The Theory of Stationary Point Processes

A b s t r a c t A n a x i o m a t i c f o r m u l a t i o n is p r e s e n t e d fo r p o i n t p r o c e s s e s w h i c h m a y b e i n t e r p r e t e d as o r d e r e d s e q u e n c e s of p o i n t s r a n d o m l y l o c a t e d o n t h e r e a l l ine . S u c h c o n c e p t s a s f o r w a r d r e c u r r e n c e t i m e s a n d n u m b e r of p o i n t s i n i n t e r v a l s a r e d e f i n e d a n d r e l a t e d i n s e t t h e o r e t i c Q) Presently at Massachusetts Ins t i tu te of Technology, Lexington, Mass., U.S.A. (2) This work was supported by the National Aeronautics and Space Adminis t ra t ion under research gran t NsG-2-59. 11 662901. Aeta mathematica. 116. Imprlm$ le 19 septembre 1966. 160 F. J . B E U T L E R AND O. A. Z. L E N E M A N terms, and conditions established under which these random variables are finite valued. Several types of stationarity are defined, and it is shown tha t these (each requiring a kind of statistical uniformity over the entire real axis) are equivalent to one another. Stationarity does not imply tha t the intervals between points are either independently or identicaUy distributed. Convexity and absolute continuity properties are found for the forward recurrence times of the stat ionary point process (s.p.p.). The moments of the number of points in an interval are described in terms of these distributions, which appear in series whose convergences are necessary and sufficient conditions for the finiteness of the moments. Local and global properties of the moments are related, and it is shown that any existent moment is an absolutely continuous function of the interval length. The distribution functions of the forward recurrences are related to the statistics of the point sequence and the interval times. Moment properties are also determined in terms of the latter. An ergodie theorem relates the behavior of individual realizations of the number of points to their statistical averages. Several classes of point processes are described, and stationarity verified where applicable, using the most convenient of the (equivalent) criteria for each case. The preceding theory is applied to the problem of calculating moments and other process statistics. 1.0. Introduction and Sllmmary A stationary point process, like a recurrent or renewal process, may be interpreted as an ordered sequence of points randomly located on the real line. The stationary point process (hereafter abbreviated s.p.p.) generalizes certain aspects of renewal processes; in particular, the intervals between points on the line need be neither independently nor identically distributed. On the other hand, the s.p.p, is required to retain a certain statistical uniformity over the entire open line, so tha t it more properly becomes a generalization of the equilibrium renewal process (see Cox [3]). The s.p.p, is not only of interest for its own sake, but also leads to applications for which the renewal process is an inadequate model. Included are the examination of randomly t imed modulations of random processes in communication theory, analyses of zero crossings of stochastic processes, and those problems in queue arrivals, traffic flow, etc., whose current behavior depends on past history. Perhaps the most direct motivation for this s tudy is a paper by McFadden [10], whose results are admittedly heuristic. Some of the properties to be demonstrated here have been stated by McFadden for processes of similar nature, and by others for renewal T H E T H E O R Y OF S T A T I O N A R Y P O I N T PROCESSES 161 processes (see Smith [14], Takacs [15], or Cox [3] for a summary). An earlier paper by Wold [16] suggests the same s ta t ionary conditions later utilized by McFadden, but fails to develop the properties consequent to the definition. McFadden's work unfortunately incurs gaps tha t make mandatory the choice of a different structure than his for a rigorous t reatment . He creates a process consisting of randomly located points tn whose indices are "floating"; the index n = 1 refers always to the point immediately to the right of various "arbitrari ly chosen" numbers t. Aside from the question of the meaning of "arbitrari ly chosen", we find that o-sets such as [(o: tn(~o ) ~< x] cannot be specified as probabili ty sets in the accepted sense. Moreover, McFadden erroneously draws the conclusion(1) (of which he makes frequent use) that the intervals between points necessarily constitute a s tat ionary random process [10] [11]. Our program is as follows. We shall define point processes axiomatically, and establish forward and backward recurrence times as well as numbers of points in intervals in terms of the probabili ty space induced by the process {tn}. Stationari ty is then introduced, using forward recurrence times. I t is shown tha t this implies a similar proper ty for the backward recurrence times. Further, McFadden's definition [10], as well as an apparent ly weaker version, are proved equivalent to the preceding stat ionari ty notions. In the second chapter, we derive convexity and absolute continuity properties for the forward recurrence distributions time of the s.p.p. The moments of the number of points in an interval are described in terms of these distributions, which appear in series whose convergence is a necessary and sufficient condition for the finiteness of the moments. Local and global properties of the moments are related; any existent moment is an absolutely continuous function of the interval length. The distribution functions of forward recurrence times are used to obtain interval statistics, and certain results on statistics of the t n. The fina] portion of this chapter presents an ergodic theorem tha t illustrates the rich structure of s.p.p. In the fourth and final chapter, several classes of point processes are analyzed. Stat ionarity is verified for these, and more specialized results obtained. 2.0. Stationary properties for point processes A stochastic point process can be intuitively described in terms of randomly located points on the real axis. Given such a process, one considers such random variables as •(t, x), the number of points falling in the interval (t, t + x], and L.(t), the t ime required (1) This was first pointed out to us by Prof. W. L. Root, who provided a counterexample that constitutes the basis for the generalization to be presented in Section 4.2. 162 F. J . B E U T L E R AND O. A. Z. L E N E M A N for the nth point after t to occur, To be meaningful, however, the process must be enunciated in strictly mathematical terms that translate these intuitive concepts into a rigorous structure. Here, the process is viewed as an ordered, non-decreasing sequence of random variables, {tn}, properly defined on a probability space. When this is done, N(t, x) and Ln(t) are specified in set-theoretic language consistent with their desired intuitive interpretation. To be a fruitful object of study, however, {tn} must be endowed with certain additional properties. For example, in renewal theory only positive indices are considered, and {tn+l-t~} is assumed mutually independent and identically distributed (at least for n = 1, 2 .. . . ) (see again Cox [3]). To us, these assumptions seem unnecessarily restrictive, and lead to an unacceptable model for many problems, including that of random sampling of random processes, which initially motivated this study. McFadden [10] has suggested by his work that the following intuitively appealing assumption should be made: {t~} is a stationary point process (s.p.p.) if the multivariate distribution functions of ~Y(tj, xj), ]=1 , 2, ..., ~Y remains invariant if, with any number h, tj is replaced by tj + h. With only this hypothesis, and occasionally the added assumption that the expected number of points in an interval is finite, we shall be able to obtain all the results which follow. Although our work makes little use of interval statistics [those of ~Y(t, x) and Ln(t ) generally being more convenient], it is advantageous to define the process by beginning with (the intuitive equivalent of) intervals. That is, we let {Vn}, n=O, + l , +2 ... . be a discrete parameter stochastic process associated with a probability space (~, :~, P). We require that each ~n be finite-valued and non-negative with probability one. De/inition 2.0.1. If {v~} is as specified above, and t~ = (2.0.1) [T0--~T~ if n ~ i , {tn} is called a stochastic point process. I t is clear from the definition that, with probability 1, {t~} is an ordered nondeereasing sequence, each of whose members is finite-valued. There is no loss of generality in supposing that these properties of {t~} hold for every o~ 6 ~, and we shall so assume henceforth. We observe also that the a-sub.fields induced on (~, 5, P) by {t=} and {~n} are identical; we may take :~ to be that a-sub-field itself, and treat (s :~, P) as the basic probability space underlying both {tn) and {vn). T H E T H E O R Y O F S T A T I O N A R Y P O I N T P R O C E S S E S 163 All sets and random variables to appear in this paper will be expressed in terms of countable set operations on the "basic building block" sets B.(t), where n is any integer, and t any real number. We define B~(t) = {co: tn(w)~0, n>~ 1, (2.1.1) In where Bk has been defined by (2.0.2) and * denotes complement. Clearly, En(t, x) is the measurable set carrying the intuitive meaning "at least n of the tj fall in (t, t + x ] . " We now take as the definition of L=(t) De]inition 2.1.1. L0(t ) = 0 for all o~ E ~. For n >~ 1, L~(t) is the random variable satisfying {r L,(t) <~x} = E,(t, x), (2.1.2) with L~(t) = ~ on [~ -limx_.r162 En(t, x)]. The sets on the right of (2.1.2) are empty for x~<0 and non-decreasing in x, so the definition makes sense. The interpretation of L~(t) as an nth recurrence time becomes more persuasive when we recall tha t (2.1.2) implies L~(t)=inf~EE~(t,x)x. The set on which L=(t) = ~ appears to be bothersome, but we shall find tha t the stationarity condition renders the set void. For future reference, we give alternate expressions for E,(t, x), as these are more useful in certain derivations. LEMMA 2.1.1. For x>~0, En(t, x) may be represented by En(t,x) = { U [Bin(t) fi B*+l(t) N Bin+n(t§ U [(A B~(t)) N (UBk(t +x))], (2.1.3) m k k in which { } and [ ] are disioint, and the indicated union in { } is itsel] disjoint over m. Another expression ]or En(t, x) is E,(t, x) = { n [B*(t) u Bm+,(t + x)]} N [ U Bk(t + x)] N [ U B*(t)]. (2.1.4) In k k 164 F. J . B E U T L E R AND O. A. Z. L E N E M A N Verification of these equalities is achieved by the usual method of showing that each side of (2.1.3) and (2.1.4) contains the other. That (2.1.3) is a disjoint union as claimed follows from Bm+k(t) n B*+l( t )=0 for k = l , 2, .... Although (2.1.3) appears to be more complicated than (2.1.1), [ ] = 0 if (tn} is an s.p.p. (see Theorems 2.1.1 and 2.2.1), so that En(t, x) is simply represented as a disjoint union. The last form of E~(t, x), (2.1.4), is of interest chiefly because it enables one to use DeMorgan's relations to obtain E*(t,x)=([.J[Bm(t) N B*+n(t § U IN B~(t + x)] IJ [NBk(t)]. (2.1.5) m k k Let us next define sets having the intuitive significance "exactly n of the tk occur in (t, t+x]" . We denote these sets by An(t, x), and define them for n>~0 by An(t, x) = En(t, x) * N En+l(t, x). (2.1.6) To complete the definition of the An, we adopt the convention E0(t, x)= ~. LEM~A 2.1.2. The An are disjoint (/or di//erent indices), and each An can be expressed as the disjoint union An(t ,x)= U[Bm(t) fl B*+l(t) flB,,+n(t+x) * N Bm+n+l(t + x)]. (2,1.7) m Proo/. From (2.1.1), En+lC En, so En+j N E*+I=O for j = l , 2, ...; hence, the An given by (2.1.5) are disjoint as claimed. The rest of the lemma is proved by substituting (2.1.3) and (2.1.5) into (2.1.6), and eliminating intersections of disjoint terms from the multiple unions. We have earlier spoken of a random variable •(t, x) as the "number of points in (t, t § x]." This notion is given formal meaning by De/inition 2.1.2. N(t, x) is the random variable satisfying (oJ: N(t, x) = n} = An(t, x) (2.1.8) for n = 0 , 1, 2, . . , and {~o: N(t, x) = oo } = ~ UAn(t, x). (2.1.9) 0 Most of our work will deal with finite-valued N(t, x), in fact with processes that guarantee N(t, x) to be finite for all t, x. As we would expect, the finiteness of N(t, x) is intimately related to the location of the limit points of {tn}. Since {tn} is non-decreasing, {t,} has exactly two limit points if we admit • The ~o-set such that the upper limit point falls on (t, t+x] is denoted by C~(t, x), and is specified by C~(t, x) = [ U B*(t)] n [ rl B~(t + x)]. (2.1.10) T~IE THEORY OF STATIONXgY POINT P~OCESSES 165 We also define Cl (t, x) = [ n B* (t)] fl [ U Bk(t § x)] (2.1.11) which is associated with the lower limit point in the sense that the set (2.1.11) contains {w: t~ [tk+~(eo) -t], which is to say o~ e limx_, oo E~(f, x) from (2.1.1). Conversely, o~ E En(t, x) TH~ THEORX OF STATION~X PORT PROCESS~.S 167 for some x implies co 6B*(t) for some m, again b y (2.1.1). Then co6 (J B*(t), as we wished to prove. COROLLARX 2.1.1. l im~_~ E~(t, x) is the same set/or each n = l , 2 . . . . . Proo/. The r ight side of (2.1.23) is the same for each such n. COROLLARY 2.1.2. Ln(t ) is (/or any n = 0 , 1, 2 . . . . ) a finite-valued random variable ill P[ U ~B*(t)] = 1. Proo/. The result follows direct ly f rom (2.1.23) and Definit ion 2.1.1. 2.2. Backward recurrence t imes and stationarity Although we have deduced some e lementary propert ies of {tn} on the basis of Definit ion 2.0.1 alone, much s t ronger propert ies accrue f rom some sort of s ta t ionar i ty assumption. McFadden [10] and others [16] have in t roduced such a notion: a process is s t a t ionary if all mul t ivar ia te distr ibutions of N(tl, xl) , /V(t2, x2) . . . . . N(tn, x~) are invar ian t under all shifts t~-->t~+h. This definition is bo th in tui t ively appeal ing and analyt ical ly fruitful. Our definition, or iented as it is toward forward recurrence statistics, seems somewhat different f rom the above, bu t will ac tual ly tu rn out to be equivalent . Definition 2.2.1. The process {tn} is said to be /orward [backward] stationary if for each posi t ive [negative] integer set /cl , k 2 . . . . . k~, and xl, x~ . . . . . x~, and any h, n E n P [ [1 kj(t, x~)] = P [ N Ekj(t + h, x~)], (2.2.1) 1 1 i.e., if the mul t iva r ia te dis t r ibut ion for each set Lk,(t), Lk,(t) . . . . . Lk~(t) is invar ian t under t ranslat ion. I n the above, backwards recurrence t imes L_n(t), n~> 1, are consistent ly specified b y Definit ion 2.1.1, ex tended to all n, with the unders tanding t h a t E-n( t , x) = [J [B*+l(t x) N'Bm+n(t)] , n = 1, 2 . . . . . (2.2.2) m Thus, L_,(t) becomes the t ime in terval between the n th point before t and t itself. I f we combine (2.1.1) wi th (2.2.2) we obta in for n~> 1 {co: L ,(t) <~x} = {co: L , ( t x ) ~ m -1. To show tha t this supposition leads to a contradiction, consider tha t E t ( n 2 m n , 2mn) fl E l ( n 2 m n , 2[m + 1]n) contains Cz(-n, 2n); the latter is demonstrated by representing E 1 and E* respectively by (2.1.3) and (2.1.5), eliminating terms having empty intersection, and comparing with (2.1.11). Hence P [ E ~ ( n + 2 ] n 2 m n , 2mn) N El(-n+2jn-2mn, 2[m+l]n)]>m -1, (2.2.8) in which the left-hand probabili ty is the same for any ~ by the forward stat ionari ty of {t~}. Further, we argue tha t the sets on the left side of (2.2.8) are disjoint for ] = 0, 1 . . . . , m 1 . We use the set identity to be proved as Lemma 2.3.2 (with A o = E * ). Then we need only show tha t E ~ ( n + 2 k n 2 m n , 2mn) N E l ( n + 2 ] n , 2 n ) = O whenever 0 ~ 1; (2.2.9) j=O this is the desired contradiction. To finish the proof, consider C,, -C as specified by (2.I.19). I f we write E ( n , 2 n ) = [(J {B in ( -n ) N B * + I ( n ) } ] [ ' )B~(n), we need to prove tha t P I E ( n , 2n)] = 0. Now E( n , 2n) differs from the monotone limit of E j ( n , 2n) by Cl(--n, 2n), which is contained in a set of probabil i ty zero. Under the assumption of forward stationarity, P [ E ( n , 2 n ) ] = lim P[ Ej( n , 2 n ) ] = lim P[ Ej( n + h , 2 n ) ] = P [ E ( n + h , 2 n ) ] . t t Furthermore, if h>~2n, E ( n , 2n) N E ( n + h , 2 n ) = O because E ( n , 2n)=B~(n) while E ( n + h , 2n)= B * ( n + h ) for some index p. 170 F. J . B E U T L E R AND O. A. Z, L E N E M A N Let us assume that there exists an integer r such that P [ E ( n , 2n) ]>r -1. From the arguments of the preceding paragraph r 1 r 1 P[ U E ( n + 21n, 2n ) ]= ~ P [ E ( n + 2~n, 2 n ) ] = r P [ E ( n , 2 n ) ] > ; t ~0 1 =0 hence P[E( n , 2n)] must be zero, and the proof is complete. (2.2.10) COROLLARY 2.2.1. I / {tn} ks /orward stationary, Ln(t) and N(t, x) are /inite-valued except on a set o/measure zero that does not depend on n, t, or x. More precisely, let A be the set specified in the theorem. Then /or any x>~O and any t, U~=oAk(t, x ) ~ A ; and /or n =0, 1, 2 ..... limy_~r y)~ ~ A . Proo/. From the first statement in the proof of Theorem 2.1.1 [UAk(t, x)]*=E(t , x) which we have just shown to be a subset of A. By (2.1.23), the second statement is equivalent to f3Bm(t)c A. Now C u ( t x , x) = [ U B * ( t x)] n [nBk(t)] c A for all t and x. Taking (monotone) limits on this expression yields [ lim UB*(v)] n [nBk(t)]cA. V---~ oo n k But lim~_,_~ B~(v) = ~ because t o is non-negative, whence n k Bk(t) c A. 2.3. Equivalent stationarity conditions In this section, we prove .that each of several stationarity conditions implies the others. To render the principal theorem more transparent, we mention two set relations that are again intuitively obvious, but require some manipulations for rigorous demonstration. LwMMA 2.3.1. Let O = k o ~ k l <~ks <~ ... <.kn be a set o/ integers, and O=xo<~X~ ~Xs < . ... <<-xn a set o/reals. De/ine m I = k t kj_ 1 and YJ = x t X j r Then n Ak~(t, xj) =jN_I A,~(t + x1_1, y,). (2.3.1) t=1 Proo/. For n = 2 , A~,(t, x~) is expressed as in (2.1.21). We then intersect both sides with Ak,(t, Xl) for the desired result. From what has been proved for n = 2 , it is possible to proceed by induction to complete the verification of the lemma. T H E T H E O R Y O F S T & T I O I ~ A R Y P O I N T P R O C E S S E S 171 L E M ~ i 2.3.2. Ar(t, x) N Ek(t + x, y) = At(t, x) n Ek+~(t, x +y). (2.3.2) Proo/. For k=0 , the result is obvious from E 0 = ~ and Er~Ar. For positive k, we verify instead the equality with Ek+, and Ek replaced by their complements. Now At(t, x) N A~(t + x, y) =Ar(t , x) N Aj+~(t, x + y) from the preceding lemma. If we take the union of both sides on ?" from zero to k 1 the desired equality is attained in view of A~(t, x) ~ Aj(t, x+y) = 0 for j 0. (3.1.1) k 0 Since the sets in the indicated union are disjoint, we obtain the probabili ty on the right as a sum. I f we then sum over m, and interchange (finite) summations, we have S~(x + h) S~(x) = ~ ~. P([A,-k( t , x)] N [Ek(t + x, h)]}. (3.1.2) k = l t = k But for z >~ y ~> 0, there is the set inequality [Aj_k(t, z) n Ek(t + z, h)] = U [Aj_k(t + (z -y), y) n Ek(t + z, h)] (3.1.3) i = k t = k from which (by stationarity and because the A's are disjoint) P[Aj-k(t , z) N Ek(t + z, h)] ~< ~ P[Aj-k(t , y) fi Ek(t + y, h)]. (3.1.4) t = k l = k I f we sum (3.1.4) over/c as in (3.1.2), the result is Sn(Z § h) Sn(z) < Sn(y + h) Sn(y). This corresponds to the usual notion of concavity; take y = x l , z = ( x 1 +x~)/2, h = ( x ~ x l ) / 2 with O <~ xl <<. x 2. Thus, Sn is for each n a monotone bounded concave function on [0, ~ ) , and its properties are precisely those derived in Section 3.18 of Hardy, Littlewood, Polya [7]. In particular, the right derivate s~ (x) =limh_~0+ [S~(x + h) Sn(x)/h] exists for all x > 0, and + if 0 < x < y < z, I S~(z) Sn(y) l -.~ sn (z) [ z y [ . That Sn meets this Lipschitz condition implies tha t S~ is absolutely continuous on any interval [(~, ~ ) , 5 >0, whence S,(x) = S,((~) + s,(u) du (3.1.5) in which s, is any Radon-Nikodym derivative of S,. But sn =s + almost everywhere, and we shall always mean s + when we speak of s~ as the derivative of S,. Then also s~ is monotone non-increasing. We may take ~ -+0+ in (3.1.5), obtaining (since Sn is continuous from the right) THEOREM 3.1.1. S~ is absolutely continuous on any [~, r ~>0 , and its derivative s n may be taken to be monot)ne non-increasing. Further Sn(x) = S~(O + ) + s~(u) du. (3.1.6) Although Sn(O+) need not be finite, s~(x) < ~ for any x > 0 . Then G , = S n S , _ I , n>~2, is absolutely continuous, and we may take gn = s ~ sn-1 for x > 0. This formulation assures 174 F . J . B E U T L E R A N D O. A . Z . LENEMAI~T that gn >/0, and that gn(0 + ) makes sense whenever s,(0 +) < oo. For future use, we state a relation between sums of g~(0 +) and derivates of the Gn and Sn, viz. Lv.MMA 3.1.2. Let sn(O + ) < c~]or each n. Then g~(O + ) = ]]mh--~o+ [Gn(h) G~(0 + )/h ] is weU-de[ined and ]inite, and o o ~ h ~. gn(0 + ) = lira [~gn( )] -lim ~ [Gn(h) Gn(O)/h], (3.1.7) 1 h ~ 0 + 1 h-->0+ 1 where the limits are not required to be ]inite. Proof. From Section 3.18 of [7], [Sn(x+h ) -Sn(x)]/h is non-decreasing in x, h, and n as x '~ , h ~ , n/~. Limits of these variables can then be freely interchanged. Moreover, the limits are finite for each n, so that these quotients and their limits can be expressed in terms of partial sums of G~ and g~. Remarlr If limx..o+Y~[Gn(x)/x] =8, 8 < ~ , Sn(O+ ) =0 and Gn(0+) =0 for each n. By an argument again based on interchanging monotone limits, s~(0+)~ 0 is possible. The answer is negative, and afort iori , S~(0 + ) = 0 and G~(0 + ) = 0 for each n. Consequently, Theorem 3.1.1 and Lemma 3.1.2 (as well as the remark immediately following) may be simplified by the omission of these terms, and Corollary 3.3.1 becomes vacuous. We show by contradiction that G 1 (0 + ) = ~ > 0 is impossible. Indeed, G 1 (0 + ) = implies P[EI(t , x)]/> ~ for each x > 0, and hence P[limx~o+ El(t, x)] ~> ~ because measures are continuous from above ([6], p. 39). Now El(t, n -1) ~ lim~_.o+ El(t, x) for each positive integer n, so that P [ N ~--1 El(t, n-l)] ~> ~. On the other hand, the limit point properties of the realizations {t,(o~)} of the s.p.p, permit us to conclude that 17 ~=1 El(t, n l ) C Ol(t , x). But P[Oz(t, x)] = 0 by Theorem 2.2.1, and the desired contradiction is attained. 3.2. Existence and global properties of moments The number of points in time intervals is of equal interest with recurrence times, and should receive equal attention. The G~ introduced earlier turn out to provide the ideal tools for the study of N(t, x) also. We call p(n, x)=P[A~(t, x)]; this is simply the probability that n occurrences fall in the interval (f, t + x]. I t is then easy to deduce from the earlier set identities that G~(x) =Z~=np(k, x) and the equivalent expression p(n, x )= G~(x)-G~+l(x ). Here all probabilities are zero for x < 0 , and Go(X ) is interpreted as unity. Our first theorem relates local moment properties of N(t, x) to global ones. T H E T H E O R Y O F S T A T I O N A R Y P O I N T P R O C E S S E S 175 TH~,OREM 3.2.1. Suppose E{[N(t, h)] k) is / ini te/or fixed k >~l and some h>0 . Then E([N(t, x)] ~} is ]inite /or all positive x, and in/act E{[N(t, x)] ~} = 0(x ~) as x ~ . Proo/. We may suppose from the hypothesis that E{[N(t, h ) ] k } = M < ~ . First, we recall that N(t, x) is non-decreasing in x for every to whence E{[N(t, x)] ~} ~< E{[N(t, mh)] ~} where we choose m = Ix~h] + 1. Next, we write the identity N(t, mh) = ~ 1N(t + kh, h) from Lemma 2.1.3. Combining these, and using the Minkowski inequality and stationarity, we obtain (E{[N(t, x)]k}) 1/k <<.m(E{[N(t, h)l~}) ~k. We now take both sides to the k power, and note that m k = O(xk). The next result characterizes finiteness of moments in terms of forward recurrence distributions, and even provides an explicit evaluation. THEOREM 3.2.2. For each k>~l, (i) ~{[N(t, x)] ~} (ii) ~. nk[G~(x) Gn+l(x)] n (iii) ~ [n ~ ( n 1) ~] Gn(x) n are equal, whether/inite or in/inite. Proo/. E([N(t, x)] k} =Y_,nkp(n, x), so that (ii) and the series for (i) are equal term-byterm. Moreover, these series (and all others of the theorem) are composed of non-negative summands, so that they converge, if only to infinity. To relate (ii) and (iii), consider their respective partial sums Un and Wn. For these Wn = nkGn+l + U~; (3.2.1) since Un ~< Wn, the proof is completed by showing that lim n k Gn +l(X) = 0 (3.2.2) n ~ follows from the finiteness of (ii). To this end, we use the identity n + p nk[Gn-Gn+,+i]+{ ~ ~k-( i -1)k]Gj-[ (n+p)~-nk]Gn+,+i}=U~+,-U~. (3.2.3) n + l The term in braces is non-negative because Gn+v+ 1 <~ G~ for j ~ n +p. Because U, converges, there is for each (~>0 an n o (not dependent on p) such that n>n o implies 0 4 n~[Gn(x)-G~+p+l(x)] <(~. Then (3.2.2) follows by taking p-~o~, provided that G~-~0. But the latter is precisely (2.1.13), which is a consequence of stationarity (of. Theorem 2.2.1). 12 662901. A c t a mathemat ica . 116. I m p r i m 6 le 19 s e p t e m b r e 1966. 176 F . J . B E U T L E R A N D O. A . Z . L E N E M A N A result of the above theorem is tha t n~Gn --> 0 with n if the kth moment is to be finite. However this condition is not sufficient, as is seen from the fact that Znk-lGn< ~ is both necessary and sufficient. I t is also easy to show tha t moments of all orders exist iff, for some x > 0 , nkG~(x) is bounded for each k = l , 2, ... as n ~ . Finally, we note tha t each te rm of (ii) and (iii) is increasing in x, so tha t convergence for any x > 0 implies uniform convergence on every [0, x0]. The latter also shows that any existent moment is continuous in x, but we shall obtain stronger properties for these moments later. 3.3. First and second moments I t is instructive to consider certain relations between the lower moments and the Gn. Indeed, the existence of the first moment already implies tha t Gn(0 + ) = 0 for all n, and tha t the gn are bounded and have a convergent sum. Knowing this, one returns to the higher moments and elicits properties of more general type. The first theorem of this section specifies completely the functional form of E[N(t, x)]. THWOR~M 3.3.1. I / N(t, X) has a/inite/irst moment, E[N(t, x)] = f x (3.3.1) /or all x >~ O, f being some non-negative constant. Proo/. From stationarity, E[N(t, x)] is a function only of x, say/(x). Now take x, y 1> 0, and write N(t, x+y)=N(t, x)+N(t+x, y) which follows from Lemma 2.1.3. Taking the expectation of both sides of (3.3.2) yields the functional equation /(x+y)=/(x)+/(y). Since ] is bounded on any subset of the positive real axis, there is only one solution (cf. [7], p. 96), namely /(x)=fix. Here fl is non-negative because N(t, x) is non-negative for every eo E ~. The theorem which follows asserts another condition from which the existence of the first moment of N(t, x) can be deduced, and the parameter f calculated. THEOREM 3.3.2. The conditions E[N(t, x)] < oo (3.3.2) and lira ~ [Gn(x)/x] < oo (3.3.3) x--~O+ 1 each imply the other. In either event, Gn(O + ) =0, n = 1, 2, ..., and