FUNCTIONAL LARGE DEVIATION PRINCIPLES FOR FIRST-PASSAGE-TIME PROCESSES

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1 The Aals of Applied Probability 1997, Vol. 7, No. 2, FUNCTIONAL LARGE DEVIATION PRINCIPLES FOR FIRST-PASSAGE-TIME PROCESSES By Aatolii A. Puhalskii ad Ward Whitt Istitute for Problems i Iformatio Trasmissio ad AT&T Bell Laboratories We apply a exteded cotractio priciple ad superexpoetial covergece i probability to show that a fuctioal large deviatio priciple for a sequece of stochastic processes implies a correspodig fuctioal large deviatio priciple for a associated sequece of first-passage-time or iverse processes. Large deviatio priciples are established for both iverse processes ad cetered iverse processes, based o correspodig results for the origial process. We apply these results to obtai fuctioal large deviatio priciples for reewal processes ad superpositios of idepedet reewal processes. 1. Itroductio. I this paper we ivestigate how a (fuctioal) large deviatio priciple (LDP) for a sequece of stochastic processes ca be used to deduce a correspodig (fuctioal) LDP for a associated sequece of firstpassage-time or iverse processes. Give a real-valued stochastic process X X t t with sample paths that are ubouded above ad satisfy X, the associated iverse process is defied by (1.1) X 1 t if s > X s > t t (We use to deote a defiitio.) Previous papers [6, 17, 25, 27, 28] have show how covergece i distributio i the fuctio space D D with oe of the Skorohod [22] topologies of X 1, where X X t t, is related to that of X 1 1. Those papers also show how covergece i distributio of the sequeces of cetered processes c X e 1 ad c e X 1 1 are related, where c as ad e is the idetity fuctio; that is, e t = t t. Our purpose here is to show that these results have fairly direct aalogs i the large deviatios cotext, with the cotractio priciple playig the role of the cotiuous mappig theorem ad a exteded cotractio priciple i [13] ad [18], Sectio 2 (also see [2] ad [24]), playig the role of extesios of the cotiuous mappig theorem i Theorem 5.5 of [1] ad o page 68 of [28]. The geeral theme of relatig LDP s to weak covergece is discussed by Puhalskii [13 16, 18, 19]. This paper exteds [3] ad [18]. Gly ad Whitt [3] established some correspodig relatios betwee oe-dimesioal LDP s Received Jauary 1996; revised November AMS 1991 subject classificatios. Primary 6F1; secodary 6G55, 6K5. Key words ad phrases. Large deviatios, large deviatio priciple, Skorohod topologies, cotractio priciple, first passage times, iverse processes, coutig processes, reewal processes, superpositios of reewal processes. 362

2 FIRST-PASSAGE-TIME PROCESSES 363 for iverse processes. Sectio 3 of [18] dealt with fuctioal LDP s for iverse processes for the weak topology ad established fuctioal LDP s for reewal processes. This paper corrects fuctioal LDP s for iverse processes correspodig to Theorems 3.2 ad 6.1 here that had bee give i a prelimiary draft of [3]. Here is how the preset paper is orgaized. I Sectio 2 we discuss fuctio space topologies ad restrictios o the limit fuctios uder which the iverse map (1.1) is cotiuous. I Sectio 3 we preset LDP s based directly o these cotiuity properties. I Sectio 4 we establish prelimiary results about superexpoetial covergece i probability, which plays the role with LDP s that ordiary covergece i probability plays with weak covergece, as i Theorem 4.1 of [1]. I Sectio 5 we use the results about superexpoetial covergece i probability to obtai LDP s for cetered first-passagetime processes. I Sectios 6 ad 7 we apply these results to obtai LDP s for reewal processes ad superpositios of reewal processes. The LDP s for cetered processes are established i a triagular array settig. The results i this paper are applied i [21]. There fuctioal LDP s are established for waitig times ad departure times i sigle-server queues with ulimited waitig space. (The results here ad i [21] are briefly summarized i [2].) Just as for the heavy-traffic diffusio limits i [5], the results for iverse processes help establish large deviatio priciples for processes stemmig from the basic etwork operatios of superpositio, splittig ad departure. We illustrate this pheomeo here by our treatmet of superpositio processes. More geerally, the large deviatio priciples are importat for determiig the probabilities of rare evets i the queueig model, such as hittig times of high levels. The large deviatio priciples also are itimately coected to the asymptotics of steady-state tail probabilities i the queueig model; for example, see [4] ad Sectio 6 of [18]. We close this itroductio by givig a illustrative applicatio. Suppose that N t t is a coutig process such that t 1 N t c > as t, ad we wat to approximate the probability P N t at, N 2t a + b t for large t, where a > c > b. Thus we are cosiderig the probability that N has uusually large values i the iterval t ad uusually small values i the iterval t 2t. By the methods here, it may be possible to show that 1 N t t, 1 obeys a LDP i fuctio space with rate fuctio I x = λ ẋ t dt for absolutely cotiuous x with x = ad I x = otherwise, where λ is a (oegative covex) local rate fuctio o R with λ c =. The we may apply the cotractio priciple with this LDP to deduce that lim sup t 1 log P t 1 N t a t 1 N 2t a + b λ a + λ b t

3 364 A. A. PUHALSKII AND W. WHITT ad lim if t t 1 log P t 1 N t > a t 1 N 2t < a + b λ a + λ b which supports the rough approximatio for large t. P t 1 N t a t 1 N 2t a + b exp t λ a + λ b 2. Fuctios ad topologies o D ad its subsets. This sectio largely follows Sectio 7 of [28], but there will be a few chages. I particular, there will be a correctio for treatig the iverse fuctio with the M 1 topology. (This correctio is relevat for covergece i distributio as well as for LDP s.) Let D be the space of all right-cotiuous real-valued fuctios x x t t o the oegative half lie with left limits everywhere i. Let E be the subset of fuctios x i D that are ubouded above ad satisfy x. Let D be the subset of odecreasig fuctios i D ad let E = E D. We are primarily iterested i the iverse fuctio, defied for ay x E by (1.1); there x is a sample path of X. Also defie the supremum fuctio for ay x D by (2.1) x t sup x s s t t Obviously, if x E, the x 1 E ad x E. We cosider the Skorohod [22] J 1 ad M 1 topologies o D ad a mior modificatio of the M 1 topology deoted by M 1. For these topologies, D is metrizable as a separable metric space. Let D have the Borel σ-field iduced by its topology. For the topologies we cosider, the Borel σ-fields coicide with the usual Kolmogorov σ-field geerated by the fiite-dimesioal projectio maps. The J 1 topology is quite familiar; it is as i [1], [7], [28]. Recall that the M 1 topology o D is defied i terms of the completed graph Ɣ x { u t R R + \ u x t x t x t x t } (2.2) x where x t deotes the left limit of x at t deotes the miimum ad deotes the maximum; see [12], [22], [27], [28]. We will call a pair of cotiuous fuctios u t u s t s s such that t s is odecreasig with t = a parameterizatio of Ɣ x if Ɣ x = u s t s s. A sequece x 1 i D coverges to x i D M 1 if there exist parameterizatios u t of x, 1, ad u t of x such that (2.3) sup u s u s + t s t s s T as for all T >. What we would like is for the iverse fuctio i (1.1) to be cotiuous o E, but we must impose costraits whe we work with the Skorohod [22] J 1

4 FIRST-PASSAGE-TIME PROCESSES 365 ad M 1 topologies with domai exteded from 1 to. We have the followig result. Lemma 2.1. (a) The supremum fuctio is cotiuous i the J 1 ad M 1 topologies. (b) The iverse fuctio i (1.1) is measurable o E, is cotiuous i the M 1 topology at those x for which x 1 =, ad is cotiuous i the J 1 topology at each strictly icreasig x. Part (a) is i Sectio 6 of [28]. The J 1 result i part (b) ad the eed for the J 1 coditio are give o page 82 of [28]. However, the M 1 coditio is missig i [27] ad Theorem 7.1 of [28]. To see that the M 1 coditio is eeded, let x t = t/, t < 1, ad x t = t 1, t 1. Clearly x x M 1 where x t =, t < 1 ad x t = t 1, t 1. However, x 1 = x 1 = 1 as, so that x 1 x 1 M 1 as. With the M 1 coditio added, the M 1 cotiuity proof is as i [27]. We look at the iverse as the compositio of the iverse ad supremum maps. Hece, it suffices to cosider the iverse map o E. Cotiuity is established by otig that, give the M 1 coditio, each parameterizatio u t of x ca serve as a parametric represetatio of x 1 whe the roles of u ad t are switched. Aother approach to the problem of the cotiuity of the iverse mappig o D is to chage the topology istead of addig the extra coditio. I [18] the cotiuity for the weak topology was proved. Here we use a weaker topology, which we call M 1 ad which is defied i the same way as M 1, except that we chage Ɣ x to (2.4) Ɣ x u t R R + u x t x t x t x t where x =. Stated aother way, Ɣ x is the exteded graph Ɣ x complemeted by addig the segmet x if x or x if x. We say that x x i D M 1 if (2.3) holds for some parameterizatios of Ɣ x ad Ɣ x. More rigorously, D M 1 is a metric space with metric d defied as follows. If x x t t ad y y t t are elemets of D, let (2.5) d k {sup x y if u s g k t p v s g k r s s k } + t s g k t s r s g k t s where g k t equals 1 for t less tha k, equals for t greater tha k + 1 ad is a liear iterpolatio betwee k ad k + 1, ad u s t s s ad v s r s s are parameterizatios of x ad y, respectively, ad the ifimum is take over all the parameterizatios. The (2.6) d x y k=1 d k x y 1 2 k metrizes M 1. It is ot difficult to show that D d is a separable metric space ad d iduces the Kolmogorov σ-field. I additio, the M 1 topology is

5 366 A. A. PUHALSKII AND W. WHITT stroger tha the M 1 topology; that is, overall the topologies are ordered by J 1 M 1 M 1. Covergece x x is equivalet for all three topologies at cotiuous x with x =. Moreover, o E with the M 1 topology x x is equivalet to poitwise covergece x t x t at all cotiuity poits except possibly for t =. The followig is the key lemma. Lemma 2.2. The supremum fuctio i (2.1) o D ad the iverse fuctio i (1.1) o E are cotiuous i the M 1 topology. Proof. The argumet for the supremum fuctio is straightforward. The claim for the iverse fuctio follows sice if u t is a Ɣ -parameterizatio of x E, the t u is a Ɣ -parameterizatio of x 1. We will eed aother basic lemma. Let e be the idetity fuctio; e t = t for t. Let c be ay real umber. Lemma 2.3. If x E, the d c x e c e x 1 d x e Proof. For ay k ad ε >, let u s t s s > ad t s t s s be Ɣ -parameterizatios of x ad e, respectively, so that sup u s t s + t s t s d k x e + ε s k Note that c u s t s t s s is a Ɣ -parameterizatio of c x e. Moreover, sice x E, c u s t s u s s is a Ɣ -parameterizatio of c e x 1. Usig these parameterizatios, we see that, for ay c, d k c x e c e x 1 sup u s t s s k sup u s t s + t s t s s k d k x e + ε Sice ε was arbitrary, d k c x e c e x 1 d k x e for each k, from which the coclusio follows. 3. Iitial large deviatio coclusios. We ow draw large deviatio coclusios from the cotiuity properties i Sectio 2. Recall that all spaces we cosider are separable metric spaces. Followig Varadha [23, 24], we say that a fuctio I x defied o a metric space S ad takig values i is a rate fuctio if the sets x S I x a are compact for all a, ad a sequece P 1 of probability measures o the Borel σ-field of S (or a sequece of radom elemets X 1 with values i S ad distributios P ) obeys the LDP with the rate fuctio I if lim sup 1 log P F if x F I x

6 for all closed F S, ad FIRST-PASSAGE-TIME PROCESSES 367 lim if 1 log P G if x G I x for all ope G S. We establish ew LDP s from previously established oes by applyig the cotractio priciple [23, 24] or a extesio [13] ad [18], Sectio 2. Here are statemets: the cotractio priciple states that if X 1 obeys a LDP with rate fuctio I ad if f is cotiuous, the f X 1 obeys a LDP with rate fuctio (3.1) I y if I x x f x =y Our exteded cotractio priciple states that if X 1 obeys a LDP with rate fuctio I, if f 1 is a sequece of measurable fuctios, if the fuctio f is cotiuous i restrictio to the sets x I x a a ad if f x f x as for all x for which x x as for all x for which I x <, the f X 1 obeys a LDP with rate fuctio (3.1). (This statemet is actually a cosequece of a more geeral result i [18]; see Theorem 2.1 ad followig Remarks 1 ad 2 there.) A importat special case is f = f, as i the cotractio priciple, where f is cotiuous at each x with I x <. I either case, if i additio f is a bijectio, the we ca write I y = I f 1 y. The applicatios here illustrate the importace of the exteded cotractio priciple. We cosider both sigle fuctios that are ot cotiuous everywhere ad sequeces of fuctios. The ext three theorems follow immediately from Lemmas 2.1 ad 2.2 ad the cotractio priciple or its extesio. Theorem 3.1. If X 1 obeys the LDP i E J 1 with rate fuctio I X, the X 1 obeys the LDP i E J 1 with rate fuctio (3.2) I X x if y E x=y I X y x E If i additio I X x = wheever x is ot strictly icreasig, the X 1 1 obeys the LDP i E J 1 with rate fuctio (3.3) I X 1 x if I X y = I X x 1 y E y y 1 =x [As a cosequece, I X 1 x = if x is ot cotiuous.] x E Theorem 3.2. If X 1 obeys the LDP i E M 1 with rate fuctio I X, the X 1 obeys the LDP i E M 1 with rate fuctio I X i (3.2). If I X x = wheever x 1 >, the X 1 1 obeys the LDP i E M 1 with rate fuctio I X 1 i (3.3). [As a cosequece I X 1 x = if x >.

7 368 A. A. PUHALSKII AND W. WHITT Theorem 3.3. If X 1 obeys the LDP i E M 1 with rate fuctio I X, the X 1 obeys the LDP i E M 1 with rate fuctio I X i (3.2) ad X 1 1 obeys the LDP i E M 1 with rate fuctio I X 1 i (3.3). I (3.3) we have used the fact that the iverse map is a bijectio o E i order to write I X 1 x = I X x 1. Remark 3.1. It may be coveiet to establish a LDP for a iverse process by applyig Theorems 3.2 or 3.3 for the M 1 or M 1 topology istead of the stroger J 1 topology, but the LDP exteds to the stroger J 1 topology from M 1 or M 1 if the rate fuctio I X 1 x is ifiite at discotiuous x ad for M 1 if i additio the rate fuctio I X 1 x = whe x. This is because covergece x x for cotiuous x with x = is equivalet for the three topologies ad we ca apply the exteded cotractio priciple to the idetity maps. Ideed, the LDP exteds to the stroger uiform topology, uder which D is oseparable, provided that X 1 remais a boafide radom elemet. However, i geeral this eed ot be the case sice the Borel σ-field is richer tha the Kolmogorov σ-field; see [1], Sectio 18. Ideed, measurability with respect to the Borel σ-field associated with the uiform topology fails eve for the Poisso process. Thus, for geeral LDP s o D it is ofte importat to work with topologies like the Skorohod topologies, for which the Borel ad Kolmogorov σ-fields coicide. X 1 I may (but ot all) cases, the rate fuctios I X x for X ad I X 1 x for have the form (3.4) I X x = λ X ẋ t dt if x is absolutely cotiuous ad x = ad I X x = otherwise, ad (3.5) I X 1 x = λ X 1 ẋ t dt if x is absolutely cotiuous ad x = ad I X 1 x = otherwise, where λ X ad λ X 1 are covex local rate fuctios o R. We ca the apply Theorems to deduce the relatio betwee the rate fuctios λ X ad λ X 1 o R, which is cosistet with what was established directly by [3]. Theorem 3.4. If X 1 obeys the LDP i E for oe of the topologies J 1, M 1 or M 1 with the rate fuctio I X satisfyig (3.4), where λ X =, the X 1 1 obeys the LDP i E J 1 with the rate fuctio I X 1 from (3.5), where (3.6) λ X 1 z = zλ X 1/z λ X 1 = If the fuctio λ X (respectively, λ X 1) is covex dowwards, the the sequece of radom variables X 1 1 (respectively, X ) obeys the LDP i R with rate fuctio λ X (respectively, λ X 1).

8 FIRST-PASSAGE-TIME PROCESSES 369 Proof. Sice λ X =, I X x = if x 1 is ot absolutely cotiuous (as follows from Lemma 3.6 i [18]). By Remark 3.1, the LDP for X 1 holds i E J 1. By Theorem 3.1 X 1 1 obeys the LDP i E J 1 with rate fuctio I X 1 which, for absolutely cotiuous x ad x 1 = y, is give by I X 1 x = I X x 1 = I X y = = = λ X ẏ x s ẋ s ds λ X 1 ẋ s ds λ X ẏ s ds sice y x = e by performig a chage of variables. Hece (3.6) holds. As idicated above, we ext apply the exteded cotractio priciple with the projectio map defied by π x = x 1 to obtai the LDP s i R. Sice λ X is covex, the ifimum over x such that x 1 = z is attaied at ẋ t = z for t 1. The relatio betwee λ X ad λ X 1 was established i [3]. It is importat to ote, however, that the rate fuctios I X of X 1 ad I X 1 of X 1 1 may ivolve fuctios with jumps, so that (3.4) ad (3.5) eed ot hold; see [9], [11], [18] ad Sectio 6 below. The the coectios to LDP s o R is more complicated, for example, because the projectio map is ot ecessarily cotiuous. Fuctios with jumps may play a role i either I X or I X 1 or both. However, Theorem 3.4 does apply to the reewal theory examples i Sectios 6 ad 7 uder regularity coditios. 4. Superexpoetial covergece i probability. As a basis for establishig relatios betwee LDP s for cetered processes ad associated cetered iverse processes, parallelig Theorems of [28], we establish some prelimiary results about superexpoetial covergece i probability. As i [16], we say that a sequece X 1 of radom elemets of a metric space S ρ coverges superexpoetially i probability to a elemet x S if, for all ε >, (4.1) lim P1/ ρ X x > ε = ad we write X P 1/ x. This mode of covergece plays a role i large deviatios similar to the role covergece i probability plays i weak covergece. We collect some simple properties of superexpoetial covergece i probability i the followig lemmas. The similarity with weak covergece should be evidet; for example, see [1], [28]. I the followig lemmas, S is the space D with ay of the topologies J 1, M 1 or M 1. Parts of the ext lemma ca obviously be exteded to geeral metric spaces; for example, see [15], [16]. Note the similarity of parts (b) ad (c) to Theorems 4.1 ad 4.4 of [1]. Part (c) follows directly from part (b) (also it is Lemma 3.3 i [16]).

9 37 A. A. PUHALSKII AND W. WHITT P Lemma 4.1. (a) X 1/ x if ad oly if X 1 obeys the LDP with rate fuctio { x = x (4.2) δ x x x x (b) If X 1 obeys the LDP i D for oe of the topologies J 1 M 1 ad P 1/ M 1 with rate fuctio I ad Y y, the X Y 1 obeys the LDP i D D for the product topology with rate fuctio I x + δ y y, ad X + Y 1 obeys the LDP i D for the same topology with rate fuctio I x y x D. (c) If X 1 obeys the LDP i D for metric m with rate fuctio I ad m X Y P1/, the Y 1 obeys the LDP i D for metric m with rate fuctio I. Lemma 4.2. (a) Let x x t t be cotiuous with x = if the P 1/ topology is M 1. The X x if ad oly if ) lim X t x t > ε = P1/( sup t T for all ε > ad T >. (b) If, for c, c X 1 obeys the LDP with some rate fuctio, P the X 1/ θ as, where θ t = t. P (c) If X has paths i E for 1 ad X 1/ e, the X 1 P 1/ e. Proof. For part (a), we do the proof oly for the J 1 topology; the proof for the other topologies is similar. By the triagle iequality, sup X t x t sup X t x λ t + x λ t x t t T t T where λ t, for t T, is ay homeomorphism of T. Hece, sup X t x t d T X x + w x d T X x t T where d T is the Skorohod J 1 metric o D T ad w x δ is the modulus of cotiuity of x as i [1], page 54. For ay x ad ε, let δ be such that w x δ ε, which is possible because x is cotiuous. The { d T X x δ ε sup t T } X t x t 2ε d T X x 2ε which implies the result. We also use the fact that a J 1 metric d o D ca be related to the metric d T o D T for large T; see (2.2) of [28]; that is, for ay ε >, d X x ε d T X x 2ε d X x 3ε for T suitably large. For (b), ote that by the defiitio of the LDP ad usig

10 FIRST-PASSAGE-TIME PROCESSES 371 that c, we have, for ε >, A >, ad metric m (associated with oe of the topologies J 1, M 1 or M 1 ), lim sup P 1/ m X θ > ε lim sup P 1/ m c X θ Aε sup exp I x x m x θ Aε ad the latter goes to as A sice the sets x I x a, a, are compact ad hece bouded. For part (c), we apply Lemmas 2.1, 2.2 ad part (a) of Lemma 4.1. We ow discuss the compositio map, deoted by. Recall that if x x t t D ad y y t t E, the x y = x y t t D; see [28]. Note the similarity of this lemma with [1], page 145. Lemma 4.3. Let X 1 obey a LDP for oe of the topologies J 1 M 1 ad M 1 with rate fuctio I, ad let Y 1 be a sequece of oegative processes with paths from E P such that Y 1/ y. If the topology is M, the let y be cotiuous. If the topology is M 1, the let y be cotiuous with y =. If I x = for discotiuous x, or y is cotiuous ad strictly icreasig, the X Y 1 obeys the LDP with rate fuctio I z if I x x x y =z z D Proof. By Lemma 4.2, X Y 1 obeys the LDP i D D with I x +δ y y. The claim the follows by Theorem 3.1 i [28] ad the exteded cotractio priciple. A aalog of Theorem 3.1 of [28] holds for M 1 ad M 1 if the limit y there is cotiuous. 5. LDPs for cetered processes. We ow apply the lemmas i Sectio 4 to deduce the followig results. For them, we assume that the processes X have paths i E ad c as. Theorem 5.1. If the sequece c X e 1 obeys the LDP i D for the J 1 topology with rate fuctio I such that I x = for fuctios x which have positive jumps or have x, the the sequece c e X 1 1 also obeys the LDP i D for the J 1 topology with the rate fuctio I. Proof. X P 1/ We follow the argumet i [28], Theorem 7.3. By Lemma 4.2(b), e ad by Lemma 4.2(c), X 1 e. Lemma 4.3 the implies that c X e X 1 1 obeys the LDP with I. Sice P 1/ c e X 1 = c X e X 1 by Lemma 4.1(b), the theorem would follow from c e X X 1 + c e X X 1 P1/

11 372 A. A. PUHALSKII AND W. WHITT which i tur, by Lemma 4.2(a), follows by (5.1) Sice (cf. [28]) sup t T sup t T c X X 1 P1/ t t X X 1 T > t t sup X t + T t X 1 where x t, for x = x t t, deotes the jump of x at t with x = x, we have for A >, ε >, that (5.2) P 1/( ) sup c X X 1 t t > ε t T P 1/ X 1 T > A + P1/( sup t A ) c X t + > ε Now it is ot difficult to see that the fuctio x sup t A x t + is cotiuous i the J 1 topology at each x with o positive jumps; for example, see [8], Chapter 6, Sectio 2. The, by the exteded cotractio priciple ad the LDP for c X e 1, lim sup P 1/( sup t A ) c X t + ε sup exp I x = x sup t A x t + ε provig that the secod term o the right-had side of (5.2) goes to as. The first term goes to as ad A sice X 1 e. Hece, the limit (5.1) has bee proved, so the theorem has bee proved. I order to obtai a result parallelig Theorem 5.1 for the M 1 topology, we first establish a result for cetered supremum processes. Theorem 5.2. If the sequece c X e 1 obeys the LDP i D for either M 1 or the M 1 topology with rate fuctio I, the c X e 1 obeys the LDP i D for the same topology with rate fuctio I. Proof. We ca use the exteded cotractio priciple with the fuctios f y = y + c e c e. Assume that y = c x e y x. Sice f y = c x e, it suffices to show that f y y wheever y y i D for the M 1 or M 1 topology, which follows by Theorem 6.3(ii) i [28]. (The proof there eeds chagig whe x has egative jumps.) Theorem 5.3. If the sequece c X e 1 obeys the LDP i D for the M 1 topology with rate fuctio I, the the sequece c e X 1 1 obeys the LDP i D for the M 1 topology with the rate fuctio I. Proof. P 1/ First apply Theorem 5.2 to see that it suffices to assume that X E for each. By Lemma 4.2(b), d X e P1/, so that d c X e c e X 1 P1/ by Lemma 2.3. Fially, apply Lemma 4.1(c).

12 FIRST-PASSAGE-TIME PROCESSES 373 The last result of the sectio is a straightforward extesio of Theorems 5.1 ad 5.3, but is quite useful i applicatios. Theorem 5.4. If the sequece c X a e 1, where a are real umbers covergig to a >, obeys the LDP i D for the M 1 topology with rate fuctio I x, the the sequece c X a e c X 1 a 1 e 1 obeys the LDP i D D for (the product topology associated with) the M 1 topology with rate fuctio I 1 x y = I x, whe y = a 1 x a 1 e, ad I 1 x y = otherwise. If the LDP for c X a e 1 holds for the J 1 topology with I equal to ifiity at fuctios x with positive jumps or with x, the the LDP for c X a e c X 1 a 1 e 1 holds for (the product topology associated with) the J 1 topology with the rate fuctio I 1. Proof. Notig that x 1 a 1 e = e a 1 x 1 a 1 e, we have as i the precedig argumet that d c X 1 a 1 e c a 1 X a e a 1 so that i the statemet of the theorem we ca replace c X a e c X 1 a 1e 1 by c X a e a 1 g c X a e a 1 e 1, where g x = x + c a e c a e. The claim follows by Lemma 4.3 ad the exteded cotractio priciple sice g x x as x x ad a a. O writig c a 1 e X 1 = c a 1 X a e X 1 + c a 1 e P1/ e X X 1 we ca apply for the case of the J 1 topology the argumet of the proof of Theorem Large deviatios for reewal processes. I this sectio, we apply the results of previous sectios to derive LDP s for sequeces of reewal processes. Correspodig results ca be established for cases i which the i.i.d. coditio is relaxed, drawig upo Zajic [29] ad refereces therei. We first assume that the X are defied by (6.1) X t 1 where ξ i, i 1, are i.i.d., oegative, Eξ 1 >. Let N t t be the reewal process with ξ 1 ξ 2 as the times betwee reewals; that is, { } k (6.2) N t max k 1 ξ i t t i=1 with N t = if ξ 1 > t, ad let N N t / t, 1. We are goig to derive the LDP for the sequece N 1. This will be doe by reducig this problem to the LDP for X 1 ad by ivokig earlier results for X 1. For this purpose, ote that (6.3) t i=1 ξ i N t / = X 1 t 1/

13 374 A. A. PUHALSKII AND W. WHITT The followig theorem is a versio of Theorem 3.1 i [18]. Part (b) is the same. Part (a) is also equivalet, because the M 1 topology here ad the weak topology i [18] coicide o E, sice both correspod to poitwise covergece at all cotiuity poits except. Theorem 6.1. Assume that E exp αξ 1 < for some α >. Let α = sup α E exp αξ 1 <. (a) The N 1 obeys the LDP i E for the topology with rate fuctio M 1 (6.4) I N x sup α<α { α ẋ l 1 t log E exp αξ 1 } dt log P ξ 1 = x l 2 where x = x l 1 + xl 2 is the Lebesgue decompositio of x with respect to Lebesgue measure; x l 1 is the absolutely cotiuous compoet with xl 1 =, xl 2 is the sigular compoet ad ẋ l 1 t is the derivative. I (6.4) it is assumed, that the product o the right-had side is if P ξ 1 = = ad x l 2 =. (b) If, i additio, P ξ 1 = =, the the LDP holds for the J 1 topology with I N x = if x is ot absolutely cotiuous, or x. Proof. By [18], Lemma 3.2, X 1 obeys the LDP o E for the weak (ad hece the M 1 topology with rate fuctio (6.5) I X x sup α<α { αẋ l 1 t log E exp αξ 1 } dt + α x l 2 where =. (Note that if α =, the I X x = wheever x is ot absolutely cotiuous, but otherwise this is ot the case.) The first part of the proof of part (a) is completed by applyig Theorem 3.3, Lemma 4.3 ad (6.3); see [18] for details. For part (b), ote that the extra coditio makes I N x = for discotiuous x or if x. The we use the fact that x x J 1 is equivalet to x x M 1 for cotiuous x with x =. Hece, we ca apply the exteded cotractio priciple with the idetity map to stregthe the topology, as oted i Sectio 3. Remark 6.1. I the settig of Theorem 6.1 assume, i additio, that E exp αξ 1 < for all α ad P ξ 1 = =. The we ca use Theorem 3.4 to get the familiar LDP s i R for the projectios at t = 1. Uder the assumptios, I X x =, whe x is either ot absolutely cotiuous, or x, or x is ot strictly icreasig for the rate fuctio (6.5), so that the coditios of Theorem 3.4 are satisfied with the local rate fuctios i R beig ad λ X z = sup αz log E exp αξ 1 α λ X 1 z = zλ X 1/z = sup α z log E exp αξ 1 α A applicatio of the theorem provides the oe-dimesioal LDP s.

14 FIRST-PASSAGE-TIME PROCESSES 375 We ow establish a LDP i D for cetered reewal processes, which is a form of moderate deviatios; see page 79 of [15]. Motivated by applicatios to queues i heavy traffic, we choose to work i the more geeral settig of triagular arrays. More specifically, we cosider a sequece of reewal processes idexed by ad deote by ξ i i 1 the times betwee reewals. We ext defie ) k (6.6) N (k t = max 1 ξ i t i=1 (6.7) N t = 1 a N a t where a /. For completeess, we first state the result of Example 7.2 [15] o the moderate deviatio ivariace priciple for partial sums of triagular arrays (prototypes for partial sums of r.v. are i [1] ad [26], Theorem 4.4.3). Lemma 6.1. Let ζ i i 1 1 be a triagular array of row-wise i.i.d. radom variables with Eζ 1 = Var ζ 1 σ 2. Defie Z t = 1 a t ζ i a i=1 Let at least oe of the followig coditios hold: (i) log a / ad sup E ζ 1 2+ε < for some ε >, (ii) for some β 1, a β / 2 β ad sup E exp α ζ 1 β < for some α >. The Z 1 obeys the LDP i D for the J 1 topology with rate fuctio 1 ẋ t 2 dt if x is absolutely cotiuous 2σ I X x 2 (6.8) ad x = otherwise The proof is i [15]. (Note that the case β = 1 which is ot icluded there is dealt with by the same argumet.) Theorem 6.2. Let N be defied by (6.6) ad (6.7). Let Var ξ 1 σ 2 ad Eξ 1 λ 1 as. Assume that at least oe of the followig coditios hold: (i) log a / ad sup Eξ 2+ε 1 < for some ε >, (ii) for some β 1, a β / 2 β ad sup E exp αξ β 1 < for some α >.

15 376 A. A. PUHALSKII AND W. WHITT The a / N e Eξ obeys the LDP i D for the J 1 topology with rate fuctio 1 ẋ t 2 dt if x is absolutely cotiuous 2σ I N x 2 λ 3 (6.9) ad x = otherwise. Proof. By Lemmas 6.1 ad 4.1(b), the sequece a / X eeξ 1 a t 1, where X t = 1/a i=1 ξ i, obeys the LDP i D for the J 1 topology with rate fuctio I X from (6.8). The proof is completed by observig that, i aalogy with (6.3), N = X 1 a 1 ad applyig Theorem 5.4 ad Lemma 4.1(b) ad the cotractio priciple. Corollary. Uder the assumptios of Theorem 6 2 a / N 1 Eξ obeys the LDP i R with rate fuctio λ N z = z2 2σ 2 λ 3 z R Proof. Apply the exteded cotractio priciple with the projectio map. By (6.9), the resultig rate fuctio is λ N z = if I N x = z2 x D 2σ 2 λ 3 x 1 =z 7. Superpositios of reewal processes. The results i Sectio 6 exted easily to superpositios of reewal fuctios provided that the compoet rate fuctios are fiite oly for cotiuous fuctios x. Otherwise, we have the difficulty that additio is ot cotiuous o D D [28]. However, from Theorem 6.1 we see that i geeral the rate fuctios ca be fiite for discotiuous x. We avoid this problem by makig additioal assumptios, as i Remark 6.1. Let ξ j i i 1, 1 j k, be k idepedet sequeces of i.i.d. oegative radom variables with Eξ j 1 >. Let Nj t t, 1 j k, be the associated k mutually idepedet reewal coutig processes, defied as i (6.2), ad let N = N N k. For each j, let X j be the ormalized partial sum process defied as i (6.1) ad let X be the ormalized partial sum process associated with the superpositio process, defied by (7.1) t X t 1 i=1 ξ i t where ξ i is the ith iterval betwee poits i the superpositio process N. Let N j ad N be associated ormalized coutig processes; that is, (7.2) N j t 1 N j t ad N t 1 N t t

16 FIRST-PASSAGE-TIME PROCESSES 377 We will derive LDP s for N 1 ad X 1. For this purpose, ote that N j t = X j 1 t 1, so that (7.3) ad (7.4) N j = Xj 1 1 N = X 1 X = N The followig theorem exteds Theorem Theorem 7.1. Assume that E exp αξ j 1 <, 1 j k, for some α >. Let α j sup α E exp αξj 1 < 1 j k ad α = k j=1 α j. Also assume that P ξ j 1 = = for all j, 1 j k. The the sequece N 1 i (7.2) obeys the LDP i E for the J 1 topology with rate fuctio (7.5) I N x = sup α ẋ t ψ α dt α<α (7.6) where (7.7) = ψ α = φ 1 α φ α = sup φ α αẋ t dt α<α k φ j α j=1 φ j α = ψj 1 α (7.8) α = mi lim 1 j k α α j ψ j α with ψ j α = log E exp αξ j 1 if x is absolutely cotiuous ad x =, while I N x = otherwise. If i additio there is oe j for which E exp αξ j 1 < for all α, the X 1 i (7.1) obeys the LDP i E for the J 1 topology with rate fuctio (7.9) I X x = I N x 1 = = sup αẋ t ψ α dt α R sup ẋ t φ α α dt α R if x is absolutely cotiuous with x =, ad I X x = otherwise. Proof. Sice the ormalized processes N j are idepedet, by Theorem 6.1 ad [9] the sequece N 1 N2 Nk 1 of radom elemets of D R k obeys the LDP for the J 1 topology with rate fuctio (7.1) I x 1 x k k j=1 sup α<α j α ẋ j t ψ j α dt whe x 1 x k are absolutely cotiuous with respect to Lebesgue measure ad x j = for all j, while I x 1 x k = otherwise. We start workig toward the first expressio i (7.5).

17 378 A. A. PUHALSKII AND W. WHITT By the exteded cotractio priciple, the superpositio N obeys the LDP for the J 1 topology with rate fuctio I N x, where I N x = if x is ot absolutely cotiuous or x. Usig a argumet as i [18] (icludig a miimax theorem o the third lie), if x is absolutely cotiuous ad x =, the (7.11) I N x = = = = if x 1 + +x k =x k j=1 if kj=1 ẋ j t =ẋ t sup α j <α j j=1 k sup α j <α j j=1 k sup α ẋ j t ψ j α dt α<α j { k sup α j <α j j=1 j=1 k { k if kj=1 ẋ j t =ẋ t j=1 { k α j α j α j ẋ t max j=1 k j=1 The required ow follows sice { ψ α = if max ψ j α j 1 j k k j=1 k j=1 } ẋ j t ψ j α j dt } ẋ j t ψ j α j dt } ψ j α j dt } k α j = α α j < α j with beig the ifimum over the empty set. [The ifimum is attaied at poits α j for which all the ψ j α j are equal, for if we have that ψ j α j > ψ j α j we ca make α j smaller ad α j larger keepig their sum uchaged.] The equality (7.6) is obvious. Turig to X, we observe that α = if there is a j for which E exp αξ j 1 < for all α >. Hece the local rate fuctio i (7.5) is at, ad a applicatio of Theorem 3.4 completes the proof. The secod equality agai is obvious. j=1 We ow establish a extesio of Theorem 6.2. Theorem 7.2. Let N t j t j = 1 k 1, be a sequece of k-tuples of idepedet reewal processes. Let ξ j i i 1 j = 1 k deote their respective iterreewal times. Assume that Var ξ j 1 σ2 j ad Eξ j 1 λ 1 j for j = 1 k as. Assume that oe of the coditios (i) or (ii) i Theorem 6.2 holds for all j, 1 j k. The the sequece a / N e k j=1 Eξ j 1 1, 1, where N t = 1/a kj=1 N a j t, obeys the LDP i D for the J 1 topology with rate fuctio (7.12) I N x 1 2 ( k σj 2 λ3 j j=1 ) 1 ẋ t 2 dt

18 FIRST-PASSAGE-TIME PROCESSES 379 for absolutely cotiuous x with x =, ad I N x = otherwise. Moreover, if ξ i i 1 are the times betwee evets i the superpositio process k j=1 N j a t ad X t = 1/a i=1 ξ i, the a / X e k j=1 Eξ j obeys the LDP i D for the J 1 topology with rate fuctio (7.13) ( k ) 3 ( k ) 1 I X x 1 λ 2 j σj 2 λ3 j ẋ t 2 dt j=1 j=1 for absolutely cotiuous x with x =, ad I X x = otherwise. Proof. By Theorem 6.2 ad i aalogy with the proof of Theorem 7.1, the sequece of processes a / N e k j=1 Eξ j i 1 1 obeys the LDP for the J 1 topology with rate fuctio I N x, which for absolutely cotiuous x has the form I N x = if k 1 x 1 + +x k =x j=1 2σj 2λ3 j = 1 2 = 1 2 ( k if k kj=1 ẋ j t =ẋ t j=1 σj 2λ3 j σj 2 λ3 j j=1 ẋ j t 2 dt 1 ) 1 ẋ t 2 dt ẋ j t 2 dt with the last lie followig from the Cauchy Schwarz iequality. By Theorem 5.4 ad Lemma 4.1(b), a correspodig limit holds for X. Remark. (7.14) The rate fuctios I N i (7.12) ad I X i (7.13) have the form I x = 1 2γ ẋ t 2 dt where γ is the asymptotic variace of the processes k j=1 N j ad X, respectively; that is, for (7.12), while for (7.13), k γ = lim a 1 Var N j a j=1 γ = lim a 1 a Var ξ i that is, the costat γ is the same as appears i the cetral limit theorems. The form of the coefficiet i the rate fuctio as a limit variace is typical i=1

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20 FIRST-PASSAGE-TIME PROCESSES 381 [26] Wetzell, A. D. (1986). Limit Theorems o Large Deviatios for Markov Radom Processes. Nauka, Moscow (i Russia; Eglish traslatio, Reidel, 1989). [27] Whitt, W. (1971). Weak covergece of first passage time processes. J. Appl. Probab [28] Whitt, W. (198). Some useful fuctios for fuctioal limit theorems. Math. Oper. Res [29] Zajic, T. R. (1993). Large deviatios for sample path processes ad applicatios. Ph.D. dissertatio, Dept. Operatios Research, Staford Uiv. Istitute for Problems i Iformatio Trasmissio 19 Bolshoi Karetii Moscow Russia pukh@ippi.ac.msk.su AT&T Bell Laboratories Room 2C-178 Murray Hill, New Jersey wow@research.att.com