Lp-intensities of random measures

Stochastic Processes and their Applications 9 (1979) 155-1611 @ North-Holland Publishing Company

Olav KALLENBERG Department of Mathematics, G6teborg University and Chalmers University of Technology.,S-412 94 Giiteborg, Sweden (Received 12 April 1978, Revised, 14 February 1979) Given a random measure q and a fixed number p > 1, the &-intensity ll~& of rl is defined as the total variation measure of the subadditive set function I[Q(. )jI,. It is shown that IIT&,can exist (be locally finite) only if the usual intensity measure Eq exists and q
1. Main results

$

Let q be a random measure on some locally compact sec:ond countable space S, and let ]I lip be the norm in L,(B), p 3 1, B being the measure on the underlying; probability space. Writirlg 3 for the class of bounded Bore1 sets in S, it is seen that the set function l]q811P,B E a, is subadditive. As such it has an associated total variation measure IIT& which may be defined as the least additive upper bound of ]lq( . )ll,,. An alternative approach is to introduce, for each set B E 98, a null-array of nested partitions {Bnj}c 3 of B (cf. [2]), and define l

(1)

IldlBB= ii& c IlrlBflillP i l

(See e.g. [lo, 00 10-151, for the classical notions of a subadditive set function on the real line and its total variation,) We shall call !lqllP the L,-intensity (measure) of q, since it generalizes the usual intensity measure ET = ilqlli. The &-intensity was first introduced and found useful in [5]. It will presently be seen that, when p > 1,llq lip can only exist (in the sense of being locally finite) if 7 is a.s. absolutely continuous (to be written <<)with respect to some fixed measure p (which may then be taken to be q), and that ]]T$, may then be dom process. lMor expressed in terms of the corresponding dens letting Jt be the class of locally finite measures with p-density f, we have 155

0, Kallenberg / Random measures

1%

1. Then l~v&,exists iff q (( Er, = p E JddCM.with a density Y Theorem 1. Let p ZZ+ satisfying 11 Yllpi EA, and in that case

IIZIIP= IsQ4a= IIYIIPP’

(2)

a implies that 79
t process 6 wdthoutfixed Theorem 2. Let 7 be the condition intensityof a simple atom and satisfying (C), and let p 1, The,y the set func s llP[&B= 1 IBc(& snd IIE[(BIBc(‘&,,B E SRI, are both subadditiue ryjithtotal uarirztionllqllp,

Another useful notion introduced in connection with flat processes and systems of non-interacting particles is that of local I&invariance [S]. A measure p on .R is said to be locallyinvariant, if for any bounded interval I, vq 3 h-’

II

J~(x”h,x)--II,(x,xfh))dx=O.

(4)

SimilarE,y, a random measure v on R is said to be locaffy L,-inwiant, if llsllpexists and moreover Fy -B h-l II Ilrl(x- h, x) - 7(x, x + h)II, dx = 0

(5)

for bounded ktervals I, For p > 1, Theorem 1 above leads to a simple criterion: Let q be a random measure orz ii and let p > 1. Then q is locally 7j is locally invariant.

0. Kalenberg / Random measures

157

An analogous result holds for random measures on R’ with arbitrary d. Note that the statement of the theorem is false for p = 1,

We shall write 11e 11instead of 11aIlawhen there is no risk for confusion. Proof st Theorem I. Let us first suppose that q = YEAwith Idyll@ q=i C I -B_’ we get by Fubini’s theorem an Hkblder’sinequality, l~~Yll;=E(pY)p==Elr.Y(~Y)‘=’ = Ep( Y(pY)“‘) = p,E(

Y(p+Y)“=‘)

~~ll~ll,llcr~>“=‘ll,~ =!4~ll,ll~~ll~‘“’ If 0~ /pY& COO,we may divide throughout by ;;he second factor on the ri ue when llkVll=goReplacin To prove the converse inequality, let {&J c: be a null-array of nested of some .I3E 98, and write B,(s) for the unique partitioning set B,,, containing so Then we get for (CLx P)-almost every (9, w)

and by applying Fatou’s lemma twice, we obtain first

and then

as desired, This completes the proof of (2). Next suppose that II& exists, and fx B E 99 with &3 B 0. Putting Y,,(s, o) = @,, (8)/&B,,(s), we get by Holder’s inequality E

IB

( Yn(s, o))l+“qp(ds) = 2: al8,~(n&J~~~j)‘/” i

irn since the p-integral on the left is non- decreasing, it follows that {Y,(s, w)} is uniformly integrable on (B, p) for almost Ieveryis).Since {I?,,}is furiher a ma;rtingale

0. Kallenberg / Random measures

on (B, p/pB), the limit Y exists a.e. ~1x P, and we get

J

Y (s, w)p (ds) = lim ,#-+a

B

J B

Y,(s)p(ds) = q(B, o)

as.

Thus 7 = Yp a.s. (cf. Exercise IV-53 in [7] or Corollary (20.57) in Cl]). Proof of Theorem 2.

To prove the subadditivity of PI(B) 5 11 {Bi} c 3 be a disjoint partition of B E 3, and conclude fro Jensen’s inequalities that

= C lIE[P[SBi = 1IBi”Sl IB’~lll

The argument for p*(B) = IIE[~BIB’[]ll is similar. By Theorem 4.2 in [4] and Jensen’s inequality PI(B)

e 142U3)

= llEhB

IB’d!

s

IIrlBII 6 IhllB

(6)

9

so the total variations of pl and JUTare bounded bY lIdI* To prove the converse inequalities, let the random measure or process w be defined as in [4], and write qnj = P[@nj= 1IB”,#l l{@nj = 0) l

9

where {Bnj) c 93 is a null-array of nested partitions of B. It may then be seen from the proof of Theorem 4.1 in [4] that ci qnj T (q - v&B, (for the monotonicity, cf. [8, Proposition 2 11). Hence by monotone convergence and Minkowski’s inequality

s lim inf Cj IIqnjII s lim C pl(Bnj) n~oO tl+OOj (7) where the integrals on the right denote the total variations on B of ~1 and ~2. In view of (6), it suffices to prove that 7 is a.s. diffuse on B whenever JBdNr < 00. But under this assumption, (7) yields 11~- r(IIB c 00,so by Theorem 1, (q-r&sEq=E[

a.s.onB,

which shows that 7 - r& is as. diffuse on B. Suppose that (vB > 0) >B0. By 2,l in [4], we may then choose a set I E 9? A B such tha s) = [I = 1 I I’(] > 0, &I\(s)) = 0) c E{s}=g

= 1 I I’~]>O}

Theorem

159

0. Kallenberg 1 Random measures

have positive probability. By Theorem 2.2 in [4], (2) implies that a.s. ?r, c 1 for all SE& so [~I=OlI’[]>O

a.s.onA,

(8)

(cf. [4, eq. (7), p. 2081). On the other hand. el>O

a.s.onA’,

because (W- &I

(9)

= 0 a.s. Since PA > 0, (8) and (9) yield the contradiction

(A’ n {[I = 0)) =

[SI =01I’~];A’]H3[

[~I=OII=~];A]>O,

so we have in fact rB = 0 a.s. Proof of Theorem 3.

Since by Jensen’s inequality IEa I s IlaII for any integrable random variable CM, it is seen that (5) implies (4) with p = Eq s llqll,provided that the latter measure exists. Suppose conversely that llqllexists and that p = ET is locally invariant. Then Theorem 1 yields q = Yp for some Y with IIYllp E .&. To prove that (4) implies (5) in this case, approximate Y by a random step function with fixed discontinuities.

3. Concluding remarks

Theorems 1 and 3 above enable us to simplify the statements of all results in [S] involving the notions of Lz=+intensityand local Lz-invariance, (i.e. the L2 versions of Theorems 4.1,4.2,4.3,5.2, and 6.2 in Es]).For a corresponding simplification of the proofs, one would need an Lz-stationary version of the density Y, the existence of which must be proved first, (cf. the remark following the proof of Theorem 4.2 in [S]). To illustrate the usefulness of L,-intensities in connection with certain particle systems, let 6 be a stationary and ergodic process of points on ‘the real line R with associated real marks, and interpret the points and marks as the positions and velocities of particles on R. Assume that the conditional intensity q of [ is such that IIT&,exists for some p > 1. Let {T,, t 3 0) be the flow according to which the particles move independently and with fixed velocities. Introduce for each I > 0 a random variable & which is independent of 5 and uniformly distributed Ion [0, I]. Then {qTirl } is relatively compact in distribution as r + 00 (cf. [2]), and every limit 5 is clearly stationary both in space and time. Since IlqT,’ II= l]qlland therefore IlqT~r’11=S Ilqll,we get 11~11 G llvllby an obvious Fatou type lemma, and hence 5 << Theorem 1 and [4, Theorem 4.21. If the velocities of 5 are assumed to be a.s. distinct, Et must attribute mass zero to every constant velocity set in the phase space, and therefore C must have the same property as. Thus it follows by Theorem 3.2 in [3] that l is a.s. invariant in space (and hence in time also, cf. [3, Lemma 2.21). Proceeding as in the proof of Theorem 4.1 [S] and in the remark fo~~~~i Theorem 4.2 in [S], we may next infer from the ergodic theorem t

0. Kallertberg/ Random measures

160

the spatial average of q. But since 6 is ergodic, so is 7. In fact, VJis, a.s. diffuse and hence solves the integral equation in [4,§ 31. Thus, for each B, @ equals the density at 6 of the reduced Campbell measure of 5 with Irespect to the distribution, and both being invariant, it follows that qTF1 =g(&TT* ) for some function g. Thus every invariant q-event has :an invariant g-preimage, and hence has probability 0 or 1, as asserted. The ergo icity of q implies that a.s. i’ = Eq = Et. Since this holds for all Emits [, we may conclude that qT$’ 2 ES, and since qTG,* is clearly uniformly integrable, this maJ be strengthened to L 1-convergence. Arguing as at the beginning of fi 6 in [5] (see also the proof of Theorem 3.2in [6]), we may next conclude that, for any sequence of r-values which goes to infinity rapidly enough,

qT,’ -, Et Writing

i

in L1 for a.e. t E [0,

1]

.

for a Poisson process with intensity Et, we get by [4, Theorem 5.21

(Th-8

for a.e. t E [0, l]

.

If pr and p denote the distributions of [T,' and i respectively, iwe thus obtain by dominated convergence, for any bounded and continuous function f,

first along the specific r-sequences, and then in general since the 'imitisfixed.~hus

tT,'+d i in the mean, in the sense of [6]. (See [6] for a fuller discussion of the notion of mean convergence.)

Acknowledgment I would like to express my sincere thanks to Fredos Papangelou for drawing my attention to the classical theory of total variation of subadditive: functions, and for suggesting the present proof of Theorem 1 (which replaces my original lengthy but more elementary argument).

eferences [l] E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer, New York, 1965). [2] 0. Kallenberg, Random Measures (Akademie-Verlag and Academic Pres::,Berlin-London, 197% 76). ’ [3) 0. Knllenberg, en the structure of stationary flat processes, Z. Wahrscneinlichkei;stheorie und Verw. Gebiete 37 (1976) 157-174.

0. Kallenberg / Random measures [4] 0. Kallenberg, On conditional intensities of point processes, Z. Wahrscheinlichkeitstheorie

[S] [6] [7] [8] [9] [IO]

161

und Vzrw. Gebiete 41 (1978) 205-220. 0. Kallenberg, On the asymptotic behavior of line processes and systems of non-interacting particles, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete ~:3 (1978) 65-95. 0. Kallenberg, Convsrgence of non-ergodic dynamical systems (1978) preprint. J. Neveu, Bases Mathematiques du Calcul des Probabilites (Masson, Paris, 1964). F. Papangelou, The conditional intensity of general point processes and an application to line processes, Z. Wahrscheinlichkeitstheorie und Vera. Gebiete 28 (19743 207-226. F. Papangelou, Some stationary random sets in homogeneous spaces, Bull. Inst. Internat. Statist. 46 (1975) 6 16-622,628-630. F. Riesz and B. Sz-Nagy, Functionai Analysis (Frederick Ungar, New York, 1955).