ARKIV FOR MATEMATIK Band 7 nr 16
1.67 01 1 Communicated 11 J a n u a r y 1967 by H. C R A ~ R and L. CARLESO~
Convergence and invariance questions for point systems in R1
under r a n d o m m o t i o n By
TORBJSRN THED~ENABSTRACT
I n section 2 we introduce and study the independence property for a sequence of two-dimen- sional random variables and by means of this property we define independent motion in section 3.
Section 4 is mainly a survey of known results about the convergence of the spatial distribution of the point system as the time t ~ co. I n theorem 5.1 we show t h a t the only distributions which are time-invariant under given reversible motion of non-degenerated type are the weighted Poisson ones. Lastly in section 6 we study a more general type of random motion where the position of a point after translation is a function ] of its original position and its motion ability.
We consider functions / which are monotone in the starting position. Limiting ourselves to the ease when the point system initially is weighted Poisson distributed with independent motion abilities, we prove in theorem 6.1 that this is the case also after the translations, if and only if the function f is linear in the starting position. I n the paper also some implications of our results to the theory of road traffic with free overtaking are given.
1. Introduction
I n t h e s t u d y of r o a d t r a f f i c t h e s i m p l i e s t c a s e is w h e n t h e cars c a n o v e r t a k e a n d m e e t e a c h o t h e r w i t h o u t d e l a y . T h e f o l l o w i n g s o - c a l l e d i s o v e l o x i c m o d e l f o r t r a f f i c
(see F . H a i g h t [8] p p . 1 1 4 - 1 2 3 ) h a s b e e n p r o p o s e d f o r t h i s case. 1
T h e c a r s a r e c o n s i d e r e d as p o i n t s o n a n i n f i n i t e r o a d w i t h n o i n t e r s e c t i o n s . T h e y c a n o v e r t a k e a n d m e e t e a c h o t h e r w i t h o u t d e l a y a n d t h e y will f o r e v e r m a i n t a i n t h e i r o n c e c h o s e n speeds. T h e t r a j e c t o r i e s in t h e r o a d - t i m e d i a g r a m w i l l t h u s b e lines. T h e i n i t i a l s p e e d s a r e i n d e p e n d e n t a n d i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e s a n d t h e y a r e also i n d e p e n d e n t of t h e i n i t i a l p o s i t i o n s of t h e cars.
I t h a s b e e n s h o w n t h a t u n d e r r a t h e r w e a k c o n d i t i o n s t h e s p a t i a l d i s t r i b u t i o n of t h e c a r s will t e n d t o a w e i g h t e d P o i s s o n d i s t r i b u t i o n (as d e f i n e d i n s e c t i o n 2) as t h e t i m e t ~ ~ (see ref. [1], [2], [4], [10] a n d [12]). F u r t h e r if t h e i n i t i a l p o s i t i o n s a r e w e i g h t e d P o i s s o n d i s t r i b u t e d t h e s p a t i a l d i s t r i b u t i o n a n d t h e i n d e p e n d e n c e c o n d i - t i o n s i m p o s e d a t t = 0 a r e c o n s e r v e d f o r a l l t > 0 (see c o r o l l a r y 6.1). I f n o w t h e m o d e l s h o u l d b e t i m e - i n v a r i a n t , i.e. t h e s p a t i a l d i s t r i b u t i o n a n d t h e i n d e p e n d e n c e c o n d i - t i o n s i m p o s e d a t t = 0 a r e c o n s e r v e d f o r all t > 0 i t will b e s h o w n t h a t t h e s p a t i a l d i s t r i b u t i o n m u s t b e a w e i g h t e d P o i s s o n o n e . I t h a s also b e e n p o s s i b l e t o s o m e w h a t r e l a x t h e c o n s t a n t s p e e d a s s u m p t i o n .
1 The model description is taken from T. Thed6en [11].
211
T. THEDI~EI'% Convergence and invariance questions for point systems
A f t e r some i n t r o d u c t o r y s t u d i e s in sections 2 - 3 , section 4, gives a s h o r t s u r v e y of t h e c o n v e r g e n c e of t h e p o i n t s y s t e m a n d its set of m o t i o n as t ~ c~. I n s e c t i o n 5 we shall give some r e s u l t s a b o u t t h e t i m e - i n v a r i a n c e of t h e s p a t i a l d i s t r i b u t i o n of a p o i n t s y s t e m u n d e r w h a t we will call r e v e r s i b l e i n d e p e n d e n t m o t i o n . T h e s e r e s u l t s c o n t a i n as a special case (corollary 5.1) t h e a b o v e m e n t i o n e d one a b o u t t i m e - i n v a r i a n t d i s t r i b u t i o n s in t h e c o n s t a n t s p e e d case. L a s t l y in section 6 we shall s t u d y a n o t h e r t y p e of i n d e p e n d e n t m o t i o n c o n s e r v i n g w e i g h t e d P o i s s o n d i s t r i b u t i o n s for t h e p o i n t s y s t e m .
T h e i n v e s t i g a t i o n s will be r e s t r i c t e d t o p o i n t s y s t e m s in R 1 b e c a u s e of t h e traffic- t h e o r e t i c a l b a c k g r o u n d of t h e p r o b l e m s . H o w e v e r i t is clear t h a t w i t h s o m e m o d i f i c a - t i o n s c o r r e s p o n d i n g r e s u l t s can b e p r o v e d for p o i n t s y s t e m s in Rk.
L a s t l y i t s h o u l d be r e m a r k e d t h a t t h e r e s u l t s of sections 5 a n d 6 in t h e special case of c o n s t a n t speeds were p r e s e n t e d b y t h e a u t h o r a t T h e T h i r d I n t e r n a t i o n a l S y m p o s i u m on t h e T h e o r y of Traffic F l o w in N e w Y o r k 1965 (see T. T h e d d e n [ l l ] ) .
2. Prdiminades
L e t {Zn} be a sequence of r a n d o m v a r i a b l e s (r.v.'s). F o r a n y ]3orel set B let N(B) = n u m b e r (no.) of Z n C B.
W e shall s a y t h a t {Zn} has no finite limit point if for a n y f i n i t e i n t e r v a l I t h e r . v . N ( I ) is p r o p e r , i.e.
P ( N ( I ) < c~) - 1.
L e t us a s s u m e t h a t t h i s is t h e case. T h e n we c a n a s s o c i a t e t o {Z~} a counting process N(x) 1 d e f i n e d b y
I
no. of Zn E (0, x], x > 0 N(x) = ~0, x = 0/
[ - no. of Z~ e (x, 0], x < 0.
T h e n a l m o s t s u r e l y (a.s.) t h e s a m p l e f u n c t i o n s of such a c o u n t i n g p r o c e s s a r e n o n - d e c r e a s i n g i n t e g e r - v a l u e d s t e p f u n c t i o n s w i t h i n t e g e r - v a l u e d j u m p s . L e t n o w {Zn}
be t h e sequence of p o s i t i o n s for p o i n t s in R 1. W e shall s a y t h a t one o r m o r e p o i n t s f o r m a cluster if t h e y h a v e t h e s a m e positions. T h i s in t u r n c o r r e s p o n d s t o a j u m p of t h e c o u n t i n g process N(x). T h e size of t h e cluster is e q u a l t o t h e size of t h e corre- s p o n d i n g j u m p of N(x). To t h e sequence {Zn} t h e n c o r r e s p o n d s a sequence of clusters c h a r a c t e r i z e d b y t h e i r p o s i t i o n s a n d sizes. W e can a.s. o r d e r t h e clusters a f t e r t h e i r p o s i t i o n s t h u s g e t t i n g t h e o r d e r e d c l u s t e r p o s i t i o n s
. . , <Z(-2) <ZI-~) <~ 0 <Z(~) <Z(2) < . . .
T h e size of a cluster w i t h p o s i t i o n Z (k~ will be called Nk. T h u s t o a sequence {Zn}
w i t h no finite l i m i t p o i n t c o r r e s p o n d s a c o u n t i n g process N(x) a n d a sequence {(g(k), Nk)}. The d i s t r i b u t i o n of N(x) a n d {(g(k~, N~)} is g i v e n b y t h e d i s t r i b u t i o n of (N(I,) ... N(Ik)) for a n y f i n i t e set of d i s j o i n t f i n i t e i n t e r v a l s 11 . . . . , I k (open, 1 To simplify the notation we shall use N(') in two senses where the actual meaning will be clear from the argument used. Notice e.g. the difference between N(x) and N({x}).
ARKIV FOR MATEMATIK. B d 7 n r 16
semi-closed or closed). (This can be shown by the same method as in Doob [5] p.
403.) This distribution is in turn given b y the generating function (g.f.).
k
~(s~ . . . sk; I~ . . . . , Ik) = E 1-I sH <x;)
j - 1
Now the indexing of a sequence {Z.} m a y depend on the sizes of the r.v.'s Zn. Thus two sequences {Z~} and {Z~} can have different distributions but their correspond- ing counting processes can nevertheless have the same distribution. We shall some- what inadequately characterize the distribution of {Z.} by that of N(x) (or {(Z (k), Nk)}). Let Ia ... Ik be any disjoint finite intervals. The following distributions given by their g.f.'s will be of interest in the following.
The distribution corresponding to a Poisson process
k
. . . ,
. . . 1-I { x i / , I ( s , - 1)}
. i = 1
where ]Ij] is the length of Ij and 2 a positive constant. We shall then say t h a t {Z=} is Poissou distributed (with the parameter 2).
The distribution corresponding to a weighted Poisson process
~(s~, ..., sk; I~ . . . Ik) = exp {~[Ijl(sj - I} dW(~)
where W(;t) is a distribution function (d.f.) on (0, ~ ) . We shall then say t h a t {Z~}
is weighted Poisson distributed (with the parameter d.f. W(;t)).
If {Z~} has any of these distributions then a.s. all the clusters have the size one.
I n the following ease clusters of larger sizes are possible.
The distribution corresponding to a weighted compound Poisson process
~ 9 ( 8 1 . . . S k ; I 1 . . .
Ik)
= e x p{ lI,
1 ) d W ( 2 )where a(s) is the g.f. of a positive integer-valued r.v. and W(2) a d.f. on (0, ~ ) . We shall then say t h a t {Zn} is weighted compound Poisson distributed.
Let us now consider a sequence {(Zn, Vn)} where {Z.} has no finite limit point and {Vn} is a sequence of r.v.'s. B y ordering the clusters of {Zn} by position and
, V ( k ) ~
the r.v. s V, in the clusters b y size we get the sequence {(Z ~), Nk; V1 (k), ..., Nk/~
where V(1 ~) ~< ~< V (k) "*" N k "
9 o , t p )
Dehnltlon 2.1. Let {(Z,, V,)} and {(Z~, Vn)~ be such that {Z,} and {Z~} have no /inite limit points. {(Z~, V~)} and {(Z~, Vn)} are said to have the same distribution but /or indexing i/ the associated sequences {(Z (k~, Nk; V(~ k), ..., V(k)~N~IS and {(Z '(k~, N~;
V~ '(~), ..., VN~)} have the same distributions.
213
T. T H E D ] ~ E N , Convergence and invariance questions for point systems
I n t h e following sections we shall often consider sequences of the t y p e {(Z~, Vn)}.
W e shall see t h a t t w o such sequences h a v i n g t h e s a m e d i s t r i b u t i o n b u t for indexing can replace each o t h e r in t h e p r o b l e m s to be considered w i t h o u t changing t h e results.
W i t h this r e m a r k in m i n d we shall n o w introduce t h e independence p r o p e r t y of
{(z., v.)}.
Definition 2.2. Let {(Z,, V'n)} be a sequence o/r.v.'s, where {Z,} has n o / i n i t e limit point, such that
(i) {Zn} and { V'~} are independent and
(ii) {V~} is a sequence o/independent identically distributed (i.i.d.) r . v . ' s with the common d./. F(v).
The sequence ((Z~, Vn)} has the independence property with the d./. F(v) i/
{(Z~, V~)} and {(Z~, V'~)} has the same distribution b u t / o r indexing.
F o r a n y Borel set B in R 2 let
M ( B ) = n o . of (Z~, V n ) E B
a n d denote the g.f. of (M(Bz) ... M(Bk) ) b y y~(s 1 ... sk; B1 ... Be) where B 1 .... , B k are Borel sets in R 2. T h e following l e m m a gives a n e q u i v a l e n t c h a r a c t e r i z a t i o n of t h e independence p r o p e r t y . L e t for a n y d.f. F(x)
A Borel set.
F ( A ) = fA dF(x),
L e m m a 2.1. ((Zn, V~)} where (Z~} has no ]inite limit point has the independence property with the d./. F(v) i/ and only i / / o r any disjoint finite intervals 11 ... Ik and /or any disjoint Borel sets BSl .... , Bj,j with ~Jni_ l B j , = R 1 , j = l , ..., lc the g./.
~)(811 . . . . , Sl ... Ski ... Sknk; I 1 • Bll .... , 11 • B1 ... Ik • Bkz ... Ik • Bkn~) =
- - ~ O ( P l l 8 1 1 -[- "'" + P l n l 8 1 . . . P k l S k l A7 "'" + p k n k S k n k ; I 1 , " " , Ik) (2.1) where
Pjv = F(Bjv), r = 1 ... nj, j = 1 ... k
Proo/. Necessity. T h e sequence {(Z~, V~)} of definition 2.2 d e t e r m i n e s g.f.'s ~o a n d ~0 which fulfil (2.1). F u r t h e r {(Zn, V,)} a n d {(Z,, V~)} h a v e t h e s a m e distribu- t i o n b u t for indexing a n d t h u s d e t e r m i n e equal g.f.'s ~o a n d % T h e n t h e g.f.'s given b y {(Zn, V,)} also fulfil (2.1).
Sufficiency. I n t h e proof we shall use a n idea f r o m D o o b [5] p. 403. N o w {Z~}
has no finite limit point. T h e n to the sequence ((Z~, V~)} corresponds a n o t h e r se- quence {(Z 0), Nj; V(1 j), 9 ..~ V ~ N i I J . W e h a v e to p r o v e t h a t this last sequence has t h e s a m e distribution as t h e sequence {(Z (jl, Nj; V~ (j) ... V~]))} of definition 2.2. T h i s is t h e case if a n d only if a n y finite set of r.v.'s f r o m the sequence {(Z r Nj; V(1 j) .... , V (j)~ has the s a m e distribution as t h e corresponding set f r o m t h e sequence Nits
"(D
{(ZO>, Nj; V~ ~ ... V~j )}. I t is easily seen t h a t it is no restriction t o choose t h e r . v . ' s f r o m {(Z (J), Nj; V(1 j) ... VNj)} (D w i t h consecutive indexes (j). I n order t o a v o i d n o t a - tional complications we shall here consider only positive indexes a n d choose t h e
ARKIV F6R MATEMATIK. B d 7 n r 16 i n d e x e s 1 . . . k. ( T h e case w i t h s o m e n e g a t i v e i n d e x e s c a n b e t r e a t e d i n t h e s a m e w a y . ) L e t for a n y B o r e l s e t B i n R z
Z s ( B ) = n o . of V<})EB Z; (B) -- n o . of Vs (j) E B
T h e d i s t r i b u t i o n of (Z (j), N i ; V(~ ", . . . Nj, j = I , ..., k) is d e t e r m i n e d b y t h a t of v ( , (Z (j), Ni; z j ( B t l ) . . . . , zj(Bjnj), ] = 1 . . . k) for all d i s j o i n t B o r e l sets B m, ] = 1 . . . k, U~La B i ~ = R 1 . T h u s we h a v e t o p r o v e t h a t (Z (j), N i ; z~(Bil), ..., z~(Bj,j), j = l . . . k) h a s t h e s a m e d i s t r i b u t i o n as (Z (j), N f z ; ( B j l ) , ..., z~(Bjn~), i = I , ..., k). L e t n o w 0 = z 0 < z 1 < . . . < z k a n d l e t atv, v = 1, ..., hi, ~ = 1, ..., k b e n o n - n e g a t i v e i n t e g e r s w i t h
~,%
~ j ~ = n j , j = l ,...,
k. P u tp(~)- l~
- -n/ ~P~~ Pm
~jn Ii = 1 0~i1! . . . 0~1n1.
I t is e a s i l y s e e n f r o m t h e d e f i n i t i o n 2.2 t h a t P(ZIJ)<~z i, N j = n i , z ~ ( B i . ) = a j . , v = 1 ... nt, j = 1 ... k) = P ( Z O) <~zs, N j = n j , j = 1 ... k).p(~).
T h e s u f f i c i e n c y of (2.I) is p r o v e d if
P = P ( Z (j) <~z i, N j =ni, zj(Bj~), v = 1 ... nj, ~ = 1 ... k)
= P ( Z (j) ~< zj, N i = hi, ~ = 1 . . . . , k).p(oO (2.2) L e t u s for j = 1 . . . k d i v i d e t h e i n t e r v a l (zi_1, zi) i n t o n i n t e r v a l s of e q u a l l e n g t h ],,=(a~, by), v = n ( j - 1 ) + l . . . n j , w h e r e t h e i n t e r v a l s a r e n u m b e r e d f r o m left t o r i g h t . N o t e t h a t
~ a , , I~r~l < ~ / . ,
(2.3)
l ~ v < ~ k n
P u t f u r t h e r
A = {zI(Bj,) = ai~, v = 1 . . . nj, j = 1, ..., k}
T h e n a p p r o x i m a t i n g P b y t h e p r o b a b i l i t y i n t h e case w h e n n o Z(J)'s fall i n t h e s a m e I v we g e t
[ P - ~ P ( Z ~ N j = hi, j = 1, ..., k, Z(*+a'E 'I,k;
A)]
V a < . , . < v I r vi<~.~n, J = 1 . . . /c
k k n
< ~ ~ P(Z<J)EG,
Z(J+I)~L)
) = 1 v = l
F r o m (2.3) a n d t h e f a c t t h a t 0 < Z (11 < . . . < Z (k+l) we g e t
k n k
Z ~
P ( Z " ) E Iv,Z,,+l, E
Iv)~< ~ P(I Z<~+I) - Z<t, [ <
z k / n ) -+ O, n--+1 = 1 v : 1 3 ~ 1
T h u s
P = l i m
n ---~ o o
P(Z(J~E/~j, N i = n j , j = l
.... ,k; Z(k+l)E'Ivk; A )
~ l < . . . ( Vk vi<~tn, 1= 1 . . . k
a n d i n t h e s a m e w a y we f i n d t h a t
(2.4)
(2.5)
2 1 5
T. THEDI~EN, Convergence and invariance questions f o r p o i n t systems P ( Z m <~ z~, N~ = n~, j = 1 , . . . , k)
Z E Iv~ (2.6)
= l i r a ~ P ( Z ( ~ ) E L r 1 6 2 1 6 2 . . . . ,k; (k+l) ,
n - - - > ~ r l < . . . < ~ / z
v ] ~ n , j = l , . . . , l~
W e shall now e s t i m a t e the s u m m a n d s of (2.5) a n d (2.6). P u t b~~ = 0 . L e t us a p p r o x i - m a t e
! .
P ( Z (j) E Lj, N j = n j , j = 1 ... k; Z (k+~) E I ~ , A) b y
P ( N ( ( b , j _ , a~j]) = 0, M ( I ~ j • Big) = o~j/~, i z = 1 . . . . , nj, j = 1 . . . k) (2.7) F r o m (2.1) we get
P(N((b~j_,, a~j]) = 0, M ( L j • Bst,) ~jz, # = 1 . . . . , nj, j = 1 . . . . , k) =
P(N((b~j_,, a~s]) = 0, N ( I ~ j ) = n j , j = 1 ... k).p(~)
(2.s)
Combining (2.7), (2.8) with (2.5) we getP = p(e) lira ~ P(N((b~j_,, a,j]) = 0, 1V(l~) = nj, j = 1 . . . k) (2.9)
n - - > r v l < , . . < ~ k
vi<~in, j = l ... k
in the s a m e w a y as we got (2.4).
L e t us f u r t h e r a p p r o x i m a t e
P ( Z I J ) E L j , N j = n j , j - 1 . . . k; Z(k+~)E "I~) b y
P(N((b~j_,, a~j]) = 0, N ( / , ~ ) = n s , j = 1 . . . k) (2.10) Using (2.4) we get f r o m (2.10) a n d (2.6) t h a t
P ( Z r <~ zj, N j = ns, j = 1 . . . . , k )
= lira ~ P ( N ( ( b ~ j _ , a~j]) = O, N(/~j) = nj, j = 1, ..., k)
vj<~]n,j~l . . . k
(2.11) (2.9) a n d (2.11) p r o v e s (2.2) a n d t h u s t h e sufficiency of (2.1) is shown.
A w e a k e r t y p e of independence p r o p e r t y is given b y t h e following.
DeTinition 2.3. A sequence {(Z~, V~)} where {Z~} has no finite l i m i t p o i n t has the w e a k independence property w i t h the d./. F ( v ) i f / o r a n y Borel sets B 1 .. . . . B k and a n y d i s j o i n t f i n i t e intervals 11, ..., Ik
~o(Sll, sl~ .... , s~l, skz; 11 • B1, 11 • 91 ... I~ • B~, I~ • Bk)
= q)(plSn § ..., pkSkl § I1 . . . . , Ik) where p j = F ( B j ) , 9j = 1 - pj, ] = 1 . . . k.
T h e relation b e t w e e n t h e independence p r o p e r t y a n d t h e w e a k independence p r o p e r t y is given b y t h e following
ARKIV FOR MATEMATIK. B d 7 n r 1 6 L e m m a 2.2.
Let
{(Z~, V~)}be a sequence with {Zn} having no finite limit point. Then
(i)i/
{(Z~, V=)}has the independence property with the d./. F(v) it also has the weak independence property with the d./. F(v)
(ii)
i/ no two el the Z~'s are equal with positive probability the weak independence property with the d./. F(v) implies the independence property with the d./. F(v).
Proo/.
(i) follows at once from l e m m a 2.1. L e t n o w {(Zn,Vn)}
have t h e weak inde- pendence p r o p e r t y with t h e d.f.F(v)
a n d suppose t h a t no two of t h e Z~'s are equal with positive probability. T h e nP ( N k = l , k - + l , + 2 .... ) = 1
a n d using t h e same technique as in the sufficiency p a r t of l e m m a 2.1, (ii) can be proved.
D e n o t e b y ~v(sl, ..., sk; B~ ... Bk) the g.f. of
N(B~) ... N(Bk),
where B1, ..., B k are Borel sets in R 1.The case when {Z~} is weighted Poisson distributed a n d
{(Zn, Vn)}
has the inde- pendence p r o p e r t y will be of particular interest in t h e sequel. L e m m a 2.4 will give a characterization of this case. I n t h e proof of t h a t l e m m a we shall need the following L e m m a 2.3.Let
{Z~}be weighted Poisson distributed with the parameter d./. W(;t).
Let/urther B 1
...B~ be disjoint Borel sets in R 1 with/inite Lebesgue measures
#(B1),.... iz(B~). Then
9)(81 . . . 8 k ; B 1 . . .
B k ) = I ~ e x p t , ~ # ( B j ) ( s j - l , t d W ( X ,
d O [. j = l J
Remark.
This simple l e m m a m a y be f o u n d in the literature b u t since t h e same technique will be used in t h e proof of l e m m a 2.4 t h e proof will be given below.Proo/.
L e t us first consider t h e case k = 1. L e t B be a Borel set w i t h / ~ ( B ) < ~ . W e shall prove t h a tN ( B )
has the g.f.~v(s; B) = qp(s; B) = f o exp (~#(B) (s - 1)}
dW(~).
(2.12) (2.12) is easily seen t o hold for B = (.J ~ j=l I j where Ij, ] = 1, 2 ... are disjoint inter- vals with ~;~ # ( I j ) < ~ . L e t ~t be t h e class of all intervals (open, semielosed a n d closed). W e k n o w t h a t/~(B) = inf
i z ( I j ) ; B = U I j , I j f ' l I k = r IjE~t,
/ = 1 , 2 . . .1 j=l
T h u s given a n y (~ > 0 there is a sequence of disjoint intervals {Ij} such t h a t
B = UIj, Bo = U I . i - B
where # ( B ~ ) < 5 .j - 1 j - 1
16:3 2 1 7
T. THEDEEN,
Convergence and invariance questions for point systems
I n t h e s a m e w a y we see t h a t t h e r e is a sequence of disjoint intervals {I~} such t h a t B ~ c U I ; ; # ( B ~ ) ~ < / ~ I < 2 8 .
J=l
N o t e t h a t
= fo~ (1 -
e-~)dW(2) H(x)
do
is a continuous d.f. with H ( 0 ) = 0 , since
W(O)=O.
T h u s given e > 0 t h e r e is a 8 > 0 such t h a tH(x)<e,
0-~<x<28. Choose 8 in accordance to this r e q u i r e m e n t . N o wB u t for 0 ~<s~< 1
~ ( s ; j ~ I s ) - q ~ v ( s ; B) I
= e x p 2 j ~ # ( I j ) (s - 1) dW(2) - ~ e x p {2#(B) ( s - 1)}
dWQ.)
I ~ I~ - exp { - , ~ ( B ~ ) } IdW(~) = H(~(B~)) < ~~<
d o
F u r t h e r for 0 ~< s ~< 1
~(s; B) -- q) (s; 5=I lJ) I = ,EsN(~) -- EsZC(B)+N(B~) ,
<EII-sN~I<P _~ I
> 0 < H ( 2 8 ) < e JT h u s I~0(s; B ) - ~ p ( s ; B)I < 2 e , 0 ~ < s ~ l which p r o v e s the ease k = l . I n t h e general case we use the s a m e a p p r o x i m a t i o n procedure for each of t h e Borel sets B 1 .... , Be.
L e m m a 2.4.
The sequence
{(Zn, Vn)}has the independence property with the d./. F(v)
a n d {Zn}is weighted Poisson distributed with the parameter d./. W(~) i / a n d only if /or any disjoint Borel sets B 1 ... B k in R 2 such that
o~ where n(Bi) = f dxdF(v), i = 1 ... k
<
3~
ithe r.v.
(M(B1) ...M(Bk) ) has the g./.
~(sl . . . sk; B1, -.., Be) = e x p u(B,) (s t - 1) d g ( 2 ) . (2.13)
Remark.
I n the sufficiency p a r t we need only (2.13) to hold for t h e BTs being p r o d u c t s of intervals.ARKIV FOR MATEMATIK. Bd 7 nr 16 Proo/. Sufficiency. P u t B j = I j x R 1 where the I j ' s are disjoint finite intervals.
Then it follows at once from (2.13) t h a t {Z~} is weighted Poisson distributed with the p a r a m e t e r d.f. W(~). Taking the B / s as products of intervals it follows from definition 2.3 t h a t {(Z~, V~)} has the weak independence p r o p e r t y with the d.f. F(v).
Since {Z~} is weighted Poisson distributed no two of the Z . ' s are equM with positive probability. The independence p r o p e r t y then follows from l e m m a 2.2 (ii).
Necessity.
1. F r o m l e m m a 2.3 it follows t h a t (2.13) holds for B~=Ai x R1, i = 1 ... k, where the A / s are disjoint Borel sets in R 1 with)u(Ai)< 0% i = 1 ... k.
2. The independence p r o p e r t y implies (2.13) to hold for disjoint products of Borel sets of finite u-measure.
3. (2.13) is easily seen to hold aiso for disjoint sets of finite u-measure in the algebra generated by all finite unions of measurable rectangles.
4. L e t B be a Borel set in R 2 with finite n-measure. I n a similar w a y as t h a t used in the proof of lemma 2.3 we can a p p r o x i m a t e this set in u-measure b y a union of disjoint products of measurable rectangles and prove (2.13) for k = 1.
5. The proof of (2.13) for a n y k is done in the same way.
3. The point system and its set o f motion
We shall consider a countable n u m b e r of points distributed on R 1 and performing a random motion in time. The positions of the points at t = 0 are given b y the se- quence of r.v.'s {Xn} the points being arbitrarily enumerated. I n the following we shall always assume t h a t {Xn} has no finite limit point. I f the position of point n a t t (t >0) is denoted b y X~(t) the positions at t (t > 0) are given b y the sequence of r.v.'s {Xn(t)}. Using the notation
y ~ ( t ) = x ~ ( t ) - x ~
we shall call { Y~(t)}, following J. Goldman [7], the set o/motion for the point, system.
The special case when for all t > 0.
Y~(t) = Un.t
will be called the constant speed ease (in this case the trajectories will be straight lines). We shall here deal with the case when the points do not interact with each other in their motions and we will thus introduce the following definitions.
Definition 3.1. {X~(t)} has (or {Y~(t)} is) an independent set o/ motion at t with the d.]. Ft(y) i/ {(X~, Y~(t))} has the independence property with the d.]. Ft(y).
Definition 3.2. {X~(t)} has (or {Yn(t)} is) an independent set o/ motion with the /amily { F t(y ) } i / i t has an independent set o/motion at twith the d./. F t(y) /or all t > 0 . I n the constant speed case we replace the family {Ft(y)} in this definition b y the d.f. of the speed G(u).
4. The asymptotic distribution o f the point system
We shall here study the a s y m p t o t i c distribution of {Xn(t)} when {Yn(t)} is an independent set of motion for all t > 0 . L e t {X ~ } be a sequence of r.v.'s with no finite 219
T. THEDi~EN, Convergence and invariance questions for point systems
limit point. Similarly to section 2 we i n t r o d u c e the following r.v.'s a n d associated g.f.'s
N ( I ) - no. of X~ E I , g.f. ~v
N t ( I ) = no. of X~(t) E I, g.f. ~vt, N~ = no. of X ~ C I, where I is a finite interval.
Definition 4.1. {Xn(t)} is said to converge in distribution to {X ~ } i I / o r any disjoint finite intervals 11 .... , I k the distribution o/ ( N t ( I i ) , . . . ,
2Vt(Ik) )
converges to that o/(N~ ..., N~ as t ~ oo.
Such a convergence will t a k e place if a n d only if we h a v e convergence of t h e corresponding g.f.'s. The convergence problem was first studied b y R. D o b r u s h i n [4] a n d G. M a r u y a m a [10]. L a t e r similar results were obtained b y L. B r e i m a n [1]
a n d [2] a n d T. Thedden [12]. These results were s u m m a r i z e d a n d completed b y J. G o l d m a n [7]. We shall in this section s o m e w h a t generalize t h e results of Goldman.
Our t r e a t m e n t will also serve as a m o t i v a t i o n a n d i n t r o d u c t i o n to t h e invariance problems dealt with in section 5.
L e t us assume t h a t {Xn(t)} has an i n d e p e n d e n t set of m o t i o n with t h e family {Ft(y)}. I t follows f r o m definition 2.2 t h a t when we consider the distribution of Nt(Ij) , j = 1 ... k, we will get the same distribution if we replace t h e last a s s u m p t i o n b y t h e following assumptions for {(Xn, Yn(t))}
(i) {X~} a n d { Yn(t)} are i n d e p e n d e n t
(ii) {Y~(t)} is a sequence of i . i . d . r . v . ' s with P(Y~(t)<~y)=Ft(y ).
W i t h these assumptions it is easily seen t h a t
~t(sl . . . sk;I 1 .. . . , I k ) : E ' f l l ~ s , E t ( I y - X n ) q - l - ~ F t ( I , - X n , t . (4.1)
n [ j = l j = l J
I t should be noted t h a t this g.f. does n o t necessarily h a v e the value one for
81 - - . . . - - 8 k ~ l .
I n order to get a n y general results a b o u t t h e convergence of ~0 t it seems n a t u r a l to s t u d y t h e g.f.
~ 2 4 7 j = l ~ J ~ j ( I j - - X n ) }
where {xn} is an infinite sequence of real numbers. This is the g.f. of a s u m of inde- p e n d e n t r.v.'s. The equivalent in this case to t h e so called ' u a n ' - c o n d i t i o n (see Lo~ve [9] p. 290) is
lira sup F t (I - xn) = 0 (4.2)
t--~or n
for all finite intervals I , If (4.2) should hold for all sequences {Xn} t h e n we will require
lim sup F t ( I - x) = 0 (4.3)
t --1,oo XfiR1
ARKIV FOR MATEMATIK. B d 7 n r 16 for all finite intervals I . Following J. G o l d m a n [7] we shall t h e n say t h a t we h a v e a spread out set o] motion.
I n t h e s t u d y of t h e convergence of at u n d e r (4.3) we shall use a slight generaliza- tion of t h e f u n d a m e n t a l l e m m a b y J. G o l d m a n [7] p. 23. First we shall give t h e following n o t a t i o n s a n d definitions. L e t us consider a r r a y s {Xt~} of k-dimensional r.v.'s, n = l , 2, ... a n d t indexed on t h e positive integers or positive real n u m b e r s where Xtn = ~ n , ~ ~:<~) ..., :~v<k)~. The r o w sums are d e n o t e d b y
Yt ~ ~ Xtn
n = l
Definition 4.2. A n array {Xt~ } will be called a null array i/
lim sup ~,.~/vcJ)t~ > O) = 0
t--->~ j , n
Definition 4.3. A Bernoulli sequence is a sequence o/independent k-dimensional r.v.' s
X 1 = ( X ( 1 ) . o . , x(k)), X 2 = (X(9.1) . . . X (k>) .. . . assuming only the values
( 0 , 0 ... 0), (1, 0, 0, ..., 0), (0, 1,0, ..., 0) ... ( 0 , 0 ... 0, 1).
Definition 4.4. A Bernoulli array ( X t , ) is an array such that/or any t (Xt~} is a Bernoulli sequence.
Definition 4.5. A r.v. X has a k-dimensional Poisson distribution with parameter (21 .... ,2k) q
P ( X = (nl, ... , nk)) = I-[ "" e-X~
j=l n / /or all non-negative integers nj.
L e m m a 4.1. Let {Xtn} be a Bernoulli null array. Then
(i) the only possible limit distributions /or Yt are the Poisson ones (including those with some 2~ equal to zero).
(ii) the distribution o/ Yt converges to a Poisson distribution with parameter (21 ... 2k) i / a n d only i/
lim ~ :~P/V(J)tn = 1 ) = 2 i j = 1, ....9 k. (4.4)
t ----> 0r n = l
Proo/. T h e sufficiency of (4.4) was p r o v e d for k = 1 b y L. B r e i m a n [2] a n d for a n y k b y T. Thedden [12]. (ii) was p r o v e d b y J. G o l d m a n [7] p. 23. T h u s we only need to prove (i). L e t
n = l
a n d fix ?" for a while. Y~+) m a y be a n i m p r o p e r r.v. More precisely Y~+) being a s u m of i n d e p e n d e n t r.v.'s m u s t be a.s. finite or infinite. W i t h t h e n o t a t i o n
221
T. THEDEEN, Convergence and invariance questions for point systems
t h e g.f. of Y~J) is
p / v o ) = 1) -- ~(~) \ ~ t n - - F t n
fl~J) (s) = f l (1 - p ~ (1 - s)).
n = l
I n order t h a t fl~'(s) should converge to a g.f. of a p r o p e r r.v. it is necessary t h a t l i m t _ ~ ~n~ vtn~(" exists a n d is equal to a c o n s t a n t e.g. 2j. Using the s a m e a r g u m e n t for j = 1 ... k we see t h a t (4.4) (for some (~1 ... 2k)) is necessary for t h e complete convergence of t h e d.f. of Yr. T h e sufficiency of (4.4) t h e n p r o v e s (i).
Following J. G o l d m a n [7] we n o w give the following definitions.
I n the following t w o definitions let {In} be a n y sequence of finite intervals, I i c I ~ c .... lim
[Inl
=oo.n"--)'r
Definition 4.6. A point system {Xn} is well-distributed with a parameter d./. W(2) i/
lim N ( I n ) / l l n ] = A a.s.
n - - ~ o o
where A is a r.v. with d./. W(2).
Definition 4.7. A set o / m o t i o n {Yn(t)} is well-distributed i/ /or any finite 2 a n d / o r any set o / n u m b e r s {xn} such that
lim (no. of
In)/llnl
: ~n . - - ) o r
we have lim
F , r zn) = lzl
t ---~oo n
/or a n y finite interval I.
J. G o l d m a n p r o v e d using his equivalent to l e m m a 4.1 (ii) t h e following t h e o r e m (this is his t h e o r e m 6.2 where we h a v e just s o m e w h a t changed t h e formulation).
: T h e o r e m 4.1. Let {Yn(t)} be a spread out set o/motion. Then a necessary and su//i- cient condition /or every initially well-distributed point system under an independent set o / m o t i o n / o r all time u > 0 to be in the limit t ~ ~ weighted Poisson distributed is that { Yn(t)} is well-distributed.
Remark. T h e case w h e n t h e weighted Poisson distribution has a p a r a m e t e r d.f.
W(2) w i t h W(0) > 0 is n o t excluded. Using l e m m a 4.1 (i) a n d (ii) we get T h e o r e m 4.2. Let {Yn(t)} be a spread out set o/motion.
T h e n a necessarg condition/or every initially well-distribvted point s~stem under an independent set o / m o t i o n / o r all time u > 0 to converge in distribution as t--> oo is that {Yn(t)} is well-distributed.
T h e proo] is omitted.
I n t h e following e x a m p l e t h e set of m o t i o n is well-distributed a n d s p r e a d out.
L e t Yn(t)= Un't (the c o n s t a n t speed case) a n d let U n h a v e a n a b s o l u t e l y conti-
ARKIV FOR MATEMATIK~ Bd 7 nr 16 nuous d.f.
G(u)
with the densityg(u).
Assumeg(u)
bounded, almost everywhere continuous and with compact support. (Cf. L. Breiman [2] and T. Thedden [12].) Let now {X~} be an initially well-distributed point system with the parameter d.f. W(2) and let {Y~(t)} be the independent set of motion above. Then b y theorem 4.1 we have convergence to a weighted Poisson distribution with the parameter d.f. W(2). Further it can be shown t h a t {(X~(t),Un)}
has the independence property with the d.f.G(u)
in the limit t ~ o~.More precisely: Let for a n y Borel set B~ in R2, i = 1 ... k, M(B~)=no. of
(Xn(t), V~) E B,
with the g.f. ~ ( s 1 ... sk; B1 ... B~).
Definition 4.8. {(X~(t), Un)}
has asymptotically the independence property with the d./. G(u) and
{Xn(t))is asymptotically weighted Poisson distributed with the parameter d./. W(,~) i / / o r any disjoint/inite intervals
I x . . . I kand intervals
J l . . .Jk
lim ~V~(Sn,
Sx2 ... ski, sk2; 11 • J1, 11 • ]1 ... Ik• Ik • ]k)
= f / exp {~ ~ ]L](G(J~)sn + G(J~)s~- l)} dW(~)
Remark.
Compare with definition 2.3, lemma 2.2 and lemma 2.4 and the remark following that lemma.B y the same method of proof as that used by L. Breiman [2] we get
Theorem 4.3.
Let
{Xn}be a well-distributed point system with the parameter d.].
W(,~) and let { Yn(t)= Unt} be an independent set o/motion /or all t
> 0(the constant speed case). The d./. o] U~, G(u), is absolutely continuous with the density g(u) being bounded, almost everywhere continuous and with compact support.
Then
{(X~(t), U~)}has as /mptoticall~ t@ independence property with the d./. G(u) and
{Xn(t)}is as]mptotically weighted Poisson distributed with the parameter d./.
W(~).
Remark.
I t should be possible to weaken the conditions ong(u).
This we have not done since the theorem is included mainly as a motivation for the condition about reversible independent motion used in section 5.5. Time-invariant distributions for point systems under reversible independent motion
The important role of weighted Poisson distributions as limit distributions for point systems with independent sets of motion stands out clearly from section 4.
Doob [5] p. 404 showed that if a point system is Poisson distributed at t = 0 and has an independent set of motion its spatial distribution will be conserved for all t > 0. The same result for weighted Poisson distributions was shown by J. Goldman [7]. From these results it is rather easily seen t h a t a point system with
E N ( I ) < oo
for a n y finite interval I has the same distribution for all t ~> 0 forall
independent sets of motion if and only if it is weighted Poisson distributed (see R. L. Dobrushin [4] and J. Goldman [7]). Here we shall t r y to characterize those distributions for 223T. TI:IEDs Convergence and invariance questions for point systems
point systems which are t i m e - i n v a r i a n t for a given i n d e p e n d e n t set of motion. L e t us first note t h a t in t h e special case t r e a t e d in t h e o r e m 4.3 the sequence {(X~(t), U~)}
has t h e independence p r o p e r t y in the limit t-+ ~ . This means t h a t in the limit we h a v e a kind of b a c k w a r d s i n d e p e n d e n t set of motion. I n the general case we h a v e the following definition of this concept.
Definition 5.1. {Xn(t)} has (or {Y~(t)} is) a backwards independent set ol motion at t with the d./. Ft(y) i~ {X~(t)} has no finite limit point and {(Xn(t), Yn(t))} has the independence property with the d./. Ft(y).
Definition 5.2. {X~(t)} has (or {Y~(t)} is) a backwards independent set ol motion with the /amily {Ft(y)} i / i t has a backwards independent set o/motion at t with the d. 1. Ft(y ) /or all t > O.
If in these definitions we h a v e t h e weak independence p r o p e r t y in place of the independence p r o p e r t y we shall s a y t h a t we h a v e a backwards weak independent set o/motion. W e can n o w i n t r o d u c e the concept of timc-invariance u n d e r a reversible set of motion.
Definition 5.3. {X,~} has an invariant distribution at t/or a d./. Ft(y) under reversible independent motion i/ {Xn(t)} has an independent set o/motion at t and a backwards independent set o/ motion at t both with the d./. Ft(y) and {X~} and {X~(t)} have the same distribution.
DeIinition 5.4. {Xn} has a time-invariant distribution under reversible independent motion/or the ]amily {Ft(y)} i/ the distribution o / { X ~ ) is invariant at t under reversible independent motion/or the d./. Ft(y )/or all t > O.
If in definition 5.3 we instead of a b a c k w a r d s i n d e p e n d e n t set of m o t i o n have a weak one we shall s a y t h a t {Xn} has a i n v a r i a n t distribution u n d e r weak reversible i n d e p e n d e n t motion. W e shall in this section show t h a t t h e distributions of {Xn}
which are t i m e - i n v a r i a n t u n d e r reversible i n d e p e n d e n t m o t i o n for a n o n - d e g e n e r a t e d family {Ft(y)} (see definition 5.5) are t h e weighted Poisson ones. W e shall need the following two simple lemmas a b o u t g.f.'s for r a n d o m vectors. L e t Y = ( Y1, -.., Yk) a n d Z = (Z 1 ... Zk) be t w o r a n d o m vectors t h e c o m p o n e n t s of which are n o n - n e g a t i v e a n d integervalued a n d let their g.f.'s be yl(sl ... sk) a n d y2(sl, ..., sk) respectively.
Lemma 5.1. I~
~21(81 . . . . , 8k) = y 2 ( 8 1 , -.., 8k)
/or s~E(a~, b~) where - 1 ~<a~<b~<l, i = l ... k, then Y and Z have the same distribu- tion.
T h e proo/ follows at once from t h e generalization of t h e i d e n t i t y principle for holomorphic functions of a complex variable to t h e case with several complex variables.
Lemma 5.2. I / Y~ <~Z~, i = 1 . . . k a.s. then
y,(sl ... sk) ~>y~(s, ... sk), s~ E [0, 1], i = 1 ... k.
ARKIV FSR MATEMITIK. Bd 7 nr 16
Proo/.
T h e given inequalities i m p l y t h a tsY1
*'" ~176176. s zk .
a.s. for s~E[0, 1], i = 1 ,. . . .
, kf r o m which inequality t h e l e m m a follows at once.
L e t for a n y Borel set B in R~
Mf.t(B)
= n o . of(X~, Y~(t))EB, M b . t ( B ) = n o .
of(X~(t), Y~(t))EB
a n d denote t h e corresponding g.f.'s b y ~0f. t a n d YJb.t respectively (cf. section 2). L e t f u r t h e r for a n y d.f.
F(y)
a n d a Borel set B in R 1 withF ( B ) > 0
FB(y) -- F ( B N ( - 0% y])/F(B)
be t h e conditional d.f. given B.L e m m a 5.3.
Let the distribution o / { X n } be invariant at t under reversible independent motion/or a d./. Ft(y ). Then it is also invariant at t under weak reversible independent motion/or the d./. Ft. B(Y).
Remark.
A c t u a l l y t h e l e m m a is t r u e also when weak is o m i t t e d b u t this we do n o t need in t h e following.Proo/.
P u tF t ( B ) - p ,
q = l - p . L e t I 1 ... I k be a n y disjoint finite intervals a n d B 1 .... , B k a n y Borel sets in R 1. P u t f u r t h e rA j = Bj N B, Cj = Bj N B, PJl = Ft(As), PJ2 = Ft(Cj), j = 1 .... , k
N o t e t h a t for the distributions considered in this section {(Xn, Yn(t))} can be a s s u m e d to fulfil the conditions (i) a n d (ii) just a b o v e (4.1). T h e n
~b. t(su, 812 ... ski, sk2; 11 x A1, 11
x 0 1 . . .Ik • Ak, Ik • Ck)
= E ~n {,=1 ~ SjlFt((Iy- Xn) N At) +,=I ~ sy2Ft((I'- Xn) N Cj)
+ 1 - ~F t ( ( I , - X n ) N B ) }
./=1
= qD(PnSu + p12s12 + q ... PklSkl + Pk2 8k2 + q;
I 1 . . .Ik)
(5.1) where t h e last equality follows f r o m t h e assumed invariance u n d e r reversible inde- p e n d e n t m o t i o n t o g e t h e r with l e m m a 2.1. P u t t i n gs[j = ps~j + q, i = 1 ... k, j = 1, 2
we get f r o m (5.1)E~n {j=IZ S;1Ft'B((I]-X~) N A j ) +
j=i ~s;2Ft'B((Iy-Xn) O Cy)+
1 - t=1 ~Ft, B ( I j - X ~ ) }
= V ( p l l S / 1 / p +
pl Sl /p
. . . p lS; l/p . . . f o rs [ j e [ q - p ,
1], i = 1 . . . k, j = 1,2. (5.2) 225T. THEDEEN, Convergence and invariance questions for point systems
U n d e r an i n d e p e n d e n t set of m o t i o n at t with the d,f. F~.~(y) we have for a n y finite interval I a n d a n y Borel set A in R~
Mb,t(I • A) - Mo.t(I • (A fl B)) a.s.
Using t h e index B for t h e g.f. of Mb.t in this case we get
%.t,.(s11, s12, ..., ski, sk2; 11 • B1, I1 x / ~ . . . Ik x B~, I k •
= % . t . , ( S u , s12 . . . ski, sk2; I1 x A1, 11 x C 1 .. . . . Ik • Ak, Ik • Ck)
= E r ] ~ sjlFt B ( ( I j - X ~ ) N A~)+ s j 2 F t , ~ ( ( I ~ - X ~ ) ~ C~)
n I . i = 1 ' j = l
- ~ 1 - i ~ I F t , B ( I , - Xn)}
(5.3)
B u t for sij=s[j, i = 1 ... lc, j ~ 1, 2, the right m e m b e r of (5.3) equals t h e left m e m b e r of (5.2). Thus
~l)b,t,B(S11'
S12 . . .Skl'
8k2; 11 • ~ 1 , 11 • ... Ik • Bk, Ik •= q~(pns~l/p +P12S12/p ... p~lskl/p +pk2Sk2/p; 11 ... Ik)
(5.4)
for s~jE[q-p, 1] a n d b y l e m m a 5.1 for s , ~ E [ - 1 , 1], i = 1 ... k, j = l , 2. N o w Ft,s(Bj) = Pn/P, Ft,~(BJ) = P~21P, J = 1 ... k
which p u t into (5.4) proves the lemma. N o t e t h a t the distribution of {X~(t)} is t h e same as t h a t of {X~} u n d e r an i n d e p e n d e n t set of m o t i o n with t h e d.f. Ft.B(y ). This is seen b y p u t t i n g B 1 . . . B k = R 1 in (5.4). The l e m m a is proved.
I n order to get our results a b o u t t i m e - i n v a r i a n t distributions for point systems we shall h a v e to require t h a t arbitrarily small displacements a n d u n e q u a l displace- m e n t s arbitrarily close to each other are possible. These requirements are m a d e precise in the following definition 5.5. L e t t h e support set S F of a d.I. F(x) be
S F = { x ; F ( x + h ) - F ( x - h ) > O , all h > 0 }
Definition 5.5. A /amily o/ d./.'s {Ft(y), 0 < t < ~ } is said to be non-degenerated i/
( i ) / o r any given e > 0 there is a t and a y such that yESF, and 0 < lYl < e and (if) /or any given ~ > 0 there is a t and Yl ~:Y2 such that Yl, y2ESFt and l Y l - Y 2 1 < e . Remark. I n t h e c o n s t a n t speed case when Y n ( t ) = U n ' t and G(u)=P(Un<~u) t h e n o n - d e g e n e r a c y of {Ft(y)} is equivalent to the d.f. G(u) being non-degenerated.
I n deciding w h e t h e r (if) holds or n o t the following l e m m a m a y be useful.
L e m m a 5.4. I / at least one o/ the Ft(y)'s is not purely discontinuous the condition (if) o/ de/inition 5.5 is/ul/illed.
The proo/follows at once f r o m t h e following
L e m m a 5.5. Let F(x) be a d.[. which is not purely discontinuous. Then /or any given e > 0 there are numbers Xl ~ X 2 such that xl, x2ES F and I x l - x 2 1 <e.
ARKIV FOR MATEMATIK. B d 7 n r 16
Proo/.
L e t D be t h e discontinuity set ofF(x)
a n d p u t C = / 3 ~ St. L e t us for a mo- m e n t assume t h a t there is a s 0 > 0 such t h a t for allXl=4=x 2
with xl,x~EC
we h a v e]xl-x~l
>s0. This implies t h a t C is countable which contradictsF(x)
being n o t p u r e l y discontinuous. This proves the lemma.L e m m a 5.6.
Let the distribution o/ (Xn) be time-invariant under reversible indepen- dent motion/or a non-degenerated family
{Ft(y)}.Then
(i)
the counting process corresponding to {Xn} has a.s. no fixed discontinuity points,
(ii)/or a monotonely decreasing (or increasing) sequence o/ intervals {ln} such that
l i m n _ ~I n = I where I is a ]inite interval we have
lim ~v(s;
In)=(v(s; I)
u n i f o r m l y for s E [ - 1 , 1];n-->~
Proo/.
(i). L e t us assume t h a t (i) does n o t hold. T h e n there is a n x 1 such t h a tP(N({xl}
) > 0) = p > 0.B y t h e n o n - d e g e n e r a c y a s s u m p t i o n given ~ > 0 there is a t a n d a y such t h a t y E SF, a n d 0 < ]y] < s . P u t B = ( - e , 0) U (0, s). T h e n
Ft(B)>O
a n d b y l e m m a 5.3 it follows t h a t t h e distribution of {Xn} is i n v a r i a n t at t u n d e r weak reversible i n d e p e n d e n t m o t i o n for t h e d.f.Ft.B(y).
This implies t h a t withA = ( x l - s ,
Xl)U (x 1, Xl+S) a n d a n i n d e p e n d e n t set of m o t i o n at t with t h e d.f. F• . . . .Nt(A) >~N({Xl}
a.s.B u t
N(A)
a n dNt(A)
have t h e same distribution. T h u sP ( N ( A )
> 0 )>~P(N( {xl) )
> 0 )a n d with a probability no less t h a n p there is a point arbitrarily near b u t separate f r o m t h e position x 1. This contradicts (Xn) h a v i n g no finite limit point a n d (i) is proved.
(ii) follows at once from (i) a n d t h e fact t h a t g.f.'s are b o u n d e d a n d continuous in [ - 1 , 1].
Theorem 5.1.
(i)
(Xn} is time.invariant under reversible independent motion/or the/amily (Ft(y)} , i/
(Xn}is weighted Poisson distributed.
(ii)
Let ( Ft(y) } be a non-degenerated/amily o/d./.'s.
Then the distribution o/ (Xn} is time-invariant under reversibe independent motion ]cr the/amily (Ft(y)} only i/ (Xn} is weighted Poisson distributed.
Remark.
I n comparison with the invariance theorems b y R. L. D o b r u s h i n [4] a n d J. G o l d m a n [7], (ii) of our t h e o r e m on t h e one h a n d d e m a n d sreversible
i n d e p e n d e n t m o t i o n b u t on the other h a n d characterize distributions for point systems which are time-invariant u n d e r reversible i n d e p e n d e n t m o t i o n for an a r b i t r a r yfixed
non- degenerated family(Ft(y)).
227
T. TIIEDEEN, Convergence and invariance questions for point systems
Proo[. (i). The proof follows from the more general s t a t e m e n t of theorem 6.1 if in t h a t theorem we p u t V~ = Yn(t), Fl(y) = Ft(y) a n d / ( x , y) =x +y.
(if) I n the proof we shall rely on the results b y H. Bfihlmann [3] 4. K a p . concerning processes on [0, 1] with exchangeable increments. I t is however easy to see t h a t his results still hold also for processes on ( - ~ , + ~ ) in which case his proofs can be done in the same way. He considers processes which are separable with the set of the diadie numbers as a separating set. I n our case it is seen from the definition of the counting process N(x) defined b y {X~} t h a t N(x) is separable with a n y count- able set which is dense in R 1 as separating set. L e t us divide R 1 into disjoint intervals I s of equal length I I I . The counting process N(x) is a process with exchangeable increments if for a n y such division the distribution of N(Lk) ... N(Ijk) where Jl #... =~]k only depends on k and the length I I I . Let now :~ be the class of distribu- tions for random processes with stationary independent increments having infinitely divisible distributions. Then Bfihlmann proved t h a t a n y process which is separable with the set of diadic numbers as a separating set a n d which has exchangeable increments has a distribution weighted over :~. I n our case the increments are non- negative and integer-valued. The only infinitely divisible distributions of non- negative integer-vMued r.v.'s are the compound Poisson ones (see e.g. Feller [6]
p. 271). We shall show t h a t N(x) has exchangeable increments (point 1 4 below).
Then it is seen t h a t {X~} m u s t have a weighted compound Poisson distribution.
Lastly in point 5 we shall prove t h a t this distribution cannot be compound. Now {X~} was assumed to be a countable set of r.v.'s, which set we tacitly assume to be non-empty. By this the case when the p a r a m e t e r d.f. W(2) has W ( 0 ) > 0 is excluded.
1. The distribution of N ( I ) where I is a finite interval is independent of the position of 1.
Let I i - I + y where yES~t for some t. For a n y intervals I + ~ I D I - we have b y lemma 5.2
~(s; I +) ~<~(s; I)~<~(s; I - ) , sE[0, 1] (5.5) B y lemma 5.6 (if) we can choose I+ and I - such t h a t given e > 0
I~(s; I - ) - ~ ( s ;
1+)1
<e, se[O, 1] (5.6) and such t h a t I + - I and I - I - are unions of intervals with positive lengths. There is an interval B with y E B such t h a tM~, t(I- x B) <~Mb, t(I 1 x B) <~Mr.t(I+ x B). (5.7) For an independent set of motion at t with the d.f. Ft. s(Y) we get from lemma 5.2, l e m m a 5.3 and (5.7)
~(s; I +) ~<~0(s; 11) <~0(s; I - ) , se[0, 1]. (5.S) (5.5), (5.6) and (5.8) together with lemma 5.1 give t h a t
q~(s; I) = ~ ( s ; I1), s E [ - 1 , 1]
and thus 2V(I) and N(I1) h a v e the same distribution. L e t n be a positive integer.
Then it follows t h a t for a n y intervals I and J such t h a t J = I + n y or J = I - n y
ARKIV F 6 R MATEMATIK. B d 7 n r 1 6 w i t h yESFt for some t t h e r . v . ' s N(I) a n d /V(J) h a v e t h e s a m e d i s t r i b u t i o n . L e t n o w Yo be a n y p o s i t i v e n u m b e r a n d p u t
I01 = I + y 0 , 102 =
I-yo.
I n t h e s a m e w a y as in l e m m a 5.6 (ii) i t can be shown t h a t g i v e n e > 0 t h e r e is a (~ > 0 such t h a t
I~(s; Ioj+h)-~(s; I0j)l <~, I~(s; I0j-h)-~(s; I0J)l <~, se[0, 1], j = l , 2, 0<h<~.
(5.9) B y t h e n o n - d e g e n e r a c y a s s u m p t i o n t h e r e is a t, a n i n t e g e r n > Yo/5 a n d a n u m b e r y such t h a t
y e S ~ a n d y o / ( n § <~yo/n L e t us first a s s u m e t h a t y > 0. T h e n
101 = I +ny+hl = I +(n + l ) y - h 2 , O<~h~ <(~, i = l , 2.
(5.10)
W e h a v e a l r e a d y p r o v e d t h a t N(I), IV(I+ny) a n d N ( I + ( n + l ) y ) h a v e t h e s a m e d i s t r i b u t i o n . T h e n t h i s f a c t t o g e t h e r w i t h (5.9), (5.10) a n d l e m m a 5.1 gives t h a t N(I) a n d N(I+yo) h a v e t h e s a m e d i s t r i b u t i o n . I f on t h e o t h e r h a n d y < 0
lo2 = I + n y - h 1 = I +(n + l)y+h2, 0 < h ~ < 5 , i = l , 2
f r o m w h i c h r e l a t i o n we b y t h e s a m e r e a s o n i n g p r o v e t h a t N ( I ) a n d N ( I - y o ) h a v e t h e s a m e d i s t r i b u t i o n .
T h u s t h e d i s t r i b u t i o n of N(I) does n o t d e p e n d on t h e p o s i t i o n of t h e i n t e r v a l I . 2. L e t D e s t a n d for t h e d i s c o n t i n u i t y set of a d . f . F . Choose for a n y ~ > 0 a t a n d yl>Y2 such t h a t Yl, Y2CSF, Yl-Y2 <~ (see d e f i n i t i o n 5.5 (ii)). I t is e a s i l y seen t h a t Yl a n d Y2 can be chosen such t h a t e i t h e r
(a) Yl, Y2 E DFt or
(b) Yl, Y2 E 'DF, a n d n e i t h e r Yl n o r Y2 a r e a c c u m u l a t i o n p o i n t s of D~t.
B y c o n s e c u t i v e i n t e r v a l s we shall in t h e following m e a n d i s j o i n t finite i n t e r v a l s n u m b e r e d f r o m t h e left t o t h e r i g h t a n d w i t h t h e i r u n i o n also being a n i n t e r v a l . L e t n o w J1 a n d J e be t w o c o n s e c u t i v e i n t e r v a l s w i t h I g i l = Ig21 =Yl-Y2. I n t h i s p o i n t we shall show t h a t (-N(JI) ,/u has t h e s a m e d i s t r i b u t i o n as (At(J2), N(J1) ) a n d t h a t t h i s d i s t r i b u t i o n is i n d e p e n d e n t of t h e p o s i t i o n of J~. P u t J = J1 + y~ = J2 + Y2 a n d choose s i m i l a r l y to p o i n t 1 of t h e p r o o f for g i v e n ~ > 0 t w o i n t e r v a l s J + a n d J - such t h a t (1) J + D J ~ J - , (2) J+--J a n d J - J - a r e u n i o n s of n o n - d e g e n e r a t e d i n t e r v a l s , (3) ~(s; J - ) - ~ ( s ; J + ) < e , sE[0, 1].
T h e r e exist d i s j o i n t i n t e r v a l s B 1 a n d B e w i t h y~ E B~ a n d Y2 E B 2 for w h i c h Mo, d J - x B~) ~Mr.dJ ~ x B~) ~Mb.dJ + x B~), i = 1, 2. (5.11) F u r t h e r
M~, d J - x B~) ~< Mb. d J x B~) ~< Mo. d J + x B~), i = 1, 2. (5.12) L e t B = B 1U B 2 a n d p u t Ft.B(B~)=p~ , 0 < p ~ < l , i = 1 , 2. I n case (a) we choose B 1 a n d B 2 d e g e n e r a t e d a t Yl a n d Y2 r e s p e c t i v e l y a n d in t h i s case p~, i - 1, 2 a r e i n d e p e n - d e n t of t h e chosen e, J + a n d J - . Consider n o w t h e case (b) a n d fix t w o n u m b e r s 229
T. THEDI~EN,
Convergence and invariance questions for point systems
p~,
0 < p i < l ,i = l ,
2, p l + P 2 = l . W e shall p r o v e t h a t we can choose B~ a n d B 2 such t h a tFt.s(B~)=pt ,
i = 1 , 2 a n d (5.11) (and (5.12))is fulfilled. Consider the intervalsBi(h~) = ( y i - h t , yi+h~],
h i > 0 , a n d p u tp~(hi)=Ft(Bi(ht)),
i = 1 , 2. There exist n u m - bers h~, h~, such t h a t forh~<~h~,
i = 1 , 2,B~(h~)
a n d B2(h2) arc disjoint a n d (5.11) holds (with Bi =Bi(hi), i = 1, 2). F o r h i > 0 the functionspi(ht)
arc non-decreasing a n d positive and limm_~0p~(hi) = 0 , i = 1, 2. N o w y~ a n d Y2 are n o t a c c u m u l a t i o n p o i n t s ofDpt.
T h u s t h e r e exists h~' such t h a tO<h~'<<.h~
a n dBi(h~') f3 D r t = ~ ,
i = 1 , 2. H e n c e (y;Pi(hi) = y
for some 0 < hi ~< h~'} = (0, pt(h~')], i = 1, 2 (5.13) F r o m (5.13) it is easily seen t h a t we can choose~ht with 0<hi~<h~, i = 1 , 2 such t h a tPl(hl)/p2(h2) = Px/P2
N o t e t h a t given a h > 0 we can in the last relation choose b o t h h i a n d h 2 less t h a n h.
T h e n
Bi=B~(h~),
i = l , 2 fulfil (5.11) (and (5.12)) a n dFt, s(Bi)=p~,
i = 1 , 2.F o r a n y i n t e r v a l I we get f r o m l e m m a 5.3
~b,t,B(81, 82; I • B 1, I
x B 2 ) =q)(plsl +p282;I)
(5.14) Using (5.14) a n d l e m m a 5.2 we get f r o m (5.11)(Pl Sl + P2 s2; J+) ~ ~(Pl Sl + P2, Pl + P2 s2; J~, J2) ~< ~(P~ sl + P2 s2; J - ) , s~, s 2 E [0, 1 ]
(5.15)
a n d f r o m (5.12)cf(p~sx +p2s2; J+) <~q~(p~sx +pzS2; J) <~cf(pls~ +p2s2;
J - ) ,s~s2E[O,
1] (5.16) L e t us n o w p u t 81=plSlA-p2 a n ds~=pl+p2s ~
in (5.15) a n d (5.16). T h e n~v(s; +s~ - 1; J ~ ) <~(s;';
s2; J1; J2) ~cf(s;+s~ -
I; J = )~(Sl +s~ - 1; J+) ~<~0(s; +s~ - 1; J ) ~<~(s; +s~ - 1; J - ) for s~ E [P2, 1], ss E [ P l , 1].
Observing t h a t Pl a n d P2 are i n d e p e n d e n t of e we get b y l e m m a 5.1
~(81,
82; J1, J~) = q9(81-~82-1; J ) , Sx,s 2 E [ - 1 ,
1] (5.17) Since b y p o i n t 1 of the proof t h e right m e m b e r of (5.17) does n o t d e p e n d on the posi- tion of J point 2 is proved.3. L e t J1 .... Jk+l be consecutive intervals with ]Ji ] = [Y~- Yl [, i = 1 ... k + 1 where Yl a n d Y2 are t h e s a m e as in point 2 of t h e proof. W e shall show b y induction t h a t
~(sl ... Sk+l; J1, ..., Jk+l) = ~V(Sl + ..- +sk+~ - k ; J ) (5.18) where J is a n y i n t e r v a l with I J I = l y 2 - y l [. L e t us a s s u m e t h a t
(p(81 . . . . , 8k; 5 1 . . . . , Jk) = ~(sl + . . - + s k - - k + l; J )
(5.19)
A R K I V FOR M A T E M A T I K . B d 7 n r 1 6
L e t t h e d i s t a n c e b e t w e e n t w o i n t e r v a l s A 1 a n d A 2 b e
d(A1, As) = inf { I x - Y I; x eA1, y eA2}
Choose t h e i n t e r v a l s J~.h, i = 1, ..., k + 1, s u c h t h a t
J ~ . h c J ~ , d(J~.h, J i + l . h ) = h , i = 1 . . . /c a n d J k . h + y l = J k + l . h + y 2
L e t J;. h = J~. h + Yl, i = 1 . . . . , k. Choose s i m i l a r l y t o p o i n t 2 i n t e r v a l s J~'.+h a n d J~'.-~
s u c h t h a t
t + 9 i _
(1) J ~ . h ~ J ~ . h D J ~ . h , i = 1 , . . . . h,
(2) g ; + -g~" h . a n d g ' ~, h - J~. h ' - a r e u n i o n s of n o n - d e g e n e r a t e d i n t e r v a l s , i = l , . . . , k,
t l_ ,
(3) Ji. h, * = 1 . . . k a r e d i s j o i n t i n t e r v a l s ,
(4) ~0(s 1, . ..,sg; Jl.h'- . . . J'k_a)--qD(sl . . . sk; Jl.h'+ . . . Jk.h)<e'+
for s,E[0,1], i = l . . . . , k (5.20)
I n t h e case w h e n Yl, Y2 e DF~ l e t u s choose B 1 = {Yl} a n d B 2 = {Y2}. I f Yl, Y2 e / ) F , we choose B t = B i ( h , ) w i t h h , < h (see t h e r e m a r k j u s t a b o v e (5.14)) as i n p o i n t 2 of t h e proof. T h e n
Mb.~(J;.-h x B I ) <~Mf.~(J~. h x B~) <~M~.t(J;.+h • i = 1, . . . , k
Mb, t (g~Th • B2) ~< M I . t ( J k + 1. h • B2) ~< Mb.t (J~+h • B2) (5.21) N ( g , . h ) ~ N ( J , . h ) ~ N ( g [ . + h ) , i = l , k (5.22) W e g e t b y l e m m a 5.3 w i t h B = B 1 U B 2 i n t h e s a m e w a y as i n p o i n t 2 f r o m (5.21)
t + t + ~§
~0(plSl q-P2 . . . p18~-1 -kP2, p18k q-P28g+1; J1. h . . . J k 1. h, J~. ~)
~< 9(p181 + P2 . . . PlSk + 1)2, 101
~-p2Sk+l;
g l . h, . . . , gk+h, gk+l. h)for
s,e[0,1],
i = l . . . k + l (5.23)L e t u s p u t s; = p l s , + P 2 , i = 1 . . . k a n d s~+l = p ~ + p 2 s ~ + l i n (5.23). T h e n
~ ( 8 1 . . . . , S k - 1 , S k - ~ S k + 1 - - 1; J1. ~ . . . J k - 1 . ~, J~. ~) ~ ~0(81 . . . . ' 8 k + 1 ; J L a . . . J*+~. a)
i = 1, . . . , k, s~+l e [Pl, 1] (5.24)
F r o m (5.22), l e m m a 5.2 a n d l e m m a 5.3 we g e t
99(81 . . . . , 8k; J1. h . . . . , J e. ~ ) <~ q)( s~ , . . . , se; J1. ~ . . . . , J e. ~ )
~<~ ~ 0 ( 8 1 , - . . , 8k; J~]a, . . . , J'~:~), s, e [0,1], i = 1 . . . k (5.25) W e g e t f r o m (5.20), (5.24) a n d (5.25) t h a t
~0(sl, ... , se+l; J1. a, -.., Je+l, a) = ~0(st . . . . , s~ 1, se + Se+l - 1; J ; . ~, . . . , I ~ 1. a, J;~. a) 2 3 1
