The standard functional central limit theorem for a renewal process with
finite mean and variance, results in a Brownian motion limit. This note shows
how to obtain a Brownian bridge process by a direct procedure that does not
involve conditioning. Several examples are also considered.Comment: 7 page

Foss, Sergey

Konstantopoulos, Takis

English

arXiv

The standard functional central limit theorem for a renewal process with
finite mean and variance, results in a Brownian motion limit. This note shows
how to obtain a Brownian bridge process by a direct procedure that does not
involve conditioning. Several examples are also considered

University of Liverpool Repository

arXiv:0708.3667v1  [math.PR]  27 Aug 2007A note on the convergence of renewal andregenerative processes to a Brownian bridgeSerguei Foss Takis Konstantopoulos12 August 2007AbstractThe standard functional central limit theorem for a renewal process with finite meanand variance, results in a Brownian motion limit. This note shows how to obtain a Brow-nian bridge process by a direct procedure that does not involve conditioning. Severalexamples are also considered.Keywords and phrases. Renewal process, Brownian bridge, functional central limittheorem, weak convergenceAMS 2000 subject classifications. Primary 60F17; secondary 60K05,60G501 The basic theoremIn proving convergence results for a stochastic ordered graph on the integers [2], we noticedthat one can obtain a Donsker-like theorem for Brownian bridge in a somewhat non-standardmanner. The result appears to be new. As it may be of potential interest in some relatedareas (statistics, large deviations), we summarise it in this short note.Consider a (possibly delayed) renewal process on [0,∞) with renewal epochs0 < R1 < R2 < · · · .We assume that {Rn+1 −Rn}n≥1 are i.i.d. with mean µ and variance σ2, both finite. LetAt := #{n ≥ 1 : Rn ≤ t}be the associated counting process. The standard functional central limit theorem for arenewal process, see, e.g., [1], states that the sequence of processes ξ1, ξ2, . . ., whereξn(t) :=Ant − µ−1nt√n, t ≥ 0,converges weakly, as n → ∞, to µ−3/2σW , where W is a standard Brownian motion on[0,∞). Weak convergence (denoted by ⇒ below) means weak convergence of probability1measures on the space D[0,∞) of functions which are right continuous with left limits,equipped with the usual Skorokhod topology (see, e.g., [3], [7]).A standard Brownian bridge [3, p. 84] W 0 is defined, in distribution, as a standard Brow-nian motion W on [0, 1], conditional on W1 = 0, i.e. as the weak limit of the sequence ofprobability measuresP (W ∈ · | 0 ≤W1 ≤ 1/n), n ∈ N,as n → ∞. Often, when Brownian bridge is obtained as a limit by a functional centrallimit theorem, there is an explicit underlying conditioning that takes place. One first provesconvergence to a Brownian motion and uses conditioning to prove convergence to a Brownianbridge. Brownian bridges appear in limits of urn processes, and also in limits of empiricaldistributions [3, Thm. 13.1].In this note we remark that it is possible to obtain a Brownian bridge from a renewal process,without the use of conditioning.Theorem 1. Define, for u > 0,ηu(t) :=R[tAu] − tu√u, 0 ≤ t ≤ 1.Considering ηu as a random element of D[0, 1] (equipped with the topology of uniform con-vergence on compacta), we haveηu ⇒ µ−1/2σW 0, as u→∞,where W 0 is a standard Brownian bridge.Here, [x] denotes the largest integer not exceeding the real number x. We remark that RAuis “close” to u, in the sense that RAu ≤ u < R1+Au . In fact, the difference u−RAu (knownas the age of the renewal process) is a tight family (over u ≥ 0) of random variables. In theabove theorem, we just introduce another parameter, t, and measure the difference betweentu and R[tAu]. When t = 0 or 1, this difference is “negligible” with respect to any power ofu. When t is between 0 and 1, then the difference is of the “order of√u” in the sense thatwhen divided by√u it converges to a normal random variable. Jointly, over all t ∈ [0, 1],we have convergence to a Brownian bridge, and this is what we show next.Proof. Consider, for u > 0,yu(t) :=R[tu] − µtu√u, t ≥ 0.From Donsker’s theorem [3] for the random walk {Rn} we have that yu ⇒ σW , where W isa standard Brownian motion. Define also, for u > 0,ϕu(t) :=tAuu.From the law of large numbers for the renewal process, Au/u→ µ−1, a.s., as u→∞. Hence,ϕu converges a.s. (and weakly) to the deterministic process {µ−1t}. Since composition is acontinuous function [3] we have that{(yu ◦ ϕu)(t)} ⇒ {σWµ−1t} d= {µ−1/2σWt}. (1)2We also have(yu ◦ ϕu)(t) =R[tAu] − µtAu√u,and soηu(t) = (yu ◦ ϕu)(t) + µtAu − µ−1u√u= (yu ◦ ϕu)(t)− t(yu ◦ ϕu)(1) − tu−RAu√u. (2)Observe now that {u−RAu , u ≥ 0} is a tight family. Indeed, from standard renewal theory(see, e.g., [1]), if R1 has a non-lattice distribution, then u−RAu converges weakly as u→∞.And if R1 has a lattice distribution with span h, then a similar convergence takes places fornh−RAnh as n→∞. Since, for all u ≥ 0, 0 ≤ u−RAu ≤ ([u/h] + 1)h−RA[u/h], the family{u − RAu , u ≥ 0} is tight even in the lattice case. Tightness implies that the last term of(2) converges to 0 in probability. From the convergence stated in (1) and the decomposition(2), we have that{ηu(t)}0≤t≤1 ⇒ µ−1/2σ{Wt − tW1}0≤t≤1.It is well known [4] that a standard Brownian bridge W 0 can be represented as W 0t =Wt − tW1, and so the process above is the limit we were looking for.2 Extensions, discussion, and examplesHere is a different version that, perhaps, makes Theorem 1 clearer: Suppose that M isa regenerative random measure on [0,∞). That is, there is some renewal process withpoints T0 < T1 < T2 < · · · such that the random measures obtained by restricting M onto[Tn, Tn+1), n = 0, 1, 2, . . . , are i.i.d. Suppose thatµ := E(T2 − T1), var(T2 − T1) <∞,α := EM([T1, T2)), 0 < varM([T1, T2)) <∞.Define the random distribution function of M byS(t) =M((0, t]), u ≥ 0.By the law of large numbers, S(t)/t → µ−1α, a.s. as t → ∞. Consider the generalisedinverseS−1(u) := inf{t ≥ 0 : S(t) > u}, u ≥ 0.Then, in some naive sense, S−1 composed with S is “approximately” the identity function,but what can we say about the composition of S−1 with a fraction tS of S where 0 < t < 1?The law of large numbers tells us that, almost surely,S(tS−1(u))u−−−→u→∞t.An extension of the previous theorem quantifies the deviation:3Theorem 2. As u→∞, the sequence of processes ηu whereηu(t) :=S(tS−1(u))− tu√u, 0 ≤ t ≤ 1,converges weakly to a Brownian bridge.The proof of this is analogous to the previous one, so it is omitted. Observe that the “tyingdown” of the Brownian motion occurs naturally at t = 0 and t = 1.The Brownian bridge has a scaling constant depending on the parameters of the process S.Note that the regenerative assumption is not crucial. All we need is to have a processfor which a Donsker theorem with a Brownian limit holds. This is then translatable to aBrownian bridge limit.If we interchange the roles of S and S−1 we still get a Brownian bridge but with differentconstant. For instance, interchanging the roles of {Rn} and {Au} in Theorem 1 we obtainthatη′n(t) :=A(tRn)− tn√n, 0 ≤ t ≤ 1,converges weakly, as n → ∞, to κW 0, where W 0 is a standard Brownian bridge and κ =σµ−1.2.1 An interpretationTo better understand the phenomenon, we cast the limit theorem as follows: We have arandom function S, composed with scaling functionsρt : x 7→ txand composed again with the inverse function S−1 and we look at the asymptotic behaviourof the family of random functionsS◦ρt◦S−1 − ρt, 0 ≤ t ≤ 1, (3)(or of S−1◦ρt◦S), as a function of the parameter t. Thus, the time parameter of the Brownianbridge obtained in the limit plays the role of a scaling factor. When t is 0 or 1, S◦ρt◦S−1−ρtis approximately zero (with respect to the normalising factor). This raises the following threequestions:(i) How much “one-dimensional” is this phenomenon?(ii) Can we replace the family ρt by a more general homotopy?(iii) Are different kind of bridges possible to obtain?With respect to the latter question, we could start with a regenerative process with finitemean but infinite variance, one that belongs to the domain of attraction of, say, a self-similarLe´vy process.42.2 Four examplesEXAMPLE 1 The first is a simple example involving a standard Brownian motion W .Let X denote the (strong) Markov processXt = (Wt − t)− min0≤s≤t(Ws − s), t ≥ 0, (4)which is the reflection of the drifted Brownian motion {Wt − t}. This process in natural inmany areas of applied probability, e.g. in the diffusion approximation of a queue. We haveX0 = 0, Xt ≥ 0. The Brownian area processS(t) =∫ t0Xrdr (5)is non-decreasing. Fix some u ≥ 0 and t ∈ [0, 1]. By continuity, there is a unique pointbetween 0 and u that splits the area S(u) into two parts with ratio t : (1 − t). Call thispoint Hu(t). Specifically,Hu(t) := min{v ≥ 0 : t∫ v0Xrdr = (1− t)∫ uvXrdr}, 0 ≤ t ≤ 1.We then claim thatηu(t) :=Hu(t)− tu√u, 0 ≤ t ≤ 1,converges weakly to a Brownian bridge as u→∞. To see this, observe thatS−1(x) = min{v ≥ 0 : S(v) = x},and henceS−1(tS(u)) = min{v ≥ 0 : S(v) = tS(u)= min{v ≥ 0 : S(v) = t(S(v) + S(u)− S(v))}= min{v ≥ 0 : (1− t)S(v) = t(S(u)− S(v))} = Hu(1− t).Apply Theorem 2 to get the result. (Notice that ηu(1 − t) also converges to a Brownianbridge.)EXAMPLE 2 Same as Example 1, but with W being a zero-mean Le´vy process. TheBrownian bridge in Example 1 was obtained not from the fact that W was Brownian, butfrom the regenerative structure of S. It is this that allows us to replaceW by a more general,say a Le´vy process, as long as we maintain the finite variance assumptions. The latter holdonce we add a strictly negative drift to a zero-mean Le´vy process W , reflect it, precisely asin (4), and integrate just as in (5). Whereas W may be discontinuous, S is continuous andthe conclusion remains the same.5EXAMPLE 3 The third example is an application of the above in proving a limit theoremfor a random digraph. We consider a random directed graph Gn = (Vn, En) on the set ofvertices Vn := {1, . . . , n} by letting the set of edges En contain the pair (i, j), i < j, withprobability p, independently from pair to pair. This is a directed version of the (nowadays)so-called Erdo˝s-Re´nyi graph.A path starting in i and ending in j is a sequence of vertices i0 = i, i1, . . . , in = j such that(i, i1), . . . (in−1, j) are edges. Amongst all paths in Gn there is one with maximum length;this length is denoted by Ln. Amongst all paths in Gn that end at a vertex j ∈ Vn there isone with maximum length; this length is called weight of vertex j. We keep track of verticeswith a specific weight and let Sn(ℓ) be the number of vertices with weights at least ℓ. (Hereℓ ranges between 0 and Ln.) So, for example, Sn(0) is the number of vertices in Vn that areendpoints of no edge in En, and Sn(Ln) is the number of paths of maximal length in Gn.Theorem 3.Sn([tLn])− tn√n, 0 ≤ t ≤ 1,converges, as n→∞, weakly to a Brownian bridge.The proof of this theorem can be found in [2, p. 453].EXAMPLE 4 Here is an illustration, of the kind of phenomenon described around (3), inStochastic Geometry. We consider a Poisson point process1 N in Rd with intensity, say, 1;that is, N is a random discrete subset of Rd such that the cardinalities of N∩B1, . . . , N ∩Bnare independent random variables whenever B1, . . . , Bn are disjoint Borel sets, for any n ∈ N,and the expectation of the cardinality of N ∩B equals the Lebesgue measure of B. For eachx in Rd we let π(x) be the point of N closest to x (there is a.s. a unique such point). Foreach point z of N , we let σ(z) be the Voronoi cell [5, 6] associated to z:σ(z) := {x ∈ Rd : ||x− z|| ≤ ||x− z′|| for all points z′ of N},where || · || is the Euclidean norm on Rd. The Voronoi tessellation of Rd is the the tilingof Rd by the Voronoi cells. If z is not a point of N we define σ(z) to be the Voronoi cellcontaining z (again this cell is a.s. unique). The distance of a closed set A ⊂ Rd from apoint x ∈ Rd isdist(A, x) = inf{||x − y|| : y ∈ A}.Consider now the processD(t, x) := dist(σ(tπ(x)), tx),where t ∈ [0, 1] and x ∈ Rd. The claim is that||x||−1/2 D(·, x)⇒ |W 0|, as ||x|| → ∞,|W 0| being the absolute value of a Brownian bridge.1More general point processes can be allowed here.6References[1] Asmussen, S. (2003) Applied Probability and Queues, 2nd. ed. Springer Verlag (to ap-pear).[2] Foss, S. and Konstantopoulos, T. (2003) Extended renovation theory and limit theoremsfor stochastic ordered graphs. Markov Processes and Related Fields 9, 413-468.[3] Billingsley, P. (1968) Convergence of Probability Measures, Wiley.[4] Revuz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion, 3rd. ed.Springer Verlag.[5] Schneider R. and Weil W. (2000) Stochastische Geometrie. Teubner.[6] Stoyan, D., Kendall, W.S. and Mecke, J. (1987) Stochastic Geometry and its Applica-tions. Wiley.[7] Whitt, W. (2002) Stochastic-Process Limits. Springer Verlag.School of Mathematical and Computer SciencesMaxwell Institute for Mathematical SciencesHeriot-Watt University, Edinburgh EH14 4AS, U.K.foss@ma.hw.ac.uk, takis@ma.hw.ac.uk7

A note on the convergence of renewal and regenerative processes to a
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A note on the convergence of renewal and regenerative processes to a Brownian bridge

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