Download pdf - Bridging the gap between a stationary point process and its Palm distribution

37

Statistica Neerlandica (1994) Vol. 48, nr. 1, pp. 37-62

Bridging the gap between a stationary point process and its Palm distribution

G. Nieuwenhuis Tilburg University, Department of Econometrics

P.O. Box 90753, 5000 LE Tilburg, The Netherlands

In the context of stationary point processes measurements are usually made from a time point chosen at random or from an occurrence chosen at random. That is, either the stationary distribution P or its Palm distribution Po is the ruling probability measure. In this paper an approach is presented to bridge the gap between these distributions. We consider probability measures which give exactly the same events zero probability as Po, having simple relations with P. Relations between P and Po are derived with these intermediate measures as bridges. With the resulting Radon-Nikodym densities several well-known results can be proved easily. New results are derived. As a corollary of cross ergodic theorems a conditional version of the well-known inversion formula is proved. Several approximations of Po are considered, for instance the local characterization of Po asa limit of conditional probability measures P,.”, n E N. The total variation distance between Po and P,, can be expressed in terms of the P-distribution function of the forward recurrence time.

Key Words 81 Phrases: Radon-Nikodym derivative, local characterization, inversion formula, ergodicity.

1 Introduction

In queueing theory it is often wanted to express characteristics of time-stationary processes in terms of characteristics of customer-stationary sequences. It turns out that the underlying theory for many problems of this type concerns the relationship between two probability measures, the distribution P of a stationary (marked) point process (the stationary distribution) and the Palm distribution Po (intuitively arising from P by conditioning on the occurrence of an arbitrary point (with some mark) in the origin). See e.g. FRANKEN et al. (1982), BACCELLI and BREMAUD (1987), and BIUNDT, FRANKEN and LISEK (1990). For this reason it is important to obtain a good understanding of the relationship between P and Po.

In this paper we will try to bridge the gap between P and Po. We will confine ourselves to unmarked point processes, although in the final section a generalization to marked point processes is sketched briefly. The approach in this paper could be called the Radon-Nikodym approach. Several “intermediate” probability measures are considered which have exactly the same null-sets as Po, having simple relations with P. The resulting Radon-Nikodym densities (i.e. the densities of these probability measures with respect to Po) are used to express Po-expectations in terms of P-expectations (and vice versa).

38 G. Nieuwenhuis

The formal definition of the Palm distribution (see (10) below) chosen in this research, as well as the classical inversion formula (see (12) below) are due to MECKE (1967). The definition presents one possibility to go from P to Po. With the inversion formula we can go the other way. We will, however, make use of intermediate probability measures to bridge the gap between P and Po. For a good understanding of this approach we will first derive some relations between stationary distribution and Palm distribution using heuristic arguments, based on an intuitive relationship between the two probability measures.

It will always be assumed that the point process is stationary (i.e. has the same distribution seen from all time points) and that there are no multiple occurrences. For the sequence (Xi) of occurrences we will use the following convention about numbering and about position of the origin:

... <x-,<x,<o<x, <xz< ... The length of the interval between Xk and x k + I will be denoted by ak. The relationship between P and Po is often described as follows:

Po arises from P by shifting the origin to an occurrence which is chosen at random. P arises from Po by shifting the origin to a time point in (- 00, + w) chosen at random.

Although these intuitive relations are only made more concrete under ergodicity conditions (see (48) and Theorem 7), they will be used here to explain the approach of this research. First note that it follows from the above random generation of P and Po that

1. under Po there is an occurrence in the origin, 2. on the average the length of the interval (between occurrences) in which the

randomly chosen time point falls will be larger than a “typical” interval length.

The last remark turns out to be the key for Theorem 1 below. To see this, we need a third probability measure, Po. It arises from P by shifting the origin to the first occurrence on its left, to Xo. Note the difference between PO and Po, in notation and in interpretation. Expectations with respect to the three distributions P, Po, and Po have to be distin- guished; denote them by E , Eo, and Eo, respectively. The intuitive random procedures for generating Po and P make plausible that, when the deterministic point process is excluded, the sequence (a i ) is stationary (i.e., has the same distribution seen from all occurrences) or non-stationary, according to whether measurements are made from an occurrence chosen at random or from a time point chosen at random. That is, according to whether Po or P is the ruling distribution. For instance, a randomly chosen time point has a higher probability to fall in a large interval between occurrences than in a small interval. So, under P the distributions of a0 and a,, n # 0, will in general be different. The same holds for the distribution of a. under P and the common distribution of the ( x i

under Po; the expectation Eao will usually be greater than Eoai. These considerations can be made more formal by using heuristic arguments.

Assume that under Po expectation p and variance u2 of a, exist. For n E No let g, and f,

Stationary point process and its Palm distribution 39

be the probability densities of (ao, a l , ..., a,) under P and Po, respectively. Starting from an origin in a randomly chosen occurrence, consider the interval [0, a. + al + ... + a,-l) where r is very large. In this interval a time point is chosen at random and the origin is moved to it. Let A be the length of the interval (one of the r intervals) in which the time point falls, and let 6r(x) be the number of the ai in [x,x + 6). Here 6 is a small positive real. Then S-'P[A E [x,x + S)] is equal to

X

As r tends to infinity and 6 to zero, r(x)./r tends to fo(x) and (zlr: a i ) / r top, provided that the sequence (a,) satisfies this strong law of large numbers. This length-biased sampling procedure (see COX and LEWIS, 1966, Section 4.2) is similar to the intuitive random generation of the stationary distribution P. So, we may expect S-'P [ A E [x,x + S)] to tend to go(x). Hence,

If a2 > 0 (i.e., if the point process is not deterministic), then E(ao /p ) > 1. That is, on the average the length of the interval in which the randomly chosen time point falls is greater than p. Using similar arguments we obtain for n E No:

A formal proof of (4) is given in Section 2. By multiple integration we derive from (4) that

for subsets B of (0, a). This result illustrates the essence of Theorem 1 which is so important for this paper. The distribution of the k'th interval length following a randomly chosen occurrence (represented by the probabilities Po[ak E B] = Eol[ , , ,E B ~ )

is transformed into the distribution of the k'th interval length following a randomly chosen time point. This is effected by multiplying the indicators I[,,, B ] byao/p. A real- ization for which a. is relatively large obtains more importance by way of the weight ao/p (recall that for the resulting distribution P these realizations should have more weight since E(ao/p) > 1). By (4) it is obvious that (5) can be generalized. In'fact we have, for functions h, and integers i(l), ..., i (k ) ,

40 G. Nieuwenhuis

Straightforward generalization of (6) to functions h on the set of realizations of the point process is not possible. For instance, the indicator function l[x,=aOl ,would be transformed into Eo(ao/p lp , = 1, while E QX, =sol = 0. This can be repaired by following up the length-biased sampling procedure by moving the origin to the last occurrence before or in the randomly chosen time point. That is, by going from P to Po. By this final operation the left-hand side of (6) is unchanged:

This equality expresses the transformation of Po into Po. We conclude that this transition and the reversed transition can be established as described in (i) and (i)’, respectively:

(i) Change the importance of the realizations ofthe point process by way of the weight function a o / g

(i)’ Change the importance of the realizations of the point process by way of the weight function p/ao.

Since a. is greater than 0, it is obvious that eventualities with zero probability under Po also have zero probabilities under PO and vice versa (see (7)). This observation, combined with (7), describes the relationship between Po and Po mentioned in Theorem 1 below, in the case that n = 0.

It is intuitively clear how transitions from P to Po or from Po to P should be effected. See (ii) and (ii)’, respectively:

(ii) Shift the origin to Xo. (ii)’ Shift the origin to a time point which is uniformly chosen in [ X o , X l ) .

The transition from Po to P, described in (ii)’, is formalized in (20) with n = 0. Trans- formation of P into Po (or Po into P) using Po as a bridge can be achieved by applying the steps (ii) and (i)’ (or (i) and (ii)’). More general the probability measure P,arising from P by shifting the origin to the n’th occurrence, n E Z, can be used as a bridge. Transition from P to Po (or vice versa) can then be effected by executing the steps (ii) and (i)’ (or (i) and (ii)’) after replacing a. in (i) and (i)’ by a-”, Xo in (ii) by X,, and [Xo, XI) in (ii)’ by [X-,, X-,+ I). See the remarks preceding Theorem 1, (18), and (20).

In Section 2 the above observations are formalized and several corollaries are derived. Some of the results are not new. The present proofs are very simple and direct. The diagram at the end of Section 2 shows that the order of the two steps in a transition between P and Po is of no importance. Interchanging the two steps only means that another probability measure is used as a bridge.

Some cross ergodic theorems are proved in Section 3 by making use of intermediate probability measures. No ergodicity conditions are assumed here. As a corollary a conditional version of the classical inversion formula (12) is derived.

Starting from the stationary distribution, some strong or pointwise approximations of


the Palm distribution are considered in Section 4. For these approximations to hold necessary and sufficient conditions are formulated. For this purpose a notion weaker than ergodicity of the point process is introduced. Some other intermediate probability measures, all having exactly the same null-sets as the Palm distribution, are considered. It is proved that a formalization of the first intuitive approximation mentioned at the beginning of the present section is uniform. It turns out that intuition seems to be false if the stationary sequence (a i ) does not satisfy the strong law of large numbers with a degenerate limit. The well-known (and intuitively clear) uniform approximation of the Palm distribution P o by conditional probability measures PI,,, usually referred to as local characterization of the Palm distribution (cf., e.g., FRANKEN et al. 1982, Th. 1.3.7), is also considered. We derive a very simple expression for the total variation distance of Po and PI,,. Conditions are given such that the rate of the resulting uniform convergence is of order l /n . Some examples are considered in Section 5. A generalization to marked point processes is briefly sketched.

At the end of this section we formalize some of the notions mentioned above and give some other definitions and notations. In the following R, R+, Z and No are the set of reals, the set of nonnegative reals, the set of integers, and the set of nonnegative integers. 9 and 9' are the Bore1 o-fields of R and R+. A pointprocess on R is a random element @ in the set M of all integer-valued measures p on 9 for which

p(B) < m for all bounded B E 9. Let AY be the a-field generated by the sets [p(B) = k]:= { p E M : p(B) = k ) , k E No and B E 9. See MATTHES, KERSTAN and MECKE (1978), KALLENBERG (1983) or DALEY and VERE-JONES (1988) for more information. Set

M':= {p E M: p(- m,O) = p(0, a) = CD; p{x} I, 1 for all x E R}, r : = ~ " n ~ .

We will always assume that @ (or rather its distribution P) is stutionuy (i.e., @(t + .) = d @ for all t E R). We also assume that @ # 0 w.p.1. that @ is simple and that the intensity I is finite; or, equivalently,

P(M") = 1 and I:= IE@(O, 13 < m. (8)

- 0 . < X - * ( d < XO(d 5 0 < X,b> <&(a) < - * *

The atoms of p E M" are denoted by X i ( p ) , i E Z, with the convention that

We interpret X,(p) as the time of the i'th arrival (point, occurrence) and ai(p):= X i + l ( p ) - X i ( p ) as the i'th interarrival time (or interval length). We have @(B):= # (i E Z : X i E B ) and [a, E B] := [ai(p) E B ] := (p E M": ai(p> E B ] , B E 9.

For t E R the time shift TI : A4 -, M is defined by T,p := p(t + -), p E M. For p E M" the atoms of Tlp are X,(y) - t, i E Z. So, T,p arises from p by shifting the origin to t (and considering p from this new position). By stationarity it is obvious that these mappings are measure preserving under P. For n E ZZ the point shut 8,: M" + M" is defined by enp:= p(X,(p) + .), p E M". The atoms of B,p are X,(p) -X,(p), i E Z. So, B,(p) arises from p by shifting the origin to the n'th occurrence. Note that

42 G. Nieuwenhuis

B,(Blp) = Bn+lp. A random sequence (ti):= (ti)isz with t i : M a + R is generated by the point shift B1 if t,(Blp) = t n + l p for all p E Ma and n E Z. See also NIEUWENHUIS (1989; p. 600). Examples of such sequences are (a i ) and (lA o O i ) , A E ,Ha. The general form is (f o B i ) , f : Ma -, Ma measurable. The distribution P, of en @ plays an important role in this paper. It arises from P by shifting the origin to the n'th arrival. Hence,

P,:=PB;l, n E Z. (9)

We now consider that Palm distribution Po of CP. An intuitive definition of Po was stated before. The formal definition of the Palm distribution Po is

Set Mo:= (p E M" : p{O} =i 1) and do := M o fld. It is obvious that Po is a probability measure on (Ma, A'") with Po(Mo) = 1. Note also that Po[ao = 03 = 0 by (lo), since CP has no multiple points w.p.1. Starting with Po, the point shift 0, induces the probability measure Poll;'. According to FRANKEN et al. (1982; Th. 1.2.7) Po has the following important property:

po=poe;l for all n E Z. (11)

Consequently, any sequence (ti) generated by B1 is Po-stationary, i.e. (tl, ...,en) and ..., tk+,) have the same distribution under Po, all n E N and k E Z. In particular,

(ai) is Po-stationary. Definition (10) allows us to express Po in terms of P. The following inversion formula

expresses P in terms of Po (cf. FRANKEN et al., 1982, p. 27). a

P(A) =,I P0"x,(p) > U ; p(u + *) E A] du, A E A. (12) 0

Substituting A = M yields

1 p = Eoao = 7

A

For p E M we define

We will sometimes write N ( t ) instead of N,. Let Q, and Q2 be probability measures on a common measurable space. We say that

Q1 is dominated by Qz (notation QI << Qz) if the Qz-null-sets are also Ql:null-sets. A Radon-Nikodym density of Ql with respect to Qz will be denoted by dQl/dQz. The attribute "Qz-almost surely" will usually be suppressed.. Q1 and QZ are equivalent (notation QI - Q2) if they dominate each other; they are singular(notation QI I Qz) if an event A exists such that QI(A) = 0 and Qz(A) = 1. The rota1 variation distance d between Q, and Qz, both dominated by a n-finite measure w and having densities h l and


h2 respectively, is defined by

d(Qi, Q2):=j I hi - h2 I do .

d(Ql ,Q2)=2su~ I Q I (A) -Q~(A) I =2(Qi[hi>hzI-Q2[hi 2h21).

(15)

(16)

Expectations with respect to the probability measures P, P,, and Po, all considered on (Ma,,&’”’’, are denoted by E, En and Eo, respectively. In particular the distinction between Po and Po and between Eo and Eo should be noted. Expectation with respect to a universal probability space (Q,.F, P) is denoted by IE. If Q is a probability measure on (Ma,&”), then Q8;’ is the probability measure induced by 8 , and EQ denotes expectation with respect to Q. Lebesque measure on R is denoted by v . “Random variable” is abbreviated to r.v., “almost surely” to as . and “almost everywhere” to a.e.

It is well-known that

A

2 Intermediate probability measures

The set of realizations with an occurrence in 0 has probability zero under P and probability one under Po. Nevertheless, certain shifts of P give exactly the same events zero probability as Po. The corresponding densities can be expressed in a very simple form. We collect formulas and conclusions that follow from this observation.

In Section 1 it was made plausible that transition from Po to Po can be established by using the weight function p/ao (see (i)’; since E (ao/p), and hence Eo(ao/p), is (usually) greater than one while Eo(ao/p) = 1, realizations for which ao/p is relatively large should obtain less importance). Recall that P,, arises from P by shifting the origin to the n’th occurrence. Seen from this new position of the origin, the interval length a0 in the above weight function is now denoted by a-,,. So, transition from P,, to Po (or Po to P,) can be effected by using the weight function p/a-,, (or a-,,/p). This is formalized in part (b) of the following theorem. See NIEUWENHUIS (1989; Th. 2.1) for a proof. Recall that p = 1/n.

THEOREM 1. Let n E Z. Then (a). P,, - PO, (b). Q,,(v):= Aa-,,(p), QI E Mo, defines a Radon-Nikodym densiv of P,, with respect to Po.

Since P,, and Po apparently have the same null-sets, it is clear that convergence w.p.1 (just as convergence in probability) holds equivalently under both probability measures. This observation leads immediately to some cross ergodic theorems. See Section 3. In NIEUWENHUIS (1989) Theorem 1 was applied to prove (under some mixing condition) the equivalence of a special type of functional central limit theorems under P and Po. The relation in (b) can serve as a tool for transforming formulas involving P into formulas involving Po and vice versa. We give some examples.

By part (b) of the theorem we obtain 1

E o f = i E , , (Lf) and E, f=AEo(a- , , f ) a-n

44 G. Nieuwenhuis

for all Po-integrable functionsf : M o -, R. The relations in (17) express transitions from P,, to Po and from Po to P,,, respectively. These transitions were described in Section 1, the adapted steps (i)' and (i). Since En g = Ego 6, and a _ , o 8, = ao, it follows from the left-hand part of (17) that

1 E o f = I E (: f of?,,).

This relation describes the transformation of P into Po when the intermediate probability measure P, is used as a bridge. It is performed by executing the adapted steps (ii) and (i)' as mentioned in Section 1. Relation (18) may be an alternative to the transition described by (10).

A transformation of Po into P follows from the inversion formula (12), which can be reformulated as

Eg=IEo ( l g o T x d x )

for all P-integrable functions g:M"+R. By (11) it follows that the right-hand part of (19) is equal to

~ E o j goT,oe-,,dx 2 - 1 ~ ' ( '- j+lgoq,dy). r x-" ,

By the left-hand part of (17) we obtain

for all P-integrable functions g : M" -, R. Relation (20) expresses a transition from P,, to P, in the interval [X-, , ,X-, ,+,) a time point is chosen uniformly and the origin is moved to it. This corresponds to the adapted step (ii)' mentioned in Section 1. By applying the right-hand side of (17), with

f = - '-r' g 0 Ty dy, a-n X-,

followed by (20) we obtain (19) again. So, the transformation of Po into P can be performed in two steps using P,, as a bridge; the steps (i) and (ii)', adapted as mentioned in Section 1.

The last two-step transformation mentioned above is of special interest if g : M" -, R is such that Eg = Ego 00. Then Eg = Eo g and step (ii)' has no influence. Consequently,

(21) if Eg= Ego Bo then Eg=IEo(aog).

If g is a function of some sequence generated by f? ,, then (21) yields an easy transformation of Po-expectation into P-expectation.

To illustrate the simplicity of this Radon-Nikodym approach we derive some quick results. The well-known relation (13) can be obtained by (18) by choosing n = 0 and


f = ao. Other formulas on (ai) can be obtained by making simple choices forf, g and n, or (probably even faster) by applying Theorem 1 directly:

1 a0

E -=A,

Eak = AEo(aoak) = Eoao + COV pO(a0, ak)/EofZo,

(cf. Cox and LEWIS, 1966, (4.2.8) and MCFADDEN, 1962, (3.

ak E -= 1, EUk =Ea-k , Eaka, = Ea-ka,-k, k: a0

k E Z,

a), n E Z.

Let n E No. If the Po-distribution of (ao, ..., a,) is dominated by Lebesgue measure with density f,, then the P-distribution of (ao, .,.., a,) is also dominated by Lebesgue measure, with density gn defined by

gn(X0, ...,xn) =Jxofn(xo, *..,xn), xO,-**,xn E (0,m). (23)

See also the heuristic arguments leading to (4). Relation (23) holds since for yo ,..., y, E (O ,m) ,A:=[ao<yo ,..., a,,sy,], and B:=Xl,o(O,yi] we have

P ( A ) ?Po(A) =AEo(aol,) = A I X O ~ , ( X O , ..., x , ) ~ x ,... dxo. B

Since Xl(T,v,) = ao(p) - u for all v, E M o and u E (0, ao(v,)), we obtain by (19) and (18) that

E [@ao E BJE (g (xi) I ao)I = E (+ao E s](g(XJ)

I = AEo I g(xi 0 T u ) l [ a , , ~ ~ 0 T, d u [:

for all B E S+ and g : R+-. R such that E I g(X,) I < a. Consequently,

the conditional P-distribution of XI given a. is uniform (0, ao). (24)

This well-known result will be applied next. By (24), Fubini's theorem and Theorem 1 we obtain

\ x

46 G. Nieuwenhuis

It follows immediately that

dPX Pxl << v and = l P o [ a o > s] v-a.e. d v

Relation (25) is (1.2.21) in FRANKEN et al. (1982).

have The following result will be applied in Section 4. By (25) and Fubini's theorem we

W

P [ X , > t ] = L Po[ao>s]ds=LEo[(ao-t)l[ , , , ,~], t E [O,m). I

By Theorem 1 we obtain for t E [0, m):

P [ X ~ > t] = P[ao > t] - LtPo[ao > t], P [ X , 5 t] = P[ao 5 t ] + I tPo[ao > t ] .

(26)

(27)

We will prove another corollary of Theorem 1 which will be useful in Section 3. Let 4 be the invariant u-field under the point shift 01, i.e.,

9:= ( A E A": BT'A = A } .

P I 2 -P01.7 . (29)

(28)

Note that P ( A ) = P I @ ) for all A E 4. Hence,

In BACCELLI and BRBMAUD (1987; p. 28) Relation (29) is proved directly from the definition of Po. We need, however, expressions for the Radon-Nikodym densities. For A E 3 we have

Hence,

Another probability measure on (M", A''") which is in a sense intermediate between P and Po is the measure P' defined by

P ' ( A ) : = i E 1 ($1.). A E J "

Note that P' is indeed a probability measure (see (22)), that P ' I Po, and that

d P d P P' - P with 7 =Lao. (32)


By (31) and Theorem 1 we obtain For A E A" and n E 24 that

Consequently,

pier1 =PO, n E Z. (33) This relation implies that random sequences on M" generated by 0 , are not only Po- stationary but also P'-stationary. IF d, is a renewal process, then the sequence (a i ) is both i.i.d. under Po and under P' (note that the P'- and the P'B{'-distribution OF ( a l , ..., a,,) are the same).

Relation (3 1) expresses a transition From P to P'. It is effected by changing the importance of the realizations by way of the weight Function p/ao; i.e. by executing step (i)' when starting from P. Similarly a transition from P' to P can be achieved by performing step (i) when starting From P'. By (33) with n = 0 it is obvious that P' can be transformed into Po by applying step (ii). Step (ii)' yields the reverse transformation, since

1 "O f o T x E'f = - E (L f ) = Eo ( - d x ) = Eo ( 7 f o T, d x )

1 a0 0 aooTx a0 0

for all P'-integrable functions f: M" -., R. Here (31) and (19) are used; E' expresses expectation with respect to P'. We conclude that transition from P to Po (or From Po to P ) can also be achieved by using P' as a bridge. It is effected by executing step (i)' followed by step (ii) (or (ii)' Followed by (i)).

The Following diagram comprises some of the above results.

P -P' eo 1 i eo

Po - PO The Radon-Nikodym density OF P with respect to P' is not affected by applying Oo (cF. Theorem 1 and (32)). By (33) and the above diagram it is obvious that the position of P' as intermediate probability measure between P and Po is similar to the position ofPo. Relations (18) and (21) can also be derived with P'. In NIEUWENHUIS (1989; Th. 7.4) the measure P' has been used to prove that some Functional central limit theorems hold equivalently under P and Po. In Section 4 a Few other intermediate probability measures will be considered. '

3 Cross ergodic theorems

Birkhoffs ergodic theorem holds For stationary sequences. If a sequence (ti) is generated by el, it is stationary under Po and not under P. Still we can derive astrong law For it under P. For the P-stationary sequence (@(i - 1, i]) we obtain a strong law under Po. In the literature these so-called cross ergodic theorems are usually Formulated under

48 G. Nieuwenhuis

ergodicity conditions, see FRANKEN et al. (1982; Th. 1.3.12), BACCELLI and BREMAUD (1987; p. 29/30), ROLSKI (1981; 8 3.3). By applying Theorem 1 we can give simple proofs for more general results without assuming ergodicity. A conditional version of the classical inversion formula (12) is derived.

We need some preliminaries first. Set

9 ' : = { A E A . P : T ; ~ A = A for all t E R } (34)

and recall the definition of 9 in (28). Part (b) of the following lemma ensures that the above invariant o-fields, one generated by the time shifts T,, t ER, the other by the point shift el, coincide.

LEMMA 2 (a). I f A E 9, then @;'A = A for all i E Z; (b). 9 = 9'.

PROOF. Let A E 9. For all w E M" we have

V E A iff B I y ~ A .

By applying this observation twice we obtain for all Q, E M",

P E A iff B I ( B o y l ) ~ ~ iff O O p e ~ .

So, @;*A = A . Since B0 = B I o we obtain by similar arguments 8I lA = A . Part (a) follows immediately by noting that 8, = 8; and 8-, = O i l for all i 2 1.

ForA E 9 ' w e have Q, E A iff T, v, E A,forall t E R. Consequently, Q, E A iff O1 v, E A . Hence, B i ' A = A and 9 ' cY . Let A €9. For Q, E M " and t E R we can take i E Z such that X,-!(Q,) I; t <X,(v , ) . Then

T , ~ , E A iff B I ( T , p ) E A iff 8 , ( p ~ A iff P E A .

The last equivalence is a consequence of (a). 0

As a consequence of Lemma 2(a) and (34) all 9-measurable functions h : M"+ R satisfy

h o 8 , = h and h o T , = h

for all i E Z and x E R.

A stationary point process @ (or its distribution P) with P(M") = 1 is called ergodic if P ( A ) E ( 0 , l ) for all A E 9' or, equivalently, if E ( f IS') = Ef P-a.s. for all f : Mm+ R with E If I < CD. Po is called ergodic if Po@) E {O, 1) for all A E 9 or, equivalently, if Eo(g 19) = Eog Po-a.s. for all g : M o d R with E o I g I < m. By (29) and Lemma 2(b) it is obvious that

P ergodic iff Po ergodic. (35) This well-known result is also proved in FRANKEN et al. (1982; Th. 1.3.9) and BACCELLI and BREMAUD (1987; p. 28/29) using different arguments.


Recall the definition of N, in (14). Let g : M" - R be P-integrable. By ergodic-type theorems we have

= I , (36) 1 and, if (ti) is Po-stationary (in particular if (ti) is generated by 8 , ) and Eo 1 to I < m,

The following theorem ensures that the convergences in (36)-(38) hold irrespective of the starting point whether the point process is considered from a time point chosen at random or from an occurrence chosen at random. We will write U := Eo(ao 19) and Y := E(Nl 13). In the proof the relations following the proof of Lemma 2 will be used repeatedly.

THEOREM 3 (a). (b). Relations (36) and (37) hold as well with Po instead of P.

is generated by 8 1 and Eo I 40 I < m, then (38) holds as well with Pinstead of P '.

PROOF. Since Po -Po, Relation (38) holds with Po as well. Part (a) follows immediately. For (b), consider

= P [ Q,(O,XI(Q,) + 11 m a > + t + 4 = 1. &(PI + t t

Since PI -Po, the first part of (b) follows. For Q, E M" we have 1 I + X O ( d 1 ,

t o t X O ( d - Sg(TJ80 VIPs = - j g(T, Q,)ds.

By this observation it is obvious that (37) is also valid with Po and thus with Po. 0

REMARKS. It is easy to prove that the events in (36)-(38) are elements of 9. This observation, combined with (29), leads to another proof of Theorem 3 (see BACCELLI and BREMAUD, 1987, p. 29/30 for the ergodic case).

Application of Theorem 3(a) with ti = g o 8 , for Po-integrable functions g : Mo-+ R yields:

50 G. Nieuwenhuis

1 - 1 g o Bi + Eo(g 19) Po- and P-as. n / = I

See also FRANKEN et al. (1982; (1.3.18)) for the ergodic case. By conditioning on 3 we obtain

E o [ a ~ l [ ~ , ~ 1 ] = E 0 [ U 1 [ ~ = o l ] = 0.

Since Po[ao = 01 = 0, we have (apply Theorem 1)

U > O Po- and P-as.

Application of Theorem 3(a) with ti = ai yields

(39)

(The last equality holds since

for all v, E M" with N,(v,) > 0. Use (39).) By (39), (30), (36) and Theorem l(a) we have

1 = E (Nl IS) Po- and P-as.

See also NAWROTZKI (1978; Ths. 1.1 and 1.2). Note the resemblance between (41) and Relations (13) and (22).

A similar almost sure limit result for

1 ' I ( t ) : = - I g o T , d s

2 0

can be derived directly from (a) and the first part of (b) of Theorem 3. If N ( t ) > 0, then Z ( t ) can be decomposed as follows:

Note that the sequences (I ; ; - , go T, d s ) and ( 5 ::-, I g o T, I d s ) are both generated by B 1 . By Theorem 3(a), the first part of (b) and (41) we have

Po- and P-as., and


Combining the limit results in (43) and the second part of Theorem 3(b) yields the following theorem. It expresses a conditional version of the inversion formula (19). It can also be proved directly from (19) by conditioning; by using the left-hand part of (17) and (41).

THEOREM 4. For functions g : M" + R with E 1 g I < CQ we have

COROLLARY 5. For functions f : Mo+ R with Eo I f I < m we have

EO(f Iy)=E(Nlls) 1 E (if oB,19) Po- and P-a.s.

PROOF. Since E o ( / i S ) = E o ( - "O 1

a0 0 f o 8 0 o T , d s 1 9 Po-andP-a .s . ,

the result follows immediately from Theorem 4 by taking g = f o 8o/ao and applying (41). 0

The corollary expresses a conditional version of (18). A conditional version of (21) can also be derived. If g:Mm+R is such that Elgl <OD and E ( g I . Y ) = E ( g o 8 0 1 9 ) P- as., then it follows from Theorem 4 that

The choice g = l/ao in (43) and Theorem 4 yield

d s -.A Po- and P-a.s., q- 1

t 0 aooTs (44)

provided that Eo(ao 19) = l / A Po-a.s. This condition is weaker than ergodicity of @; see also Section 4.

4 Approximations of Po

In this section we will consider several expressions tending in some sense to Po as n + 03. For this purpose a notion is introduced which is weaker than ergodicity of 0. Several new intermediate probability measures are defined, all equivalent to Po. The corresponding Radon-Nikodym densities are used to approximate P o starting from P. A formalization of the intuitive random generation of Po (mentioned in Section 1) as a limit result for CBsaro means turns out to be uniform. If the stationary sequence (ai)

52 G. Nieuwenhuis

does not satisfy the strong law of large numbers with a degenerate limit, then intuition seems to be false.

According to FRANKEN et al. (1982; (1.3.21)) the distribution P,, of B,,@ tends to Po pointwise (that is, P,,(A) + Po@) for all A E A'"), if Po is mixing (ergodic-sense), i.e. if

Po([e,,p E A] n c) P O ( A ) P O ( C ) , A , c E A-. (45)

Relation (45) could equivalently be formulated as (cf. NIEUWENHUIS, 1989, Section 5)

POB;' --+ Po pointwise, independently of Am.

We can obtain a stronger result under a weaker condition. The implication ( a ) a (b) of the following theorem ensures that pointwise convergence of POB;' to Po, independently of the o-field o(ao) in A", implies pointwise convergence of P, =PO;' to Po, independently of a(ao). Consequently, P,, tends to Po if hypothesis (a) (which is weaker than (45)) is satisfied. See also MATTHES, KERSTAN and MECKE (1978; Th. 9.4.5) and MIYAZAWA (1977; Th. 3.2').

THEOREM 6. The following statements are equivalent: (a). Po[ao(p) E B and 8,p E A ] + Po[ao E B]Po(A) for all B E 9' and A E do, (b). P[ao(p) E B and B,u, E A ] - + P[ao E B]PO(A) f o r all B E S+ and A E A ' O .

PROOF. Assume (a). For all B E S+ and A E do we have (cf. Theorem 1)

Pbo E BI n [b-J E A1 = ~ ~ o [ ~ o l [ , , . B ] ~ [ B , ~ E A ] l

m

= A j Po([ao > X ] n [a0 E B] n [en p E A ] ) ~ x 0

m

+ A Po[ao > x and a0 E B ] d x P o ( A ) as n co 0

because of (a) and dominated convergence. This limit is equal to

which proves (b). The implication (b)*(a) can be proved the same way. 0

Hypothesis (a) is weaker than mixing property (45); hypothesis (b) could equivalently be formulated as

P,, = P[B,,p E .]-Po pointwise, independently of a(ao).

Next we consider strong approximation ofPo. For n E N the empirical distributionp,, is defined by

. "


Since the sequence ( lAo e i ) is generated by 81, we obtain by (38) and Theorem 3(a) that

p,,(A) + E 0 ( l A IS) Po- and P-as. (47)

Note that for each u, E M" fixed, p,,(-, u,) is a probability measure on (M", A") and that p,,(A) is a Po-unbiased estimator of Po(A). The next statement follows immediately from (35) and (47). It characterizes strong approximation of Po by pn under P.

@ is ergodic iff p,,(A) + Po(A) P-as. for all A E ,H". (48)

Starting with (47) under P we obtain:

1 " EP,,(A) = - C Pj(A) -* EIEo(lA IS)] =: Qo(A), A E A'".

n i = I (49)

Qo is a probability measure on ( M " , A " ) having Qo(Mo) = 1, since E 0 ( l M o I S ) = 1 Po- and P-as. (cf. Th. l(a)). If we consider the limit result in (49) as a formalization of the intuitive procedure for the random generation of Po (see Section l), we might conclude now that intuition was false. Qo and Po need not be the same.

LEMMA 7. Qo and Po are equivalent. The Radon-Nikodym densiiy of Qo with respect to PO is:

PROOF. By Theorem 1 we have:

Qo(A) = IEo[aoE0(lA I 9)] = IEo[Eo(ao I .Y)Eo( lA I .Y)] =dEo[E0(lAEo(ao1.Y) IY)] =IEo[lAEo(ao l , ~ ) ] .

In the second equality we conditioned on 9. Since Po[Eo(ao 13) = 01 = 0 by (39), the El conclusions of the lemma follow immediately.

By (50) we obtain

(5 1) 1 1 Qo = Po iff €'(ao IS) = - Po-as.

If @ is ergodic, then Eo(ao 19) = Eoao = ,I-' Po-a.s. Relation (49) could then be takenas a definition of Po. If, however, @ is not ergodic, then it is possible that Qo #Po.

EXAMPLE 1. A bank has two settlements, the first in the center of a city and the second. in a suburb. The queueing systems of arrivals of customers for the two locations are modelled as renewal processes with uniform (0,l) and uniform (0,2) distributed interarrivals, respectively. Let Pi" be the corresponding Palm distribution and l / L i the mean interarrival time for system i; i = 1,2. The probability measure

PO:=pPP + (1 - p ) &

54 G. Nieuwenhuis

p E (0, l), is a mixture of the two distributions. It describes in a sense the probabilistic characteristics of the arrivals within the bank as a whole. Natural choices for p seem to be Al/(Al + A,) and A;/(Aj + A$). For allp E (0,l) the sequence (ai) is stationary under the mixture, with (2 - p ) / 2 as mean interval length. According to FRANKEN et al. (1982; Th. 1.3.4) it is indeed a Palm distribution (in the sense that there exists exactly one stationary distribution P on (M",A?") such that the above defined mixture is the Palm distribution associated with P). Set B1 := [ai(p) E (0,l) for all i E Z] and B2:= [ai(p) E (0,2) for all i E 23. Note that Po(Bl) = p , Po(B2 n Bf) = 1 - p , and that Bl, B2 n Bt E 9. Since 9 is trivial under Pp and P!, we have

Po[Eo(aolY) = f l s , + I B , , ~ =pPP[Eo(ao19) =f ] + (1 -p)P,"[Eo(ao14) = 11 = p + (1 - p ) = 1.

Consequently, Po is not ergodic and (cf. Lemma 7) Qo # Po, although Pp and P! are ergodic.

DEFINITION 1. A stationary point process # with P[# E M"] = 1 a n d l E (0, w) is called pseudo-ergodic i f Eo(ao 19) = I - ' Po-a.s.

An ergodic point process is pseudo-ergodic. A pseudo-ergodic point process need not be ergodic.

EXAMPLE 2. Consider the mixture of Example 1 if Pp is the Palm distribution of the deterministic arrival process with interarrivals identically equal to one. P! is unchanged. Then Po is again a Palm distribution, and (a i ) is Po-stationary with Eoao = 1. Since A , := [ai(p) = 1 for all i E Z] E 4 and Po(A1) = p $ {O, l), this mixture Po is not ergodic. Note, however, that Eo(ao 19) = lA, + lg2 = 1 Po-a.s. So, a stationary point process which has Po as Palm distribution is pseudo-ergodic and not ergodic.

Since EP,, << Po with Radon-Nikodym density A n - ' ~ y ~ l a_, (see Theorem l), we obtain by (50) that d(EP,,, Qo) = AEo I n-' 1 a-l - Eo(ao 19) I (recall the definition of d in (15)). We want to prove that this last expression tends to 0 as n -, 00.

A sequence (Y,,),,N of integrable r.v.'s is uniformly integrable if

or, equivalently,

sup IEI Ynl = M < w and for every e > O there exists 6>0 n e N

such that for all events A with P ( A ) c 6 we have: sup IEI YnllA <&.

(53)

n e N

If (Yn),,eN is uniformly integrable, then so is (n-' Y,)neN as is obvious by (53). A random sequence with identically distributed elements is uniformly integrable.


Consequently, (n-' 1 1=1 E~ is uniformly Po-integrable. Since a _ , -, Eo(ao 13) Po-a.s., we obtain that d(Epn, Qo) + 0 as n -, m (cf. e.g. Th. T26 in BREMAUD, 1981). We conclude that the convergence in (49) is uniform in A:

1 " sup 1- 1 Pi(A) - Qo(A) I+ 0.

ACE.#" n i = I (54)

The consequences of this observation for the Palm distribution are explained in the next theorem.

THEOREM 7 . For stationary point processes with P(A4") = 1 and I E (0, m) the following statements are equivalent:

(i) yll Pi@) -* Po(A) for all A E J", (ii) ~ U P A E p I tZ y-1 PiM) + ~ ~ ( 4 I -, 0,

(iii) 0 is pseudo-ergodic, (iv) P ~ [ A ~ ? = , a i+f ] = 1 , (v) P[ jN , - , I ] = 1 ,

(vi) PO = P on 4.

PROOF. Relations (49), (54), (S l ) , and (30) imply (i) e. (ii), (i) e. (iii), and (iii) e (vi). The equivalence of (iii) and (iv) is an immediate consequence of Birkhoffs ergodic theorem. The implication (iv) - (v) is a corollary of Theorem 3(a) and observations as in (40), with U replaced by I - ' . Theorem 3(b) and

yield the implication (v) - (iv). 0

An important conclusion from Theorem 7 is that it is not always correct to define Po as the limit of n-' ?= I Pi, attractive as it may be. This is also discussed in NAWROTZKI (1978). It is, however, possible to obtain Po(A) as another limit without any restraint, uniformly in A E Am. Note that

P O ( A ) = E O [ E O ( ~ ~ I S ) ] = I - ' E [ ~ ; ' E ' ( I ~ IS) ] = I - ' E [ E ( ~ ; ' ~ S ) E ~ ( ~ , ~ Y ) ] =I - 'E[E0( lAE(a; ' IY)IY)]. (55)

Since the sequence (I- l lA(Oi.)E(a;' lY))iez is generated by B , , we obtain by Theorem 3(a) that

so (cf. (55)) ,

Q,(A) := i E [. ( I Y) pn(A)]+ Po@), A E A'". a0

(57)

56 G. Nieuwenhuis

By Relation (22) Q, is a probability measure. By Theorem 1, (41), (1 l), and the observation following the proof of Lemma 2 we have

Hence, Q, -Po and

For B E 9' we have for k E Z.

So, the random sequence (a-i/Eo(ao 19)) is identically Po-distributed and hence (n-' a-i/Eo(ao IS)),,, is uniformly Po-integrable (cf. the arguments preceding (54)). By (58) it is obvious that the convergence in (57) is uniform in A E Am. Note that Q, = n-l 1

According to (48) the sequence (P,,), considered as a sequence of estimators of Po, is strongly P-consistent iff # is ergodic. By Theorem 7 it is asymptotically P-unbiased iff 0 is pseudo-ergodic. It is an easy exercise to prove that E ($,(A) - Po@))', the mean squared error under P, tends to 0 iff # is ergodic.

In the next theorem we examine for sequences (ti) generated by d l the asymptotic P-unbiasedness of the estimator n-l

P, = EP, iff # is pseudo-ergodic.

5, of ,?'to.

THEOREM 8. Suppose that (ti) is generated by 8 , and that Eoai v E o t i < 03. Then . "

If # is pseudo-ergodic, then-n-' t i is asymptotically P-unbiasedfor E'to.

PROOF. By Theorem 1 we have

Since

E o ~ a o t n I l [ l a , c , l > a ~ ~ ~ o I a o t n ~ l [ a ~ > a ~ +EO~aotn I 1[e:>a]

5 ~ ~ o " . ~ ~ ~ a ~ ~ a ~ I ~ o t i ~ 1 ' 2 ~ ~ o ~ t i ~ ~ ~ ~ ~ a ~ I ~ o ~ i ~ 1 ' z and since this upper bound tends to zero as a + 03, it is obvious that (ao tn )nEW and (n-' I,"=, a. are uniformly Po-integrable (see (52) and the arguments following (53)). Note also that by (38),

i n


By (60) and BRBMAUD (1981; T26) Relation (59) follows immediately. The limit in (59) is equal to

1 1 E o [ E o ( a ~ 1 9 ) E o ( ~ ~ 1 ~ ) 1 = l l E o [ t ~ E o b ~ l y ) 131 = E @ ( < O ) .

By (51) the last part of the theorem follows. 0

COROLLARY 9. Suppose that E0a$ < 03. The estimator n - I z : = , ai of E o a o =A-' is asymptotically P-unbiased iff 0 is pseudo-ergodic.

PROOF. The if-part is a consequence of Theorem 8. If n-' xr=l a, is asymptotically P-unbiased, then we obtain by (59) that Eo[aoEo(ao I Y)] = (E0ao)*. Consequently,

0

A stationary point process associated with Po in Example 1 satisfies A-I = 1 - f p and E o ( a o IS) = f l ~ , + l ~ , n B ; Po-a.s. SO (cf. (59)),

4 - 3p

VarpoEo(ao19) = O and E o ( a o 1 9 ) = 1/11 Po-a.s.

2E0(E0(ao I S))2 -- - 1 " - 1 E a , + n r = l 2 - P 4-2p'

This limit is indeed not equal to 1-' = 1 - f p .

n E N, defined by It is well-known that P o can also be approximated by the probability measures

P1,,(A):=P B l p EAIX1(p)5; , A E d # - . (61) [ I 1 In FRANKEN et al. (1982; Th. 1.3.7) it is proved that Pl, ,(A) tends to P o ( A ) uniformly in A, or equivalently, d(PlSn, Po) -, 0 as n - co. We will, however, express d(P1,, , Po) in terms of F, the distribution function of XI under P.

THEOREM 10. L e t ' 0 be a stationary point process wirh P(M") = 1 and 1 E (0, 03). Then

PROOF. By (12) we obtain

Hence PI,"(A) = l l E o [ l A ( t ~ c ~ - ~ ) ] / F ( f ) , which proves (i). By (15) it is obvious that d(P1,, , Po) = E o I on - 1 I . We will 'express this Po-expectation in terms of F. First we note that (cf. Theorem 1)

58 G. Nieuwenhuis

Set h(n) := F(l /n) /A . By (62), Theorem 1 and (27) we obtain

Eola , , - l I =-Eo - A a o - h ( n ) htn) 1;

F(h(n) ) + Ah(n) -A/. + l / n - h(n) = 2 - 2 - Ah(n) Ah(n) h(n)

F(h(n) ) M n ) F ( l / n )

F (F ( l / n ) / 4 - 2 - 2 =2-2--

To convergence to 0 follows immediately since F ( x ) =Ax + o ( x ) as x-+ 0, cf. e.g. FRANKEN et al. (1982; Th. 1.2.12). 0

Because of part (ii) it is possible to determine in many situations the rate at which PI," tends to Po. If 0 is a Poisson process, then it is an easy exercise to prove that d(Pl,,,Po) = f F ( t ) + o(F( t ) ) = fA /n + 00) as n - r m . This rate l / n is not universal; it turns out that the renewal process with Po[ao 5 x ] = x ' - ~ for 0 .c x < 1,p E (0, l ) , satisfies d (PI,,,, Po) = cn-( '-P) + o(n-(l-P)) as n -r m. (Here c E (0, w) is some constant, not depending on n.) We can, however, give conditions such that the rate l / n is satisfied.

COROLLARY 11. Set G ( x ) := Po[ao i; x] , x E [0, 03). Suppose that G is differentiable on (0, 8 ) f o r some E > 0 with bounded derivative. Then

PROOF. Set g := G' on (0, &). Recall the definition of F preceding Theorem 10. Because of the continuity of G it is obvious (see (25)) that F is differentiable on ( 0 , ~ ) with F ' = L ( I - G). By the mean value theorem we have for n sufficiently large:


for some I], in the interval (0, F ( l / n ) / l ) . Since F(0) = 0 and F(l/n) 5 1/n, see (62), we obtain by Theorem 10,

Another application of the mean value theorem yields for n sufficiently large:

for some 6, E (0, t). Here c := sup {g(x) : x E (0, E ) ) , not depending on n. 0

Because of (25) the condition in Corollary 1 1 may equivalently be stated in terms of F, the distribution function of X I under P, as follows:

(63) F is twice differentiable on (0, E ) for some E > 0 with bounded second derivative F".

5 Examples and generalization

Many of the results in this paper are beyond ergodicity. To illustrate some of the theorems the nonergodic mixture of Example 1 is reconsidered. A shot-noise process is considered as an illustration of the a(:) rate of convergence in local characterization of the Palm distribution. The results of Sections 1 to 4 can be generalized to marked point processes. This extension is briefly indicated. At the end of this research some remarks are made, especially on simulation applications.

EXAMPLE 3. The Palm distribution P o of the non-pseudo-ergodic point process of Example 1 in Section 4 satisfies

Po[ao 5 tl = P P X Q O 5 tl + ( 1 - ~ > ~ 2 0 [ a o 5 tl = p ( t A 1) + (1 - p ) f t ,

t E ( 0 , 2 ) . Since 1 = 2/ (2 - p ) , we obtain by Theorem 1 for the corresponding stationary distribution P,

Relation (27) yields

p [ x , 5 tl =

1 + P t - - t2

2 - p 4 - 2 p 2 - if O < t < l

- P + - t - - 2 - 2 P l - p t * if 1 < t < 2 . 2 - p 2 - p 4 - 2 p

60 G. Nieuwenhuis

Recall the definition of Qo in (49). Since P is not pseudo-ergodic, we have

I - C Pj(A) + Qo(A), n i = l

A E A?",

where Qo # Po, cf. (49) and (51). The probability measure Qo can easily be expressed in terms of the Palm distributions P/ and Pi. By Lemma 7 we obtain

Qo(A> = l E o ( E o ( a o I y ) l ~ ) = lEo(( f l s , + ls,ns$l~)

A E .Ha, which is a weighted average of PP(A) and @(A) . Especially,Qo(BI) = p / ( 2 - p ) and Qo(B2 fl B;) = (2 - 2 p ) / ( 2 - p ) .

By Corollary I1 it is obvious that the local characterization supremum is @(:). By Theorem lO(ii) and (64) it is, however, an easy exercise to calculate this supremum. It is equal to

Several modern point processes (for instance stationary Markov renewal processes) are defined starting from an arbitrary arrival in the origin, i.e., they are considered under Po. Then it is usually not difficult to check the condition on G in Corollary 11. Cox processes and cluster processes, however, are often considered under P. Checking (63) may take some time.

EXAMPLE 4. Let @ be the Cox process with intensity process A := ( A ( t ) : t E R} defined by

A ( t ) : = 1 Y , f ( t - U,). l € Z

The U,, i E Z, are the (unobserved) occurrences of a stationary Poisson process with intensity cr E (0, a), the sequence (Y,) consists of nonnegative, independent and identically distributed r.v.3, and f is a nonnegative function, vanishing on (- m, 0), with fFf(u)du < m. It is assumed that (U,) and (YJ) are independent and that A ( t ) is finite.

@ is called a shot-noise or trigger process. The term Y , f ( t - U,) can be interpreted as the contribution to A ( t ) by a point process (a cluster) created by a "trigger point" Ui. In VERE-JONES and DAVIES (1966) the above model was used to describe earthquake occurrences. Forf(t) the special choice Qe-P'l[O,m)(t) was made. Here Q is some suitable constant in (0, m). The general shot-noise model is analyzed in DALEY and VERE-JONES (1988; Example 8.5(a)). It is shown that the process can also be regarded as a Neyman- Scott Poisson cluster process. In Exercise 8.5.1 of the last reference the finiteness of A ( t ) is considered.

Assume further that E( Y2) < m (and hence IE Y < m, which guarantees the finiteness of A ( t ) ) , that j ,"f2( t )d t < a, and thatf is differentiable on (0, a) withjr If'(t) I d t < a.


(Note that the exponential function mentioned above satisfies these conditions.) With the help of the probability generating functional of @ (see the above reference) it is an easy exercise to determine po(x) := P[p(O,x] = 01:

Here y(s) := IE(e-"i> defines the Laplace-Stieltjes transform of Yi . Since F(x) = P [ X , 5 x] = 1 - po(x), we can derive F' and F". For x > 0 we obtain

F'b) =-vo(x) -m

For s > 0 we have I y(s) I 5 1, 1 y'(s) I 5 JE Yi and I y"(s) I 5 IE( Y;). These observations, combined with the above assumptions for f, yield the boundedness of F" (see (63)). Hence, the conclusion of Corollary 11 holds.

The results of Sections 1 to 4 can be generalized to marked point processes with (complete and separable) mark space K and hence to stochastic processes with an embedded point process. In these more general situations a mark ki, usually describing the state at time Xi, is attached to the occurrence Xi for all 'i E Z.

Suppose that we are interested in the occurrences with marks in a fixed subset L ofK, the so-called L-points. By replacing B i by Bi ,L , the point shift to the i'th L-point, and I by I @ ) , the intensity of the L-points, we obtain in (10) the Palm distribution Pj of P with respect to L. Intuitively Pf arises from P by shifting the origin to an L-point chosen at ramdom. If, moreover, Xi is replaced by Xi" (the position of the i'th L-point), ai by a! := Xi", I -Xi", and Pi by P,L := PBYL, then all results remain valid. Only some of the proofs need an argument. Especially Theorem 1, the fundamental theorem for this research, generalizes to

In a forthcoming publication some of the generalized results will be considered more thoroughly.

At the end of this section we make some remarks on further research. Stationarity of the point process was the most important condition. It seems natural

to ask how this condition can be relaxed. For non-stationary point processes we have

62 G. Nieuwenhuis

to relate P with a whole family of Palm distributions. Generalization of some of the theorems in this paper seems to be possible under additional conditions.

The approach in this research was based on Theorem 1. It lead to a two-step transition method between Palm distribution and stationary distribution. It seems to be possible to use this procedure for simulation purposes; generating the Palm process when starting with simulations from the stationary process (and vice versa).

In the context of stationary marked point processes a similar two-step procedure can be developed for transitions between two Palm distributions P: and Pi,, where L and L' are disjoint sets of marks. Under some conditions the Palm L-process (with distribution Pi) can be generated when starting with simulations from the Palm L'-process.

Papers on some of these subjects are being prepared.

Acknowledgement

The author would like to thank the referees and the editor for their helpful comments. Especially the detailed suggestions of the editor have greatly improved the paper.

References BACCELLI, F. and P. BRBMAUD (1987), Palm probabilities and stationary queues, Springer, New

York. BRANDT, A., P. FRANKEN and B. LISEK (1990), Stationary stochastic models, Wiley, New York. BRBMAUD, P. (1981), Point processes and queues, Springer, New York. Cox, D. R. and P. A. W. LEWIS (1966), Thestatistical analysis ofseries of events, Chapman and Hall,

London. DALEY, D. J. and D. VERE-JONES (1988), An introduction to the theory ofpoinrprocesses, Springer,

New York. FRANKEN, P., D. KONIG, U. ARNDT and V. SCHMIDT (1982), Queues andpointprocesses, Wiley, New

York. KALLENBERG, 0. (1983), Random measures, 3rd edition, Akademie-Verlag and Academic Press,

New York. MATTHES, K., J. KERSTAN and J. MECKE (1978), Infinitely divisible point processes, Wiley, New

York. MCFADDEN, J. A. (1962), On the lengths of intervals in a stationary point process, Journal of the

Royal Statistical Society 24, 364-382. MECKE, J. (1967), Stationare zufallige Masse und lokalkompakten Abelschen Gruppen, Zeitschrift

fur Wahrscheinlichkeitstheorie und vetwandte Gebiete 34, 199-203. MIYAZAWA, M. (1977), Time and customer processes in queues with stationary inputs, Journal of

Applied Probabifity 14, 349-357. NAWROTZKI, K. (1978), Einige Bemerkungen zur Verwendung der Palmschen Verteilung in der

Bedienungtheorie, Mathematische Operationsforschung und Statistik, Series Optimization 9,

NIEUWENHUIS, G. (1989), Equivalence of functional limit theorems for stationary point processes and their Palm distributions, Probabilio Theory and Related Fields 81, 593-608.

ROLSKI, T. (1981), Stationary random processes associated with point processes, Springer, New York.

VERE-JONES, D. and R. B. DAVIES (19661, A statistical survey of earthquakes in the main seismic region of New Zealand. Part 2, time series analysis, New Zealand Journal of Geology and Geo-

24 1-253.

physics 9, 251-284. Received: August 1991. Revised: July 1992.