - v a r i a te p o i n t p r o c e s s e s w hich a r e p th o r d e r s t a t i o n a r y b u t
n o t s t r i c t l y s t a t i o n a r y o r ev en ( p + l ) t h o r d e r s t a t i o n a r y . (H ow ever, s e e § 5 .6 .) F o r a f i r s t - o r d e r s t a t i o n a r y u n i v a r i a t e p o i n t p r o c e s s we can d e f i n e :
( 3 .6 ) ( i )
Into.nA'Ltitu
d 9 d-j^ * ^ 2 ’ * ■byu = M [ 0 ,l ) f ii a
[ 0 ,1 ) (k =
w here c l e a r l y y :
=
V
S in c e M(A)a s t a n d a r d r e s u l t t h a t
M(A) = y | A| , ii \ | A | (k
w here ] A| d e n o te s th e L ebesgue m easu re o f A £ ß. ( i i )
VcUwmU<M
A, A( l ) , A( 2 ) , . . . ,A(m) byA = lim t “ 1 P { N [ 0 ,t) > 0 } , A ^ = lim t “ 1 P { N ^ [ 0 , t ) > 0} ( 3 . 7 )
tto
t + 0(k = 1 , 2 , . . . , m) . The e x i s t e n c e o f t h e s e l i m i t s ( f i n i t e o r i n f i n i t e ) fo llo w s sim p ly as f o r
t
§ 1 . 3 10
any f i r s t - o r d e r s t a t i o n a r y u n i v a r i a t e p o i n t p r o c e s s ( K h i n c h i n , 1955 , §7; s e e a l s o L e a d b e t t e r , 1968; and § 2 .2 o f t h i s t h e s i s ) . I n g e n e r a l X <_ ^Av
/ . \
We r e f e r t o X as t h e t o t a l p a r a m e t e r and t h e XK as m a r g i n a l p a r a m e t e r s , ( i i i ) Ch.o&k-poJi(m\<L£(ltii> X ^ * ^ ( i 4 j , i , j = l , 2 , . . . , m ) by
X^1,,3) = li m t “ 1 P { N ^ [ 0 , t ) > 0 , N ^ [ 0 , t ) > 0> ( i 4 j , i , j e K ) . ( 3 . 8 ) t * 0
The e x i s t e n c e o f su ch c r o s s - p a r a m e t e r s f o l l o w s e a s i l y from ( i i ) when X < °°. C l e a r l y X^ <_ X" + X'J i n g e n e r a l . F o r b i v a r i a t e p o i n t p r o c e s s e s when X i s f i n i t e we have X^*^ = X ^^ + X ^^ - X. We rem ark t h a t i f we assume M(. ) i s a B o r e l m easure i t i s im m ed iate t h a t su ch i n t e n s i t i e s
and p a r a m e t e r s must b e f i n i t e .
A f i r s t - o r d e r s t a t i o n a r y m - v a r i a t e p o i n t p r o c e s s we c a l l mcUigMlCLlZy j
0/ide.SiMj i f i t s m a r g i n a l s a r e o r d e r l y i n t h e u s u a l s e n s e i . e . i f
P { N ^ ^ [ 0 , t ) > l ) = o ( t ) as t + 0 (k = 1 , 2 , . . . , m) ( 3 . 9 )
( K h i n c h i n , 195 5 , p . 1 2 ) . I t i s c a l l e d A& iong£y oft-dunJU) i f
p { N [ C ,t) > 1} = o ( t ) as t \ 0 ( 3 . 1 0 )
m ( V ^
i . e . i f t h e s u p e r p o s e d p r o c e s s N = Z _ N' i s o r d e r l y i n t h e u s u a l s e n s e . C l e a r l y , s t r o n g o r d e r l i n e s s i m p l i e s m a r g i n a l o r d e r l i n e s s . F or a s t r o n g l y o r d e r l y m - v a r i a t e p o i n t p r o c e s s i t i s im m e d ia te ly o b v io u s t h a t x ( 1 >i) = 0 f o r a l l i , j e K.
S in c e t h e i n t e n s i t y i s g r e a t e r th a n o r e q u a l t o t h e p a r a m e t e r f o r any
f i r s t - o r d e r s t a t i o n a r y u n i v a r i a t e p o i n t p r o c e s s we have
m m / . x
y = I u. > Z XU j > X ( 3 . 1 1 )
i = l 1 i = l
f o r any f i r s t - o r d e r s t a t i o n a r y m - v a r i a t e p o i n t p r o c e s s . C l e a r l y u n d e r m a r g i n a l o r d e r l i n e s s
m m / . V
y = Z y. = Z XU ;
i = l 1 i = l
§1.3 11
s t r o n g o r d e r l i n e s s a f u r t h e r a p p l i c a t i o n o f t h i s th e o re m ( t o t h e f i r s t - o r d e r s t a t i o n a r y p r o c e s s N) y i e l d s y = X and e q u a l i t y th r o u g h o u t ( 3 . 1 l ) . When y < 00 th e c o n v e rs e s h o ld ( c f . Z i t e k , 1957» f o r u n i v a r i a t e p o i n t
p r o c e s s e s ) . F o r f u r t h e r r e l a t e d d is c u s s i o n s e e C h a p te r 2 .
F or a u n i v a r i a t e p o i n t p r o c e s s N we c a l l x a y-ixcd cctom o f N i f
P{N ({x}) > 0} > 0 . (3.1 2)
The s e t o f su ch p o i n t s i s a t m ost c o u n t a b le , w ith f i n i t e e x p e c te d number in a b o un ded i n t e r v a l when th e same i s t r u e o f N, and i s em pty f o r f i r s t -o r d e r s t a t i -o n a r y p -o i n t p r -o c e s s e s ( P y ll- N a r d z e w s k i, 1961; M a tth e s , 1 9 6 3 b ). K h in c h in (1 9 5 6 a) ( s e e a l s o fty’l l - N a r d z e w s k i, 1953) h a s shown t h a t th e f i x e d
atoms o f a p o i n t p r o c e s s a r e atoms o f i t s f i r s t moment m easure when t h i s i s a B o r e l m easu re and v i c e - v e r s a . By a f i x e d atom o f a m u l t i v a r i a t e p o i n t p r o c e s s we u n d e r s ta n d a p o i n t w hich i s a f i x e d atom o f any o f th e com p o nen ts.
We say t h a t a se q u e n c e o f m - v a r i a te p o i n t p r o c e s s e s {N^} conv&SiggA i n d i b i A i b u i i o n t o an m - v a r i a te p o i n t p r o c e s s N and w r i t e ^ ( o r P -> P) i f th o s e f i d i d i s t r i b u t i o n s o f N f o r w hich th e B o r e l s e t s
n %n
in v o lv e d hav e no f i x e d atoms o f N on t h e i r b o u n d a r ie s ( t h i s r e q u ir e m e n t
n
-i s c o n s -id e r e d f u r t h e r -i n C h a p te r U ) a l l co n v erg e t o th e c o r r e s p o n d in g f i d i d i s t r i b u t i o n s o f N i . e . i f
%
P{N ( A. ) = k . s i = 1 , 2 , . . . , s } P{N(A. ) = k . ; i = 1 , 2 , . . . , s } ( 3 .1 3 )
r\,n 1 o-i « 1 01
f o r a l l s e Z w here f o r e a c h i = l , 2 , . . . , s , k. e Z and A. i s a
o - i 1
B o r e l s e t w ith no f i x e d atoms o f N on i t s b o u n d a ry . I n an a p p e n d ix t o o
t h i s t h e s i s we o u t l i n e a p r o o f o f a r e s u l t w hich a p p e a rs t o have b een
assum ed i n th e l i t e r a t u r e v i z , t h a t th e co n v e rg e n c e o f ( 3 .1 3 ) i s e q u i v a l e n t t o th e same ty p e o f co n v e rg e n c e w ith th e A^’ s i n t e r v a l s i n s t e a d o f g e n e r a l B o r e l s e t s .
12
§ 1 . U
p o i n t p r o c e s s e s as a d o u b le se q u e n c e i = l , 2 , . . . , s n ; n = 1 , 2 , . . . . (w here s^ -*• 00 as n 00) w here th e p r o c e s s e s i n eac h row a re in d e p e n d e n t. F or e a c h n we d e f in e an m - v a r i a te p o i n t p r o c e s s as t h e s u p e r p o s i t i o n
s
o f th e p r o c e s s e s in t h e n th row o f th e t r i a n g u l a r a r r a y i . e . ^ . I n th e u s u a l w ay, hy s u p e r p o s in g m a r g i n a ls , we can a l s o d e f in e N . =
n , i
F.m and N = Zm . ( n o te a l s o K = Z?n - N . ) .
j = l n , i n j = l n n i = l n , i
A u s u a l s u p e r p o s i t i o n p ro b le m c a n th e n b e e x p r e s s e d f o r m - v a r ia te
p o i n t p r o c e s s e s as : f i n d c o n d i tio n s u n d e r w hich W co n v erg e s i n d i s t r i b u t i o n a>n
t o a s u i t a b l e l i m i t N as n •► °°. Such p ro b lem s a r e i n v e s t i g a t e d i n C h a p te r 6 . The q u e s ti o n as t o w hat p r o c e s s e s can a r i s e i n th e l i m i t u n d e r c e r t a i n c o n d i tio n s i s lo o k e d a t in C h a p te r 5.
We c a l l a m u l t i v a r i a t e p o i n t p r o c e s s
COMpl.QXe.Zy fumdom
i f t h e random v e c t o r s W(A^), N( A^ ) , . . . , ^( A^) a r e m u tu a lly in d e p e n d e n t f o r a l l f i n i t e c o l l e c t i o n s o f d i s j o i n t s e t s A^, A^ , . . . ,A^ ( c f . K ingm an, 1 9 6 7 , who was i n t e r e s t e d i n random m e asu re s w ith t h i s p r o p e r t y ; s e e a l s o K h in c h in , 1 9 5 6 a, and F i e g e r , 19 6 5 , b o th o f whom lo o k e d a t u n i v a r i a t e p o i n t p r o c e s s e s w ith t h i s p r o p e r t y w hich th e y c a l l e d ’w ith o u t a f t e r - e f f e c t ? ) . M u l t i v a r i a t e p o i n t p r o c e s s e s w ith t h i s p r o p e r t y a r e c o n s id e r e d b r i e f l y i n C h a p te r 5» I f N i s c o m p le te ly random th e n i t i s c l e a r t h a t e a c h o f i t s m a rg in a ls m ust b e a c o m p le te ly random u n i v a r i a t e p o i n t p r o c e s s b u t th e c o n v e rs e does n o t h o ld in g e n e r a l ( s e e § 5 * 6 ) .1
. h
Some Exam ples o f B i v a r i a t e P o in t P r o c e s s e sOne o f th e s i m p l e s t c l a s s e s o f b i v a r i a t e p o i n t p r o c e s s e s , and t h e one i n w hich we s h a l l be m ost i n t e r e s t e d , i s t h a t c o m p ris in g b i v a r i a t e P o is s o n p r o c e s s e s .
L et us o b s e rv e t h a t by a ( u n i v a r i a t e )
PoZiAon ptiOCQAb
N we mean a c o m p le te ly random p o i n t p r o c e s s ( i n P^) f o r w h ic h , f o r e a c h bounded§ 1 . U 13
th e f i r s t moment m easure o f th e p r o c e s s ( c f . L ee, 19 68 , who c a l l e d su ch p r o c e s s e s ’ g e n e r a l i z e d P o i s s o n ') . The P o is s o n p r o c e s s h as b een c o n s id e r e d i n v a r io u s form s by many a u th o r s e . g . R y ll-N a rd z e w sk i (1 9 5 3 , 195*0,
K h in c h in (1955» 1 9 5 6 a ,b ), F ie g e r (196*1, 1 9 6 5 ). We s h a l l b e c o n te n t t o E iention j u s t a r e s u l t o f R enyi ( 1 9 6 7 ) , t h a t th e r e q u ir e m e n t o f co m p lete random ness i s re d u n d a n t i f , i n f a c t , th e m easure X( . ) i s n o n -a to m ic and, f o r I any f i n i t e u n io n o f f i n i t e i n t e r v a l s , W( l ) i s a P o is s o n random v a r i a b l e w ith p a r a m e te r X( l ) . I n g e n e r a l th e m easu re X( . ) may have ato m s. These m ust be f i x e d atoms o f th e c o r r e s p o n d in g p r o c e s s ( R y11-W ardzew ski, 1953; K h in c h in , 1956a) and a t su ch p o i n t s m u l t i p l e o c c u r r e n c e s a r e c l e a r l y
p o s s i b l e b e c a u s e P o is s o n num bers o f e v e n ts o c c u r (K ingm an, 1 9 6 7 ). T h u s, i f a P o is s o n p r o c e s s h a s no m u l t i p l e o c c u r r e n c e s , th e n i t s f i r s t moment m easure X( . ) m ust be n o n - a to m ic . U nder t h e a ssu m p tio n o f f i r s t - o r d e r s t a t i o n a r i t y we f i n d a g a in t h a t X( . ) m ust be n o n -a to m ic and i n f a c t t h a t X( l ) = X111 f o r a l l bo u nded i n t e r v a l s I . Such a s t a t i o n a r y P o is s o n
p r o c e s s h a s no m u l t i p l e o c c u r r e n c e s and we c a l l i t a
AÄmpZc Vo^AAon pAOCCAA
w ith p a r a m e te r X. Many c h a r a c t e r i z a t i o n s o f t h i s p r o c e s s a r e known e . g .
K h in c h in (1 9 5 5 ) ; R y ll-N a rd z e w sk i (1 9 5 * 0 .
We c o n t r a s t th e sim p le P o is s o n p r o c e s s whose o n e - d im e n s io n a l d i s t r i b u t i o n s a r e a l l ( s im p le ) P o is s o n d i s t r i b u t i o n s w ith p r o c e s s e s whose o n e -d im e n s io n a l d i s t r i b u t i o n s a r e compound P o is s o n i n th e s e n s e o f F e l l e r (1 9 6 8 , p . 2 8 8 f f .) . Such p r o c e s s e s w i l l be lo o s e l y c a l l e d
compound Po-tAAon pAoccAACA
. H ow ever, we n o te t h a t th e name ’ compound P o is s o n p r o c e s s ’ i s u s u a l l y g iv e n t o as p e c i a l s u b c la s s o f t h i s c l a s s o f p r o c e s s e s ( s e e W e s t c o t t , 1 9 7 1 b ).
By a
b'LvctfU.cutc PoÄAAon pAOCCAA
we s h a l l mean a b i v a r i a t e p o i n t p r o c e s s whose m a r g in a l p r o c e s s e s a r e b o th P o is s o n p r o c e s s e s . The m ost e a s i l yP o is s o n p r o c e s s e s . H ow ever, as we s h a l l s e e in §5* 6 , an i n f i n i t e l y
d i v i s i b l e ( § 5 . 2 ) , b i v a r i a t e P o is s o n p r o c e s s can b e c o m p le te ly s p e c i f i e d by t h r e e m e asu re s w hich a r e e q u i v a l e n t t o th e f i r s t two moment m easu res and th e f i r s t cro ss-m o m en t m e a su re . I n t h i s c a s e th e l a t t e r m easu re c o m p le te ly sums up t h e dependence b etw ee n t h e two m a r g in a l p r o c e s s e s .
A s p e c i a l i n f i n i t e l y d i v i s i b l e , b i v a r i a t e P o is s o n p r o c e s s i s one whose f i r s t com ponent i s a s t a t i o n a r y P o is s o n p r o c e s s w ith p a r a m e te r A and whose se c o n d com ponent i s th e s t a t i o n a r y P o is s o n p r o c e s s d e r iv e d from
th e f i r s t by s u b j e c t i n g eac h p o i n t i n i t t o a random d is p la c e m e n t ( p o s i t i v e o r n e g a t iv e ) t h e s e d is p la c e m e n ts b e i n g in d e p e n d e n t and i d e n t i c a l l y
d i s t r i b u t e d w ith d i s t r i b u t i o n f u n c ti o n F (D oob, 1 9 5 3 , pp.*+0U-i+07). We c a l l su ch a p r o c e s s a
randomly tsianAiat&d PodAAon ptiocdAA.
I t s dependence s t r u c t u r e i s s im p le and th e p r o c e s s i s d e te r m in e d c o m p le te ly by th e p a r a m e te rA and t h e d i s t r i b u t i o n f u n c ti o n F. As we s h a l l s e e i n C h a p te r T i f F i s c o n c e n tr a te d on [ 0 , °°) t h e m a rg in a ls o f t h i s p r o c e s s a r e j u s t t h e i n p u t and o u tp u t o f an M/G/00 q u e u e in g s y ste m .
A n o th er s p e c i a l i n f i n i t e l y d i v i s i b l e , b i v a r i a t e P o is s o n p r o c e s s h as
b een c o n s id e r e d by Dwass and T e ic h e r ( 1 9 5 7 ) , and M a r s h a ll and O lk in ( 1 9 6 7 a ,b ) . I t i s s im p ly c o n s t r u c t e d as fo llo w s : ta k e t h r e e in d e p e n d e n t s t a t i o n a r y
P o is s o n p r o c e s s e s ^i »^2 ’^12 w ith r e s p e c t i v e p a r a m e te r s A^, A^, A ^ and c o n s id e r th e b i v a r i a t e ( P o is s o n ) p r o c e s s w ith com ponents
and + N £. -^Sa ^n d ep en d ence s t r u c t u r e i s s im p le b e in g c o m p le te ly s p e c i f i e d , i n f a c t , by A ^ w hich i s e a s i l y s e e n t o b e t h e c r o s s - p a r a m e t e r .
The w hole p r o c e s s i s u n iq u e ly d e te rm in e d by A^, A^, and A ^> We s h a l l c a l l th is
p r o c e s s a f e ta tio n a r y ) COMZiatiLd b iv a A ic tiz Po^U^on p r o c e s s by a n a lo g y w ith t h e c o r r e l a t e d b i v a r i a t e P o is s o n d i s t r i b u t i o n (C a m p b e ll, 193*+; Ahmed, 1961; H o lg a t e , 196*+).
I n f i n i t e l y d i v i s i b l e , b i v a r i a t e p o i n t p r o c e s s e s and i n p a r t i c u l a r
§1. u 15
Though we do n o t i n v e s t i g a t e them f u r t h e r h e r e , we m e n tio n t h r e e o th e r ( o f c o u rs e t h e r e a r e many m ore) i n t e r e s t i n g c l a s s e s o f b i v a r i a t e p o i n t p r o c e s s e s .
( i )
BtvcvUcutd SKLndLOaZ psiocdAAdt
by w hich we mean b i v a r i a t e p o i n t p r o c e s s e s whose m a rg in a ls a r e re n e w a l p r o c e s s e s e . g . t w o - s t a t e sem i-M arkov p r o c e s s e s . Cox and Lewis (1970) p o i n t o u t t h a t th e o n ly sem i-M arkov p r o c e s s e s w ith in d e p e n d e n t re n e w a l m a rg in a ls a r e th o s e whose m a rg in a ls a r e a l s oP o is s o n .
( i i )
B4.vcvU.CLtd doubUj AtockabtUc Po-c&Aon ptiocdAAdA f o r w hich, c o n d i t i o n a l
on th e v a lu e (A ( t ) , A,p(t)) o f a r e a l - v a l u e d b i v a r i a t e s t o c h a s t i c p r o c e s s ^ ( t ) , th e m a rg in a ls o f th e b i v a r i a t e p o i n t p r o c e s s a r e in d e p e n d e n t P o is s o n
p r o c e s s e s w ith r e s p e c t i v e f i r s t moment m easu res
J
A ^ ( t ) d t , i = 1 , 2 ( e . g . Cox and L e w is , 1 9 7 0 ).( i i i ) B i v a r i a t e p o i n t p r o c e s s e s a r i s i n g from a s e l e c t i v e i n t e r a c t i o n s i t u a t i o n i . e . w here an e v e n t o f one p r o c e s s i n h i b i t s t h e n e x t e v e n t o f
2 . SIMPLE PROOFS OF SOME THEOREMS ON MULTIVARIATE POINT PROCESSES 2 . 1 I n t r o d u c t i o n
T h ro u g h o u t t h i s c h a p t e r we s h a l l b e c o n c e rn e d w ith a f i r s t - o r d e r s t a t i o n a r y ( § 1 . 3 ) , b i v a r i a t e p o i n t p r o c e s s i n R^ whose m a r g in a ls may be n o n - o r d e r l y . The r e s u l t s w i l l a l l b e c a p a b le o f e x te n s i o n t o m u l t i v a r i a t e p o i n t p r o c e s s e s b u t f o r s i m p l i c i t y and c l a r i t y o f e x p r e s s io n we c o n s id e r
o n ly th e b i v a r i a t e c a s e . Some r e a s o n a b le ’ f i n i t e n e s s ' c o n d i tio n w i l l alw ays be n e c e s s a r y and we i n t r o d u c e su ch c o n d i tio n s l a t e r as r e q u i r e d .
The p ro b lem s c o n s id e r e d h e r e a r o s e o u t o f an a tte m p t t o g e n e r a l i z e ( b e f o r e i t was r e a l i z e d t h a t t h i s h a d a l r e a d y b een done) t o s t a t i o n a r y , n o n - o r d e r l y , u n i v a r i a t e p o i n t p r o c e s s e s t h e w ell-k n o w n th e o re m o f K o rolyu k
( K h in c h in , 1955 p . 1+1-2; a l s o Z i t e k , 1 9 5 7 , and. L e a d b e t t e r , 1968) w hich s t a t e s t h a t f o r s t a t i o n a r y o r d e r l y u n i v a r i a t e p o i n t p r o c e s s e s
lim t" * ^ [l - p ( 0 ; t ) ] = y ( l . l )
t-VO
w here y i s t h e i n t e n s i t y d e f in e d i n § 1 .3 . I n th e n o n - o r d e r ly c a s e th e r e s u l t a n a lo g o u s t o ( l . l ) , v i z
lim t ~ 1 [ l - p ( 0 ; t ) ] a = Aa = y ( 1 . 2 )
tIO
w here a i s t h e mean group s i z e , h as b een p ro v e d in d e p e n d e n tly by B e u t l e r and Leneman ( 1 9 6 6 ) , F ie g e r ( 1 9 6 5 ) , and S liv n y a k (1 9 6 2 , 1966) and i s i m p l i c i t i n th e work o f M atth es (1 963b) ( s e e § 2 . 4 ) . S l i v n y a k 's m ethods w ere m easu re t h e o r e t i c m aking heavy u se o f h i s fu n d a m e n ta l fo rm u la (1 9 6 2 , e q u a tio n 8; 1 9 6 6 , e q u a tio n 1 3 ) . F i e g e r , who was a l s o i n t e r e s t e d i n a n a lo g u e s o f ( l . 2 ) f o r n o n - s t a t i o n a r y p o i n t p r o c e s s e s , em ployed th e o re m s from t h e th e o r y o f t h e B u r k h i l l i n t e g r a l . The argum ents o f B e u t l e r and Leneman depended on e x te n s i o n o f th e e le m e n ta r y c o n v e x i t y - s u b a d d i t i v i t y p r o p e r t i e s n o t i c e d by K h in c h in (1 9 5 5 ). A ls o , from th e r e s u l t s o f B e u t l e r and Leneman i t can be shown f o r p o i n t p r o c e s s e s i n R"*- t h a t th e l i m i t
§2 . 2 IT
l i m t 1 E { ( N [0, t ) ) 2} = Ab ( 1 . 3 )
t! 0
w h e re b i s th e mean s q u a re g ro u p s i z e , w i t h s i m i l a r r e s u l t s f o r th e
h i g h e r - o r d e r m om ents. S im p le p r o o f s o f t h e s e r e s u l t s w e re p r o v id e d i n
t h e p a p e r o f th e a u t h o r ( M i l n e , 1 9 7 1 ) u s in g e x t e n s io n s o f a t e c h n iq u e
s u g g e s te d b y L e a d b e t t e r (196 8) . The p r o o f s w e re g iv e n f o r f i r s t - o r d e r
s t a t i o n a r y u n i v a r i a t e p o i n t p r o c e s s e s i n Rn t o i l l u s t r a t e t h e a d v a n ta g e
n
o f t h e t e c h n iq u e s i . e . t h a t t h e y g e n e r a liz e d e a s i l y t o R w h e re n o n e o f
t h e a p p ro a c h e s o f p r e v io u s a u t h o r s h a d seem ed c a p a b le o f s u c h e a s y e x t e n s io n .
R e c e n t ly , L e a d b e t t e r ( 1 9 7 0 ) , e x t e n d in g id e a s s u m m a riz e d i n B e ly a e v (1 9 6 8 )
a nd e x p a n d e d i n B e ly a e v ( 1 9 7 0 ) , h a s shown t h a t f o r u n i v a r i a t e p o i n t p ro c e s s e s
i n v e r y g e n e r a l s p a c e s t h e c o r r e s p o n d in g r e s u l t s may be p r o v e d u s in g s i m i l a r
m e th o d s .
R a th e r t h a n j u s t r e p r o d u c e t h e r e s u l t s c o n t a in e d i n M iln e (1 9 7 1 ) i t
h a s seem ed u s e f u l t o p r e s e n t th e c o r r e s p o n d in g th e o re m s f o r b i v a r i a t e
p o i n t p r o c e s s e s and t o i n d i c a t e some a d d i t i o n a l r e s u l t s .
2 . 2 I n t e n s i t i e s , P a r a m e t e r s , and G roup S iz e s
21" We f i r s t a d o p t some s u i t a b l e n o t a t i o n . By R ^ , ( k , £ ) e , we
d e n o te t h e u n iv a r ia t e p o i n t p r o c e s s c o m p r is in g th o s e e p o ch s o f t h e o r i g i n a l
b i v a r i a t e p o i n t p ro c e s s a t w h ic h e x a c t l y k e v e n ts o f R ^ ^ an d £ e v e n ts
(2)
o f R o c c u r . So as t o s a v e e x c lu d in g i t a lw a y s as a tr o u b le s o m e s p e c i a l
case we make t h e c o n v e n t io n t h a t t h e p o i n t p r o c e s s R ^ h a s n o p o i n t s . L e t
R v a n d R v b e as d e f i n e d i n § 1 .2 w i t h R = Rv + R v a n d n o te t h a t
/ v OO CO . CO CO
/ M l ) P ) ,
N = E E R p a n d , IV ' = Z Z R . (2.1)
k = l £=0 K k = 0 £= 1 K36
A
We s t r e s s t h e d i f f e r e n c e b e tw e e n t h e s u p e rp o s e d p r o c e s s R as d e f in e d
above a nd t h e p r o c e s s d e r i v e d fr o m t h e s u p e rp o s e d p r o c e s s R b y t h e
c o l l a p s i n g o p e r a t io n ( i . e . f o r a b i v a r i a t e p o i n t p r o c e s s t h e o p e r a t io n s
o f c o l l a p s i n g and s u p e r p o s in g t h e m a r g in a ls do n o t c o m m u te ). T h is p ro c e s s
+
z
2=
z 2 - { 0 }§2.2
18
we choose to denote hy N and observe that
CO CO
5 = E
k=0
z
£=0
k£
(2.2)
For the most part we shall assume that E N[0,l) < <» which (because of stationarity) clearly implies that E N(A) < °o and hence that N(a) < <» a.s. for all bounded A e 8. In Theorem 2.3 the stronger assumption
E{(N(l) [0,l))a (N^[0,l))^} < co (a , ß >■ l) will be made.
Our first theorem extends the result of Theorem 1 of Leadbetter (1968). This theorem provided a neat proof of both Khinchin’s result about the
existence of a parameter for a stationary (univariate) point process and Korolyuk’s theorem.
Theorem 2.1 Assume that E S[0,l) < °o. Then the limits
lim t“1 P { N ^ [0,t) > 0} , lim t -1 P{NV '[0,t) > 0}(2) (2.3)
tio t+0
lim t”1 P { N ^ [0,t)
(o')
> 0 or ir ;[0,t) > 0} (2.4) t+0
lim t”1 P { N ^ [0,t) > 0, N ^ t O j t ) > 0} (2.5) t+0
exist and are equal to E f^^[0,l), E N ^ [ 0 , l ) , E N[0,l), and
E{N^^[0,l) + N^^[0,l) - N[0,l)} respectively. Also, for each k £ Z 2
'Xj 0
the limit
lim t-1 p(k; t) t+0 " ^
exists and is equal to E N [o, l).
kr"2
(2 .6)
Proof. Consider a subdivision of the interval [0, l) into n equal subintervals and define indicator functions by
( 1 if [-, - ^ ) = k xln>(k)
0 otherwise
for i = 0,1,...,n-1 and j = 1,2. Then .( J ) ri i+1
V E k=r
§2 . 2 19
and
n - 1
(1) (2)
N,(k) = E
( £
x - n(A)) ( E x:^ (m))
i= 0 £=k1 m=k.
in
= th e number o f s u b i n t e r v a l s c o n t a in in g a t l e a s t M e v e n ts o f ty p e i , i = 1 , 2 .
We now p ro v e t h a t as n ■+ 00 00 00
N (k ) -> £ E IJ0 [ 0 ,1 ) a . s .
n o , 7 . . . £mL
£=k m=k2
I n d e e d , s i n c e w ith p r o b a b i l i t y one t h e r e a r e o n ly f i n i t e l y many p o i n t s o f th e p r o c e s s N i n [ 0 , l ) , e a c h o f t h e s e m ust be an i s o l a t e d p o i n t and h en ce w ith p r o b a b i l i t y one t h e r e e x i s t s an n^ su ch t h a t
00 00
N (k ) = £ I N0 [ 0 , 1 )
*=k l m=k2 *“
f o r a l l n > n _ . S in c e N (k ) < N [0 , l ) f o r a l l k and n and s in c e
u n 'u — %
E l [ 0 , l ) < 00 i t fo llo w s by d o m in ated c o n v e rg e n c e t h a t as n 00
Eh (k ) + E E E N [ 0 , 1 ) .
'Xj £=k m=k2 Äm
B ut
n -1
E»n ( | ) = P{N( j ) [ i , i i i ) > k ^ ; J = 1 , 2 }
J = 1 , 2 }
i = 0 “ “ 3
= n P{N( <j ) [ 0 , ^ ) > k . ;
by s t a t i o n a r i t y . Thus we have
/ . \ CO 0 0
l i m n P { r J ' [ 0 , ~ ) > k ; j = 1 , 2 } = E E E N [0, l ) .
n-*» 3 £=k m=k2 36
Wow, from th e m o n o to n ic ity o f P { f t ^ [ 0 , t ) _> k . ; j = 1 , 2 } as a f u n c ti o n J
o f t , we can deduce i n e q u a l i t i e s a n a lo g o u s t o th o s e o f L e a d b e tt e r ( 1 9 6 8 ) , and M in e ( 1 9 7 1 ) e q u a tio n ( 2 . U) . I t fo llo w s from su ch i n e q u a l i t i e s t h a t
, , 0 0 o o
lim t ‘ P { r J ' [ 0 , t ) > k . ; j = 1 ,2 } = E E E N [ 0 , l )
t'I'O 3 £=k^ m=k2 m
2
f o r a l l k £ and h en ce t h a t
lir a t - 1 p ( k ; t ) =
