A R K I V F O R M A T E M A T I K B a n d 7 n r 4 0
1.68 06 4 Communicated 10 April 1968 b y H a r a l d Cram6r a n d L e n n a r t Carleson
P o i s s o n processes as renewal processes invariant under translations
B y MURALI RAO a n d HANS ~TEDEL
Introduction
L e t ( X n : n = + 1, __ 2 . . . . } be a sequence of r a n d o m variables such t h a t a.s.
, , , X _ ~ < X _ I < 0 < X 1 < X 2...
P u t Y o = X _ I , Y I = X 1 Y ~ = X , - X ~ _ I , n 4 0 , 1.
A s s u m e t h a t :
(i) {(Yo, Y1), Yn, n ~ O . 1} is a set of i n d e p e n d e n t r a n d o m variables.
(ii) { Y~ : n 4= O, 1}, are independent, identically distributed positive r a n d o m v a r - iables with P[ Y~ < y] = F(y),
F ( O ) = O a n d E [ Y n ] = - < c ~ . 1
m
L e t { ~ , n = -!-_ 1, ___ 2, ...} be a set of i n d e p e n d e n t r a n d o m variables which is inde- p e n d e n t of {X~} a n d ~n, n = _ 1, • 2 . . . , be identically distributed with the s a m e n o n - d e g e n e r a t e d distribution G. P u t Zn = Xn + ~,~ a n d let N(I) = n u m b e r of X~ E I a n d _N(I) = n u m b e r of Z~ E I .
D o o b has shown [1, pp. 404-407] t h a t if all Y~ h a v e a n exponential distribution t h e n N(I) a n d /~(I) h a v e t h e s a m e distribution. Thed6en p r o v e d t h e converse of this s t a t e m e n t , n a m e l y t h a t e v e r y Y~, n 4= 0, 1, h a s a n exponential distribution if N(I) a n d ~ ( I ) h a v e the s a m e distribution a n d if
(iii') P[Yo >Yo, Y1 >Y] = mfyo+ul (1 - F(s)) de.
W e shall here p r o v e t h a t t h e w e a k e r conditions E[N(I))] = m ]I] a n d E[N(I)N(J)] = E[.N(I)I~'(J)] are sufficient t o i m p l y e x p o n e n t i a l distributions of Y.. Our proof is a t t h e s a m e t i m e a simplification of Thed6en's proof.
L e t X~ a n d Y~ be as in t h e introduction a n d i n s t e a d of (iii') p u t (iii) E[N(I)] = m III where II] denotes t h e L e b e s g u e m e a s u r e of I .
539
M. RAO, H. WEDEL, Poisson processes as renewal processes
T h e n (iii) is e q u i v a l e n t t o P [ Y i > u ] = ~ ( 1 - F(t))dt for i =0, 1, see [2], pp. 354.
L e t n o w {~n, n = + 1, • 2, ...} be a sequence of r a n d o m variables which is inde- p e n d e n t of the sequence {Xn}. W e shall a s s u m e t h a t for all n, m, n4=m, (~, ~ ) h a v e t h e s a m e joint distribution G a n d t h a t t h e s u p p o r t g r o u p of G. i.e. t h e g r o u p g e n e r a t e d b y t h e s u p p o r t of G, has a n e l e m e n t of t h e f o r m (0, d) w i t h d > 0; if ~.
a n d ~ are i n d e p e n d e n t a n d h a v e a n o n d e g e n e r a t e distribution t h e n this is certainly true.
P u t Z~ = Xn + ~n, n = _+ 1, • 2, . . . , a n d 1~(I) = n u m b e r of Z~ E I .
Theorem. Let X~, ~ , g , be as above. I / E [ N ( I ) 57(J)] = E[N(I)N(J)] /or all I, J then {Xn} is Poisson i.e. F ( y ) = 1 - m e --my.
P r o # . P u t O ( I , J ) = E[N(I) N ( J ) ] - E [ N ( I N J ) ] = ~ . m P[Xn E I, Xm E J]. Using independence of { ~ } a n d {Xn} we get E [ i ~ ( I ) R ( J ) ] - E [ 2 ~ ( I N J ) ] = ~ O ( I - u , J - v) dG(u, v). T h e condition B[N(I)] = m [I[ implies E[2V(I)] = m ] I { . T h u s E[2V(I) 2~(J)] - E [ N ( I N J ) ] = E[N(I) N ( J ) ] - E [ N ( I N J ) ] : which gives 9 = 9 ~+ G.
A simple consequence of t h e renewal t h e o r e m is t h a t for a n y p a i r of finite inter- vals I , J , we see t h a t E [ N ( I + h ) N ( J + k)] is a b o u n d e d function of (h, k). T h e C h o q u e t - D e n y t h e o r e m [3, p. 152] applies a n d we deduce t h a t e v e r y p o i n t of t h e s u p p o r t of G is a period for r T h e set of periods for (I) is a group a n d this g r o u p contains the e l e m e n t (0, d) a n d hence (0, kd) where k is a n y positive integer (in- deed a n y integer). T h u s for all I , J a n d all positive integers k, (I)(I, J ) = r J + kd). T a k e I = (0, x] w i t h x < kd. T h e n I fl (I + kd) = r
Also
@(I,I+kd)= ~
P [ X ~ e l , X , ~ e I + k d ] = ~. P [ X ~ e I , X m e I + k d ]n4-m rn,n>~I
= n=l ~ m>n ~" P[Xn EI, Xm E I + kd]= n=l ~ f [ H ( I + k d - u ) d ( F o - ) e F (n-l)*) (u)
= f [ H ( I + kd - u) du,
where H(x) = ~ ~ 1 F k* (x) a n d (iii) implies m x = ~ - o Fo-)e F k* (x), x > O.
Similar calculations give (I)(I, I ) = 2 ~ H ( x - u) clu = 2 ~ H(u) du. T h u s
yo f:
2 H(u) du = H ( I + kd - u) du = [H(x + ]cd - u) - H(kd - u)] du
fo
= [H(t~d + u) - I-I(kd - u)] du.
This equality for all x < kd implies 2 H(u) = H(kd + u) - H(]cd - u), u < kd.
Suppose d o is a positive n u m b e r such t h a t F(do)>0 a n d F(d o - ) = 0 . T h e n F n*
has a n a t o m a t nd 0 a n d t h u s H has a m a s s p o i n t a t e v e r y positive integral multiple of do, b u t H ( u ) = 0 u < d o. Choose ]c so t h a t kd > d o. F o r u < d o H ( u ) = 0 a n d t h e functional e q u a t i o n for H shows t h a t H ( k d - u ) = H(kd + u ) u < do. Since e v e r y in- 540
ARKIV FOR MATEMATIK. B d 7 n r 4 0
t e r v a l of l e n g t h I a r g e r t h a n d 0, c o n t a i n s a m u l t i p l e of d o a n d H h a s a p o s i t i v e m a s s a t s u c h a p o i n t , w e s e e t h a t H ( k d " u) < H(kd + u) f o r s o m e 0 < u < d 0. T h i s is a c o n t r a d i c t i o n a n d t h u s F c e r t a i n l y c a n n o t b e a r i t h m e t i c . A s k - ~ c ~ B l a c k w e l l ' s t h e o r e m [2 p. 347] s h o w s t h a t H ( k d + u) - H ( k d - u) -~ 2 um, a n d t h u s 2 H(u) = 2 urn.
T h i s is e q u i v a l e n t t o F b e i n g e x p o n e n t i a l . Q . E . D .
A C K N O W L E D G E M E N T
I t was H. BergstrSm who pointed out t h a t our conditions were sufficient. We are grateful to to him and P. Jagers for valuable discussions.
University o] G6teborg, Department of Mathematics, GSteborg, Sweden
R E F E R E N C E S
1. DooB, J. L., Stochastic Processes. Wiley, 1953.
2. FELLER, W., An Introduction to Probability Theory and its Applications, vol. I I . Wiley, N e w York 1966.
3. M_EY~R, P., Probability and Potentials. Blaisdell, 1966.
4. T H ~ D ~ , T., On stochastic stationary of renewal processes. Arkiv fSr mathematik 7, H~fte 3, 249-263 (1967).
Tryekt den 30 augusti 1968 Uppsala 1968. Almqvist & Wiksells Boktryckeri AB
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