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

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<ul><li><p>37 </p><p>Statistica Neerlandica (1994) Vol. 48, nr. 1, pp. 37-62 </p><p>Bridging the gap between a stationary point process and its Palm distribution </p><p>G. Nieuwenhuis Tilburg University, Department of Econometrics </p><p>P.O. Box 90753, 5000 LE Tilburg, The Netherlands </p><p>In the context of stationary point processes measurements are usually made from a time point chosen at random or from an occurrence chosen at ran- dom. 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 mea- sures 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 con- ditional 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. </p><p>Key Words 81 Phrases: Radon-Nikodym derivative, local characterization, inversion formula, ergodicity. </p><p>1 Introduction </p><p>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 under- standing of the relationship between P and Po. </p><p>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). </p></li><li><p>38 G. Nieuwenhuis </p><p>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. </p><p>It will always be assumed that the point process is stationary (i.e. has the same distri- bution 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: </p><p>... </p></li><li><p>Stationary point process and its Palm distribution 39 </p><p>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 </p><p>X </p><p>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, </p><p>If a2 &gt; 0 (i.e., if the point process is not deterministic), then E(ao /p ) &gt; 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: </p><p>A formal proof of (4) is given in Section 2. By multiple integration we derive from (4) that </p><p>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) &gt; 1). By (4) it is obvious that (5) can be generalized. In'fact we have, for functions h, and integers i(l), ..., i (k ) , </p></li><li><p>40 G. Nieuwenhuis </p><p>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: </p><p>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: </p><p>(i) Change the importance of the realizations ofthe point process by way of the weight function a o / g </p><p>(i) Change the importance of the realizations of the point process by way of the weight function p/ao. </p><p>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, com- bined with (7), describes the relationship between Po and Po mentioned in Theorem 1 below, in the case that n = 0. </p><p>It is intuitively clear how transitions from P to Po or from Po to P should be effected. See (ii) and (ii), respectively: </p><p>(ii) Shift the origin to Xo. (ii) Shift the origin to a time point which is uniformly chosen in [ X o , X l ) . </p><p>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 nth 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). </p><p>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. </p><p>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. </p><p>Starting from the stationary distribution, some strong or pointwise approximations of </p></li><li><p>Stationary point process and its Palm distribution 41 </p><p>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 con- vergence is of order l /n . Some examples are considered in Section 5. A generalization to marked point processes is briefly sketched. </p><p>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><p>p(B) &lt; 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 </p><p>M':= {p E M: p(- m,O) = p(0, a) = CD; p{x} I, 1 for all x E R}, r : = ~ " n ~ . </p><p>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><p>P(M") = 1 and I:= IE@(O, 13 &lt; m. (8) </p><p>- 0 . &lt; X - * ( d &lt; XO(d 5 0 &lt; X,b&gt; E B ] , B E 9. </p><p>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 </p></li><li><p>42 G. Nieuwenhuis </p><p>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 impor- tant role in this paper. It arises from P by shifting the origin to the n'th arrival. Hence, </p><p>P,:=PB;l, n E Z. (9) </p><p>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 </p><p>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: </p><p>po=poe;l for all n E Z. (11) Consequently, any sequence (ti) generated by B1 is Po-stationary, i.e. (tl, ...,en) and </p><p>..., tk+,) have the same distribution under Po, all n E N and k E Z. In particular, (ai) is Po-stationary. </p><p>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). </p><p>a </p><p>P(A) =,I P0"x,(p) &gt; U ; p(u + *) E A] du, A E A. (12) 0 </p><p>Substituting A = M yields </p><p>1 p = Eoao = 7 </p><p>A </p><p>For p E M we define </p><p>We will sometimes write N ( t ) instead of N,. Let Q, and Q2 be probability measures on a common measurable space. We say that </p><p>Q1 is dominated by Qz (notation QI </p></li><li><p>Stationary point process and its Palm distribution 43 </p><p>h2 respectively, is defined by </p><p>d(Qi, Q2):=j I hi - h2 I do . </p><p>d(Ql ,Q2)=2su~ I Q I (A) -Q~(A) I =2(Qi[hi&gt;hzI-Q2[hi 2h21). </p><p>(15) </p><p>(16) </p><p>Expectations with respect to the probability measures P, P,, and Po, all considered on (Ma,,&amp;, 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,&amp;), 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. </p><p>It is well-known that </p><p>A </p><p>2 Intermediate probability measures </p><p>The set of realizations with an occurrence in 0 has probability zero under P and prob- ability 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. </p><p>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 nth 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. </p><p>THEOREM 1. Let n E Z. Then (a). P,, - PO, (b). Q,,(v):= Aa-,,(p)...</p></li></ul>


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