WEAK QUENCHED LIMITING DISTRIBUTIONS FOR TRANSIENT ONE-DIMENSIONAL RANDOM WALK IN A RANDOM ENVIRONMENT

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1 WEAK QUENCHED LIMITING DISTRIBUTIONS FOR TRANSIENT ONE-DIMENSIONAL RANDOM WALK IN A RANDOM ENVIRONMENT JONATHON PETERSON AND GENNADY SAMORODNITSKY Abstract. We consder a one-dmensonal, transent random walk n a random..d. envronment. The asymptotc behavour of such random walk depends to a large extent on a crucal parameter κ > 0 that determnes the fluctuatons of the process. When 0 < κ < 2, the averaged dstrbutons of the httng tmes of the random walk converge to a κ-stable dstrbuton. However, t was shown recently that n ths case there does not exst a quenched lmtng dstrbuton of the httng tmes. That s, t s not true that for almost every fxed envronment, the dstrbutons of the httng tmes centered and scaled n any manner converge to a non-degenerate dstrbuton. We show, however, that the quenched dstrbutons do have a lmt n the weak sense. That s, the quenched dstrbutons of the httng tmes vewed as a random probablty measure on R converge n dstrbuton to a random probablty measure, whch has nterestng stablty propertes. Our results generalze both the averaged lmtng dstrbuton and the non-exstence of quenched lmtng dstrbutons. 1. Introducton A random walk n a random envronment RWRE s a Markov chan wth transton probabltes that are chosen randomly ahead of tme. The collecton of transton probabltes are referred to as the envronment for the random walk. We wll be concerned wth nearest-neghbor RWRE on Z, n whch case the space of envronments may be dentfed wth Ω = [0, 1] Z, endowed wth the cylndrcal σ-feld. Envronments ω = ω x } x Z Ω are chosen accordng to a probablty measure P on Ω. Gven an envronment ω = ω x } x Z Ω and an ntal locaton x Z, we let X n } n 0 be the Markov chan wth law Pω x defned by Pω x X 0 = x = 1, and ω x y = x + 1 Pω x X n+1 = y X n = x = 1 ω x y = x 1 0 otherwse. Snce the envronment ω s random, Pω x s a random probablty measure and s called the quenched law. By averagng over all envronments we obtan the averaged law P x = Pω x P dω. Ω Snce we wll usually be concerned wth RWRE startng at x = 0, we wll denote P 0 ω and P 0 by P ω and P, respectvely. Expectatons wth respect to P, P ω, and P wll be denoted by E P, E ω and E, respectvely. Throughout the paper we wll use P to denote a generc probablty law, separate from the RWRE, wth correspondng expectatons E. We wll make the followng assumptons on the dstrbuton P on envronments Date: December 6, Mathematcs Subject Classfcaton. Prmary 60K37; Secondary 60F05, 60G55. Key words and phrases. Weak quenched lmts, pont processes, heavy tals. J. Peterson was partally supported by Natonal Scence Foundaton grant DMS G. Samorodntsky was partally supported by ARO grant W911NF and NSF grant DMS at Cornell Unversty. 1

2 2 JONATHON PETERSON AND GENNADY SAMORODNITSKY Assumpton 1. The envronments are..d. That s, ω x } x Z s an..d. sequence of random varables under the measure P. Assumpton 2. The expectaton E P [log ρ 0 ] s well defned and E P [log ρ 0 ] < 0. Here ρ = ρ ω = 1 ω ω, for all Z. In Solomon s semnal paper on RWRE [Sol75], he showed that Assumptons 1 and 2 mply that the RWRE s transent to +. That s, Plm X n = + = 1. Moreover, Solomon also proved a law of large numbers wth an explct formula for the lmtng velocty v P = lm X n /n. Interestngly, v P > 0 f and only f E P [ρ 0 ] < 1, and thus one can easly construct examples of RWRE that are transent wth zero speed. Soon after Solomon s orgnal paper, Kesten, Kozlov, and Sptzer [KKS75] analyzed the lmtng dstrbutons of transent RWRE under the followng addtonal assumpton. Assumpton 3. The dstrbuton of log ρ 0 s non-lattce under P, and there exsts a κ > 0 such that E P [ρ κ 0 ] = 1 and E P [ρ κ 0 log ρ 0] <. Kesten, Kozlov, and Sptzer obtaned lmtng dstrbutons for the random walk X n by frst analyzng the lmtng dstrbutons of the httng tmes T x := nfn 0 : X n = x}. Let Φx be the dstrbuton functon of the standard normal dstrbuton, and let L κ,b x be the dstrbuton functon of a totally skewed to the rght stable strbuton of ndex κ 0, 2 wth scalng parameter b > 0 and zero shft; see [ST94]. Theorem 1.1 Kesten, Kozlov, and Sptzer [KKS75]. Suppose that Assumptons 1-3 hold, and let x R. 1 If κ 0, 1, then there exsts a constant b > 0 such that lm P Tn n 1/κ x = L κ,b x. 2 If κ = 1, then there exst constants A, b > 0 and a sequence Dn A log n so that lm P Tn ndn x = L 1,b x. n 3 If κ 1, 2, then there exsts a constant b > 0 such that lm P Tn n/v P x = L κ,b x. n 1/κ 4 If κ = 2, then there exsts a constant σ > 0 such that lm P Tn n/v P σ n log n x = Φx. 5 If κ > 2, then there exsts a constant σ > 0 such that lm P Tn n/v P σ x = Φx. n Theorem 1.1 s then used n [KKS75] n the natural way to obtan averaged lmtng dstrbutons for the random walk tself, but for the sake of space we do not state the precse statement here. It should be noted that a formula for the scalng parameter b > 0 appearng above when κ < 2 has been obtaned recently n [ESZ09b, ESTZ10].

3 WEAK QUENCHED LIMITS 3 It was not untl more recently that the lmtng dstrbutons of the httng tme and the random walk were studed under the quenched dstrbuton. In the case when κ > 2, All proved a quenched central lmt theorem for the httng tmes of the form 1 lm P Tn E ω T n ω x = Φx, x R, P a.s., σ 1 n where σ1 2 = E P [Var ω T 1 ] < [Al99]. The envronment-dependent centerng term E ω T n makes t dffcult to use 1 to obtan a quenched central lmt theorem for the random walk, but ths dffculty was overcome ndependently by Goldshed [Gol07] and Peterson [Pet08] to obtan a quenched central lmt theorem for the random walk also wth an envronment-dependent centerng. When κ < 2 the stuaton s qute dfferent. Even though one could reasonably expect that, smlarly to 1, a lmtng stable dstrbuton of ndex κ exsted possbly wth envronmentdependent centerng or scalng, ths has turn out not be the case. In fact, t was shown n [PZ09, Pet09] that quenched lmtng dstrbutons do not exst when κ < 2. For P -a.e. envronment ω, there exst two random subsequences n k = n k ω and m k = m k ω so that the lmtng dstrbutons of T nk and T n k under the measure P ω are Gaussan and shfted exponental, respectvely. That s, and lm k P ω lm k P ω T mk E ω T mk Varω T mk T nk E ω T nk Varω T nk x = x = Φx, x R, 1 e x 1 x > 1 0 x 1, x R. These subsequences were then used to show the non-exstence of quenched lmtng dstrbutons for the random walk as well [PZ09, Pet09]. These results of [PZ09, Pet09] are less than completely satsfyng because one would lke to be able to say somethng about the quenched dstrbuton after a large number of steps. Also, the exstence of subsequental lmtng dstrbutons that are Gaussan and shfted exponental begs the queston of whether and what other types of dstrbutons are possble to obtan through subsequences. The proof of the non-exstence of quenched lmtng dstrbutons n [Pet09] mples, for large n, the magntude of the httng tme T n s determned, to a large extent, by the amount of tme t takes the random walk to pass a few large traps n the nterval [0, n]. Moreover, as was shown n [Pet09, Corollary 4.5], the tme to cross a large trap s approxmately an exponental random varable wth parameter dependng on the sze of the trap. Therefore, one would hope that the quenched dstrbuton of T n could be descrbed n terms of some random dependng on ω weghted sum of exponental random varables. Our man results confrm ths by showng that the quenched dstrbuton vewed as a random probablty measure on R converges n dstrbuton on the space of probablty measures to the law of a certan random nfnte weghted sum of exponental random varables. Before statng our man result, we ntroduce some notaton. Let M 1 be the space of probablty measures on R, BR, where BR s the Borel σ-feld. Recall that M 1 s a complete, separable metrc space when equpped wth the Prohorov metrc 2 ρπ, µ = nfε > 0 : πa µa ε + ε, µa πa ε + ε A BR}, π, µ M 1, where A ε := x R : x y < ε for some y A} s the ε-neghbourhood of A. By a random probablty measure we mean a M 1 -valued random varable, and we denote convergence n dstrbuton of a sequence of random probablty measures by µ n = µ; see [Bl99]. Ths notaton does carry the danger of beng confused wth the weak convergence of probablty

4 4 JONATHON PETERSON AND GENNADY SAMORODNITSKY measures on R, but we prefer t to the more proper, but awkward, notaton L µn = L µ wth L µ beng the law of a random measure µ. Next, let M p be the space of Radon pont processes on 0, ]; these are the pont processes assgnng a fnte mass to all sets x, ] wth x > 0. We equp M p wth the standard topology of vague convergence. Ths topology can be metrzed to make M p a complete separable metrc space; see [Res08, Proposton 3.17]. For pont processes n M p we denote vague convergence by v ζ n ζ. An Mp -valued random varable wll be called a random pont process, and, as above, we wll use the somewhat mproper notaton ζ n = ζ to denote convergence n dstrbuton of random pont processes. We defne a mappng H : M p M 1 n the followng manner. Let ζ = 1 δ x, where x s an arbtrary enumeraton of the ponts of ζ M p. We let Hζ to be the probablty measure defned by P 3 Hζ = 1 x τ 1 1 x2 < δ 0 otherwse, where, under a probablty measure P, τ s a sequence of..d. mean 1 exponental random varables. Note that the condton 1 x2 < guarantees that the sum nsde the probablty converges P-a.s. It s clear that the mappng H s well defned n the sense that Hζ does not depend on the enumeraton of the ponts of ζ. We defer the proof of the followng lemma to Appendx A. Lemma 1.2. The map H s measurable. We are now ready to state our frst man result, descrbng the weak quenched lmtng dstrbuton for the httng tmes centered by the quenched mean. Theorem 1.3. Let Assumptons 1-3 hold, and for any ω Ω let µ n,ω M 1 be defned by Tn E ω T n 4 µ n,ω = P ω n 1/κ. Then there exsts a λ > 0 such that µ n,ω = HN λ,κ where N λ,κ s a non-homogeneous Posson pont process on 0, wth ntensty λx κ 1. Remark 1.4. The Gaussan and centered exponental dstrbutons that were shown n [Pet09] to be subsequental quenched lmtng dstrbutons of the httng tmes are both, clearly, n the support of the random lmtng probablty measure obtaned n Theorem 1.3. Indeed, lettng ζ k = kδ k 1/2 M p we see that Hζ 1 s a centered exponental dstrbuton, and the central lmt theorem mples that lm k Hζk s a standard Gaussan dstrbuton. Remark 1.5. One can represent the non-homogeneous Posson process N λ,κ as N λ,κ =, j=1 δ λ 1/κ Γ 1/κ j where Γ j j 1 s the ncreasng sequence of the ponts of the unt rate homogeneous Posson process on 0,. In partcular, the ponts of N λ,κ are square summable wth probablty 1 f κ < 2 and square summable wth probablty 0 f κ 2. Furthermore, the random lmtng dstrbuton n Theorem 1.3 can be wrtten n the form 5 HNλ,κ = P λ 1/κ j=1 Γ 1/κ j τ j 1 and we recall that the probablty n 5 s taken wth respect to the exponental random varables τ j, whle keepng the standard Posson arrvals Γ j fxed.,

5 WEAK QUENCHED LIMITS 5 The random probablty measure L = HN λ,κ above has a curous stablty property n M 1 : f L 1,..., L n are..d. copes of L, then 6 L 1... L n law = L /n 1/κ for n = 1, 2,.... To see why ths s true, represent each L as n 5, but usng an ndependent sequence of Posson arrvals for each = 1,..., n. Then the n-fold convoluton L 1... L n has the same representaton, but the sequence of the standard Posson arrvals has to be replaced by a superposton of n such ndependent sequences. Snce a superposton of ndependent Posson processes s, once agan, a Posson process and the mean measures add up, we conclude that L 1... L n law = P λ 1/κ j=1 Γ 1/κ j τ j 1 where Γ j j s the ncreasng sequence of the ponts of a homogeneous Posson random measure on 0, wth ntensty n. Snce the sequence Γ j /n j also forms a Posson random measure wth ntensty n, 6 follows. Snce we know that when κ < 2 there s no centerng and scalng that results n convergence to a determnstc dstrbuton, we have some flexblty n choosng what centerng and scalng to work wth. For example, f we use the averaged centerng and scalng n Theorem 1.1, then a slghtly dfferent random probablty dstrbuton wll appear n the lmt. Before statng ths result we need to ntroduce some more notaton. Defne mappngs H, H ε : M p M 1, ε > 0, as follows. For ζ = 1 δ x, Hζ and H ε ζ are the probablty measures defned by P 1 7 Hζ = x τ f 1 x < δ 0 1 x =. and 8 H ε ζ = P x τ 1 x >ε}. 1, As was the case n the defnton of H n 3, the defnton of Hζ does not depend on a partcular enumeraton of the ponts of ζ. Furthermore, an obvous modfcaton of the proof of Lemma 1.2 shows that the map H s measurable. The maps H ε are even almost contnuous, as wll be seen n Secton 7. Theorem 1.6. Let Assumptons 1-3 hold. For λ, κ > 0 let N λ,κ be a non-homogeneous Posson pont process on 0, wth ntensty λx κ 1. Then for every κ 0, 2 there s a λ > 0 such that the followng statements hold. 1 If κ 0, 1, then Tn µ n,ω = P ω n 1/κ = HN λ,κ. 2 If κ = 1, then µ n,ω = P ω Tn ndn n [ ] = lm Hε N λ,1 δ cλ,1 ε 0 + ε, where c λ,1 ε = 1 ε λx 1 dx = λ log1/ε, and Dn s a sequence such that Dn A log n for some A > 0.

6 6 JONATHON PETERSON AND GENNADY SAMORODNITSKY 3 If κ 1, 2, then Tn n/v P µ n,ω = P ω n 1/κ where c λ,κ ε = ε λx κ dx = λ κ 1 ε κ 1. [ ] = lm Hε N λ,κ δ cλ,κ ε 0 + ε, Remark 1.7. The lmts as ε 0 + n the cases 1 κ < 2 n Theorem 1.6 are weak lmts n M 1. The fact that these lmts exst s standard; see e.g. [ST94]. As we show n Secton 7, fxng a Posson process N λ,κ on some probablty space for example, as n Remark 1.5, even convergence wth probablty 1 holds. The lmtng random probablty measures obtaned n the dfferent parts of Theorem 1.6 also have stablty propertes n M 1, smlar to the stablty property of HNλ,κ descrbed n Remark 1.5. Specfcally, f L 1, L 2,..., L n are..d. copes of the lmtng random probablty measure L n Theorem 1.6, then the stablty relaton for the convoluton operaton 6 stll holds f κ 1. In the case κ = 1, the correspondng stablty relaton s L 1... L n law = L /n λ log n. The proof s smlar to the argument used n Remark 1.5. We omt the detals. The statement and proof of the weak quenched lmts wth the quenched centerng Theorem 1.3 s much smpler than the correspondng result wth the averaged centerng Theorem 1.6. However, n transferrng a lmtng dstrbuton from the httng tmes T n to the locaton of the random walk X n t s easer to use the averaged centerng. Corollary 1.8. Let Assumptons 1 3 hold for some κ 0, 2, and let λ > 0 be gven by Theorem If κ 0, 1, then for any x R, P ω Xn n κ < x = HN λ,κ x 1/κ,. 2 If κ = 1, then there exsts a sequence δn n/a log n wth A > 0 as n the concluson of Theorem 1.6 such that for any x R, Xn δn P ω n/log n 2 < x = lm H ε N λ,1 δ cλ,1 ε 0 + ε A 2 x,. 3 If κ 1, 2, then for any x R, Xn nv P P ω n 1/κ < x = lm H ε N λ,κ δ cλ,κ ε 0 + ε xv 1 1/κ P,. Remark 1.9. The type of convergence n Corollary 1.8 s weaker than that n Theorems 1.3 and 1.6. Instead of provng that the quenched dstrbuton of X n centered and scaled converges n dstrbuton on the space M 1, we only prove that certan projectons of the quenched law converge n dstrbuton as real valued random varables. We suspect that, wth some extra work, the technques of ths paper could be used to prove a lmtng dstrbuton for the full quenched dstrbuton of X n, but we wll leave that for a future paper. Some results n ths drecton have prevously been obtaned n [ESZ09a] Remark Theorem 1.6 and Corollary 1.8 generalze the stable lmtng dstrbutons under the averaged law [KKS75]. For nstance, when κ 0, 1, [ ] Tn P n 1/κ x Tn = E P P ω n 1/κ x E[HN λ,κ, x]], and t s easy to see that E[HN λ,κ, x]] = L κ,b x for some b > 0.

7 WEAK QUENCHED LIMITS 7 The structure of the paper s as follows. In Secton 2 we ntroduce some notaton and revew some basc facts that we wll need. Then, n Secton 3 we outlne a general method for transferrng a lmtng dstrbuton result for one sequence of random probablty measures to another sequence of random probablty measures by constructng a couplng between the two sequences. The method developed n Secton 3 s then mplemented several tmes n Secton 4 to reduce the study of the quenched dstrbuton of the httng tmes T n to the quenched dstrbuton of a certan envronment-dependent mxture of exponental random varables. Then, these envronment-dependent mxng coeffcents are shown n Secton 5 to be related to a nonhomogeneous Posson pont process N λ,κ. In Secton 6 we complete the proof of Theorem 1.3 by provng a weak quenched lmtng dstrbuton for ths mxture of exponentals. The proof of Theorem 1.6 s smlar to the proof of Theorem 1.3, and n Secton 7 we ndcate how to complete the parts of the proof that are dfferent. Fnally, n Secton 8 we gve the proof of the Corollary 1.8. Before turnng to the proofs, we make one remark on the wrtng style. Throughout the paper, we wll use c, C, and C to denote generc constants that may change from lne to lne. Specfc constants that reman fxed throughout the paper are denoted C 0, C 1, etc. 2. Background In ths secton we ntroduce some notaton that wll be used throughout the rest of the paper. For RWRE on Z, many quenched probabltes and expectatons are explctly solvable n terms of the envronment. It s n order to express these formulas compactly that we need ths addtonal notaton. Recall that ρ x = 1 ω x /ω x, x Z. Then, for j we let j j j 9 Π,j = ρ x, R,j = Π,k, and W,j = Π k,j. Denote x= 10 R = lm j R,j = k= k= Π,k and W j = lm W,j = k= j k= Note that Assumpton 2 mples that R and W j are fnte wth probablty 1 for all, j Z. The followng formulas are extremely useful see [Ze04] for a reference Π k,j. 11 P x ω T > T j = R,x 1 R,j 1 and P x ω T < T j = Π,x 1R x,j 1 R,j 1, < x < j, 12 E ωt +1 = 1 + 2W, Z. 13 As n [PZ09, Pet09], we defne the ladder locatons ν of the envronment by ν 0 = 0, and ν = nfn > ν 1 : Π ν 1,n 1 < 1}, 1. Snce the envronment s..d., the sectons of the envronment ω x : ν 1 x < ν } between successve ladder locatons are also..d. However, the envronment drectly to the left of ν 0 = 0 s dfferent from the envronment to the left of ν for > 1. Thus, as n [PZ09, Pet09] t s convenent to defne a new probablty law on envronments by 14 Q = P Π, 1 < 1, all 1 ; by Assumpton 2 the condton s an event of postve probablty. Two facts about the dstrbuton Q wll be mportant to keep n mnd throughout the remander of the paper. Under the measure Q the envronments statonary under shfts by the ladder locatons ν.

8 8 JONATHON PETERSON AND GENNADY SAMORODNITSKY Snce, under P, the envronment s..d., the measure Q concdes wth the measure P on σω x : x 0. Often for convenence we wll denote ν 1 by ν. It was shown n [PZ09, Lemma 2.1] that the dstrbuton of ν whch s the same under P and Q has exponental tals. That s, there exst constants C, C > 0 such that 15 P ν > x = Qν > x C e Cx, x 0. In partcular ths mples that lm ν n /n = ν := E Q ν = E P ν, both P and Q - a.s.. In contrast, t was shown n [PZ09, Theorem 1.4] that, under Assumpton 3, the dstrbuton of the frst httng tme E ω T ν has power tals under the measure Q. That s, there exsts a constant C 0 such that 16 QE ω T ν > x C 0 x κ, x. 3. A General Method for Transferrng Weak Quenched Lmts Our strategy for provng weak quenched lmts for the httng tmes wll be to frst prove a weak quenched lmtng dstrbuton for a related sequence of random varables. Then by exhbtng a couplng between the two sequences of random varables we wll be able to conclude that the httng tmes have the same weak quenched lmtng dstrbuton. The second of these steps s accomplshed through the followng lemma. It apples to random probablty measures on R 2, whch are smply random varables takng values n M 1 R 2. The latter space s the space of all probablty measures on R 2 whch can be turned nto a complete, separable metrc space n the same way as t was done to the space M 1 n Secton 1. The two maps assgnng each probablty measure n M 1 R 2 ts two margnal probablty measures are automatcally contnuous. Lemma 3.1. Let θ n, n = 1, 2,... be a sequence of random probablty measures on R 2 defned on some probablty space Ω, F, P. Let γ n and γ n be the two margnals of θ n, n = 1, 2,.... Suppose that for every δ > 0 17 lm P θ n x, y : x y δ } > δ = 0. If γ n = γ for some γ M 1, then γ n = γ as well. Remark 3.2. Generally the space Ω wll be the space of envronments and P wll be the measure Q on envronments defned n 14. However, n one applcaton Lemma 4.2 below we wll use slghtly dfferent spaces and measures and so we need to state Lemma 3.1 n ths more general form. Proof. The defnton of the Prohorov metrc ρ n 2 mples that, f θ n x, y : x y δ } δ, then ργ n, γ n δ. Therefore, the assumpton 17 mples that ργ n, γ n 0 n probablty. Now the statement of the lemma follows from Theorem 3.1 n [Bl99]. The followng s an mmedate corollary. Corollary 3.3. Under the setup of Lemma 3.1, assume that 18 E θn X Y 0, n P-probablty here X and Y are the coordnate varables n R 2 and E θn s expectaton wth respect to the measure θ n. If γ n = γ for some γ M 1, then γ n = γ as well. Proof. The clam follows mmedately from Lemma 3.1 and Markov s nequalty va P θ n X Y δ δ PE θn X Y δ 2.

9 WEAK QUENCHED LIMITS 9 Remark 3.4. By the Cauchy-Scwarz nequalty, a suffcent condton for 18 s 19 E θn X Y 0 and Var θn X Y 0, n P-probablty. 4. A Seres of Reductons In ths secton we repeatedly apply Lemma 3.1 and Corollary 3.3 to reduce the problem of fndng weak quenched lmts of the httng tmes T n to the problem of fndng weak quenched lmts of a smpler sequence of random varables that s a random mxture of exponental dstrbutons. Frst of all, nstead of studyng the quenched dstrbutons of the httng tmes, t wll be more convenent to study the httng tmes along the random sequence of the ladder locatons ν n. Snce by 15, the dstance between consecutve ladder locatons has exponental tals, and ν n /n ν = E P ν 1 the quenched dstrbuton of T n should be close to the quenched dstrbuton of T νᾱn wth ᾱ = 1/ ν for ease of notaton we wll wrte νᾱn nstead of ν ᾱn. Based on ths, we wll reduce our problem to provng a quenched weak lmt theorem for T νn = n T ν T ν 1. Secondly, as mentoned n the ntroducton, the proof of the non-exstence of quenched lmtng dstrbutons for httng tmes n [Pet09] hnged on two observatons. The frst of these says that, for large n, the magntude of T νn s manly determned by the ncrements T ν T ν 1 for those = 1,..., n for whch there s a large trap between the ladder locatons ν 1 and ν. The second observaton s that, when there s a large trap between ν 1 and ν, the tme to cross from ν 1 to ν s, approxmately, an exponental random varable wth a large mean. That s, T ν T ν 1 may be approxmated by β τ where 20 β = β ω = E ν 1 ω T ν = E ω T ν T ν 1, and τ s a mean 1 exponental random varable that s ndependent of everythng else. When analyzng the httng tmes of the ladder locatons T νn the measure Q s more convenent to use than the measure P snce, under Q, the envronment s statonary under shfts of the envronment by the ladder locatons. In partcular, β } 1 s a statonary sequence under Q. The man result of ths secton s the followng proposton. Proposton 4.1. For ω Ω, suppose that P ω s expanded so that there exsts a sequence τ whch, under P ω, s an..d. sequence of mean 1 exponental random varables. Let σ n,ω M 1 be defned by 1 21 σ n,ω = P ω β τ 1, n 1/κ Q P where β = β ω s gven by 20. If σ n,ω = HNλ,κ then µ n,ω = HN λ/ ν,κ, where µ n,ω s defned n 4. Lemma 3.1 says that weak mts for one sequence of M 1 -valued random varables can be transferred to another sequence of M 1 -valued random varables f these random probablty measures can be coupled n a nce way. We pursue ths dea and prove Proposton 4.1 by establshng the seres of lemmas below. All of these results wll be proved usng Lemma 3.1 and Corollary 3.3. Lemma 4.2. If µ n,ω Q = HNλ,κ then µ n,ω P = HN λ,κ. Lemma 4.3. For ω Ω, let φ n,ω M 1 be defned by Tνn E ω T νn 1 φ n,ω = P ω n 1/κ = P ω n 1/κ T ν T ν 1 β.

10 10 JONATHON PETERSON AND GENNADY SAMORODNITSKY If φ n,ω Q Q = HNλ,κ then µ n,ω = HNλ/ ν,κ. Lemma 4.4. If σ n,ω Q = HNλ,κ then φ Q n,ω = HNλ,κ. Proof of Lemma 4.2. Recall that P and Q are dentcal on σω x : x 0. We start wth a couplng of P and Q that that produces two envronments that agree on the non-negatve ntegers. Let ω be an envronment wth dstrbuton P and let ω be an ndependent envronment wth dstrbuton Q. Then, construct the envronment ω by lettng ω x ω x x 1 = ω x x 0. Then ω has dstrbuton Q and s dentcal to ω on the non-negatve ntegers. Let P be the jont dstrbuton of ω, ω n the above couplng. Gven a par of envronments ω, ω, we wll construct coupled random walks X n } and X n} wth httng tmes T n } and T n}, respectvely, so that the margnal dstrbutons of X n } and X n} are P ω and P ω respectvely. Let P ω,ω denote the jont dstrbuton of X n } and X n} wth expectatons denoted by E ω,ω, and consder random probablty measures on R 2 defned by [ Tn E ω,ω T n θ n = P ω,ω n 1/κ, T n E ω,ω T n ] n 1/κ. We wsh to construct the coupled random walks so that 22 lm n 1/κ E ω,ω T n E ω,ω T n T n E ω,ω T n = 0, P a.s. Ths wll be more than enough to satsfy condtons 18 of Corollary 3.3, and the concluson of Lemma 4.2 wll follow. We now show how to construct coupled random walks X n } and X n}. Snce the envronments ω and ω agree on the non-negatve ntegers, our couplng wll cause the two walks to move n the same manner at all locatons x 0. Precsely, on ther respectve th vsts to ste x 0, they wll both ether move to the rght or both move to the left. To do ths, let ξ = ξ x, } x Z, 1 be a collecton of..d. standard unform random varables that s ndependent of everythng else. Then, gven ω, ω and ξ, construct the random walks as follows: X n + 1 f X n = x, #k n : X k = x} =, and ξ x, ω x X 0 = 0, and X n+1 = X n 1 f X n = x, #k n : X k = x} =, and ξ x, > ω x and X 0 = 0, and X n+1 = X n + 1 f X n = x, #k n : X k = x} =, and ξ x, ω x X n 1 f X n = x, #k n : X k = x} =, and ξ x, > ω x. Havng constructed our couplng, we now turn to the proof of 22. It s enough n fact to show that 23 sup n E ω,ω T n T n <, and sup E ω,ω T n E ω,ω T n <, P-a.s. n To show the second nequalty n 23, we use the explct formula 12 for the quenched expectatons of httng tmes, so that E ω T n = n + 2 W = n + 2 W 0, + Π 0, W 1 = n + 2 W 0, + 2W 1 R 0,n. =0 =0 =0

11 WEAK QUENCHED LIMITS 11 Smlarly, wth the obvous notaton for correspondng random varables correspondng to ω E ω T n = n + 2 W 0, + 2W 1R 0,n 1 = n + 2 W 0, + 2W 1R 0,n, =0 where the second equalty s vald because ω x = ω x for all x 0. Thus, sup n E ω,ω T n E ω,ω T n = sup 2R 0,n W 1 W 1 = 2R 0 W 1 W 1 <, n Turnng to the frst nequalty n 23, let L n := T n k=0 1 Xk <0}, L n := T k=0 =0 1 Xk <0}, P-a.s. be the number of vsts by by the walks X n } and X n}, correspondngly, to the negatve ntegers, by the tme they reach ste x = n. The couplng of T n and T n constructed above s such that T n T n = L n L n. Therefore, E ω,ω T n T n = E ω,ω L n L n E ω L n + E ω L n. Lettng L = lm L n and L = lm L n denote the total amount of tme spent n the negatve ntegers by the random walks X n } and X n}, respectvely, we need only to show that E ω L + E ω L <, P-a.s. To ths end, note that L = G U where G s the number of tmes the random walk X n } steps from 0 to 1 and the U s the amount of tme t takes to reach 0 after the th vst to 1. Note that G s a geometrc random varable startng from 0 wth success parameter P ω T 1 = > 0, and that the U are ndependent and ndependent of G wth common dstrbuton equal to that of the tme t takes a random walk n envronment ω to reach 0 when startng at 1. Thus, by frst condtonng on G, we obtan that Smlarly, [ ] E ω L = E ω G E 1 ω T 0 = E 1 P ω T 1 < ω T 0 P ω T 1 =. E ω L = E 1 P ω T 1 < ω T 0 P ω T 1 =. Ths completes the proof snce Eω 1 T 0 and E 1 ω T 0 are fnte, P-a.s. by 12. Proof of Lemma 4.3. For ω Ω, let ˆφ n,ω M 1 be defned by Tνᾱn E ω T νᾱn ˆφ n,ω A = P ω n 1/κ A = φ n 1/κ ᾱn,ω ᾱn 1/κ A. Snce n 1/κ / ᾱn 1/κ ᾱ 1/κ = ν 1/κ as n, t follows for example, by Lemma 3.1 that φ n,ω = Q HN λ,κ mples that ˆφn,ω = Q HN λ,κ ν 1/κ Now, t follows from 5 that HN λ,κ ν 1/κ Law = HN λ/ ν,κ. Therefore, the clam of the lemma wll follow once we check that that Q Q 24 ˆφn,ω = HNλ,κ mples that µ n,ω = HNλ,κ To show 24 we wll verfy condton 19 of the remark followng Corollary 3.3. Snce both ˆφ n,ω and µ n,ω are mean zero dstrbutons on R, t s enough to show that 25 lm Q n 2/κ Var ω T n T νᾱn > δ = 0, δ > 0.

12 12 JONATHON PETERSON AND GENNADY SAMORODNITSKY To ths end, note that f νᾱn n ν k then Var ω T n T νᾱn = n x=νᾱn+1 Var ωt x T x 1 Var ω T νk T νᾱn. A smlar nequalty holds f ν k n νᾱn. Usng ths, we obtan that for any ε > 0 Q Var ω T n T νᾱn > δn 2/κ Q n νᾱn > εn + Q Var ω T ν[ᾱn]+[εn] T νᾱn > δn 2/κ Var ω T νᾱn T ν[ᾱn] [εn] > δn 2/κ 26 + Q = Q n νᾱn > εn + 2Q Var ω T νεn > δn 2/κ, where the last equalty s due to the fact that, under the measure Q, the envronment s statonary under shfts of the ladder locatons. The frst term n 26 vanshes snce νᾱn /n 1, Q-a.s., by the law of large numbers. For the second term n 26, recall that n 2/κ Var ω T νn has a κ-stable lmtng dstrbuton under Q [Pet09, Theorem 1.3]. Thus, there exsts a b > 0 such that lm Q Var ω T νεn > δn 2/κ = 1 L κ,b δε 2/κ. Snce the rght hand sde can be made arbtrarly small by takng ε 0, we have fnshed the proof of 25 and, thus, also of the lemma. Proof of Lemma 4.4. The proof of the lemma conssts of showng that we can couple the standard exponental random varables of Proposton 4.1 wth the random walk X n } n such a way that condton 19 of the remark followng Corollary 3.3 holds. Snce the relevant random probablty measures have zero means, we only need to ensure that n 2/κ Var ω 27 lm Q T νn E ω T νn β τ 1 > δ = 0, δ > 0. We wll perform the couplng n such a way that the sequence of pars T ν T ν 1, τ s ndependent under the quenched law P ω. Snce E ω T νn = n β, ths wll mply that Var ω T νn E ω T νn β τ 1 = Var ω Tν T ν 1 β τ. As n [PZ09], for any defne 28 M = maxπ ν 1,j : ν 1 j < ν }. The utlty of the sequence M s that t s roughly comparable to β and Var ω T ν T ν 1, but M s an..d. sequence of random varables see [PZ09, equatons 15 and 63] for precse statements regardng these comparsons. In [PZ09, Lemma 5.5] t was shown that for any 0 < ε < 1, lm Q 1 n 2/k Var ω T ν T ν 1 1 M n 1 ε/κ } > δ = 0, δ > 0. A smlar argument see also the proof of [PZ09, Lemma 3.1] mples that lm Q 1 n 2/k β 2 1 M n 1 ε/κ } > δ = 0, δ > 0. Then, snce Var ω T ν T ν 1 β τ 2 Var ω T ν T ν 1 + 2β 2, n order to guarantee 27 t s enough to perform a couplng n such a way that for some 0 < ε < 1, 29 lm Q 1 n 2/k Var ω T ν T ν 1 β τ 1 M >n 1 ε/κ } > δ = 0, δ > 0.

13 WEAK QUENCHED LIMITS 13 Recall that we separately couple each exponental random varable τ wth the correspondng crossng tme T ν T ν 1. For smplcty of notaton we wll descrbe ths couplng when = 1, and we wll denote ν 1, β 1 and τ 1 by ν, β and τ, respectvely. Frst, note that T ν can be constructed by dong repeated excursons from the orgn. Let T 0 + = nfn > 0 : X n = 0} be the frst return tme to the orgn, and let F j } j 1 be an..d. sequence of random varables all havng the dstrbuton of T 0 + under P ω T 0 + < T ν. Also, let let S be ndependent of the F j } and have the same dstrbuton as T ν under P ω T ν < T 0 +. Fnally, let N be ndependent of S and the F j } and have a geometrc dstrbuton startng from 0 wth success parameter p ω = P ω T ν < T 0 +. Then we can construct T ν by lettng N 30 T ν = S + F j. Note that 31 β = E ω T ν = E ω S + 1 p ω E ω F 1 p ω Gven ths constructon of T ν, the most natural way to couple T ν wth τ s to provde a couplng between τ and N. We set 1 32 N = c ω τ, where c ω = log1 p ω, so that N s exactly a geometrc random varable wth parameter p ω. For ths couplng, we obtan the followng bound on Var ω T ν βτ. Lemma 4.5. Let T ν and βτ be coupled usng 30 and 32. Then, 33 Var ω T ν βτ E ω S 2 + E ωf Var ω T ν E ω F 1 2 Var ω N. 3 Proof. Frst of all, note that N Var ω T ν βτ = Var ω S + F j βτ 34 j=1 j=1 N = Var ω S + Var ω F j βτ j=1 = Var ω S + Var ω F 1 E ω c ω τ + Var ω c ω τ E ω F 1 βτ Snce c ω τ s ndependent of c ω τ c ω τ, we can use the dentty for β n 31 to wrte, wth the help of a bt of algebra, Var ω c ω τ E ω F 1 βτ = E ω F 1 2 Var ω c ω τ + β 2 2E ω F 1 β Cov c ω τ, τ = E ω F 1 2 2E ω F 1 β/c ω Var ω c ω τ + β 2 = E ω S 2 + 2E ω SE ω F 1 1 p ω p 2 p ω + log1 p ω ω + E ω F p ω p 2 2 p ω p ω log1 p ω. ω p ω.

14 14 JONATHON PETERSON AND GENNADY SAMORODNITSKY Usng a Taylor seres expanson of log1 p for p < 1, one can show that for any p [0, 1, p k p + log1 p = k 0, and 1 p p 2 k=2 2 p p log1 p = 1/3 p k=1 4p k k + 1k + 2k Therefore, Var ω c ω τ E ω F j βτ E ω S 2 + E ωf Recallng 34, we obtan that Var ω T ν βτ Var ω S + E ω S 2 + E ωf Var ω F 1 E ω c ω τ 3 Snce 30 mples that N Var ω T ν = Var ω S + Var ω F = Var ω S + E ω F 1 2 Var ω N + Var ω F 1 E ω N, the bound 33 follows. The utlty of the upper bound n Lemma 4.5 s that E ω F 1 and E ω S are relatvely small when M 1 s large. Lemma 4.6. For 0 < ε < 1, 35 Q E ω S > n 6ε/κ, M 1 > n 1 ε/κ = on 1, and 36 Q E ω F 1 > n 6ε/κ, M 1 > n 1 ε/κ = on 1. The bound 35 on the tal decay of E ω S was proved n [PZ09, Corollary 4.2]. The proof of 36 s smlar and nvolves straghtforward but rather tedous computatons usng explct formulas for quenched expectatons and varances of httng tmes condtoned on extng an nterval on a certan sde. We defer the proof to Appendx B. We now proceed to fnsh the proof of Lemma 4.4 by extendng the couplng of T ν wth τ to all crossng tmes and showng that the resultng couplng satsfes 29. As was done for T ν n 30 we may decompose T ν T ν 1 so that, wth the obvous notaton, Lemma 4.5 tells us that Var ω Tν T ν 1 β τ 1M >n 1 ε/κ } E ω S 2 + E ωf N T ν T ν 1 = S + F j. j=1 + Var ω T ν T ν 1 E ω F 1 2 Var ω N 1 M >n 1 ε/κ }.

15 WEAK QUENCHED LIMITS 15 An mmedate consequence of Lemma 4.6 s that for any 0 < ε < 1, on an event of probablty convergng to one, all the E ω S and E ω F 1 wth n are less than n 6ε/κ when M 1 > n 1 ε/κ. Thus, by choosng 0 < 12ε/κ < 2/κ 1 we obtan that lm Q 1 n 2/κ E ω S 2 + E ωf > δ = 0, δ > 0. M>n1 ε/κ} Therefore, to prove 29 t s enough to show 37 lm Q 1 n 2/κ Var ω T ν T ν 1 E ω F 1 2 Var ω N 1 M >n 1 ε/κ } > δ = 0, δ > 0. In [PZ09], t was shown that, when M 1 s large, β1 2 = E ωt ν 2 s comparable to Var ω T ν1. In fact, as was shown n the proof of Corollary 5.6 n [PZ09], lm Q n 2/κ Varω T ν T ν 1 β 2 1M >n 1 ε/κ } > δ = 0, δ > 0. Therefore t only remans to show that lm Q n 2/κ β 2 E ω F 1 2 Var ω N 1 M >n 1 ε/κ } > δ = 0, δ > 0. Note that by 31 β 2 E ω F 1 2 Var ω N = E ω S 2 + 2E ω SE ω F 1 E ω N E ω F 1 2 E ω N 2 E ω S 2 + 2E ω SE ω T ν. On the event where E ω S n 6ε/κ for all n wth M > n 1 ε/κ we have β 2 E ω F 1 2 Var ω N 1 M >n 1 ε/κ } n1+12ε/κ + 2n 6ε/κ E ν 1 ω T ν = n 1+12ε/κ + 2n 6ε/κ E ω T νn. Agan, applyng Lemma 4.6 wth 0 < 12ε/κ < 2/κ 1, we see that for any δ > 0, lm sup Q n 2/κ β 2 E ω F 1 2 Var ω N 1 M >n 1 ε/κ } > δ lm sup Q n 2/κ+6ε/κ E ω T νn > δ 2 and so the proof wll be complete once we show that n 2/κ+ε E ω T νn = n 2/κ+ε n β converges n probablty to 0 for ε > 0 small enough. If κ < 1, then snce n 1/κ E ω T νn converges n dstrbuton [PZ09, Theorem 1.1], choosng ε < 1/κ works. If κ > 1 then snce E ω T νn = n β and the β are statonary and ntegrable under Q see 16, the ergodc theorem mples that n 1 E ω T νn converges and, hence, choosng ε < 2/κ 1 works. Fnally, when κ = 1 t follows from 16 that for any 0 < p < 1, E Q n β p a p n for some a p 0,, so choosng ε < 1 works. We conclude ths secton by notng that wth a few mnor modfcatons of the proof of Proposton 4.1 we can obtan the followng analog n the case of the averaged centerng.,

16 16 JONATHON PETERSON AND GENNADY SAMORODNITSKY Proposton 4.7. For ω Ω, suppose that P ω s expanded so that there exsts a sequence τ whch, under P ω, s an..d. sequence of mean 1 exponental random varables. Let σ n,ω M 1 be defned by P 1 n ω n 1/κ β τ κ < 1 38 σ n,ω = P 1 n ω n β τ D n κ = 1 P 1 n ω n 1/κ β τ β κ 1, 2, where D n = E Q [β 1 1 β1 νn}] C 0 logn and β = E Q [β 1 ] = E Q [E ω T ν ]. Let c λ,κ ε be as n Theorem 1.6, and set c λ,1 ε = ν ε λx 1 dx = c λ,1 ε + λ log ν. If HN λ,κ κ < 1 Q σ n,ω = lm ε 0 + H ε N λ,1 δ cλ,1 ε κ = 1, lm ε 0 + H ε N λ,κ δ cλ,κ ε κ 1, 2 then µ n,ω P = where µ n,ω s as n Theorem 1.6. HN λ/ ν,κ κ < 1 lm ε 0 + H ε N λ/ ν,κ δ cλ,κ ε κ [1, 2, Remark 4.8. In the case κ = 1, the relaton between the sequences Dn and D n can be gven by Dn = n/ ν n D n/ ν = n/ ν n E [ ] Q β1 1 β1 ν n/ ν }. 5. Analyss of the crossng tmes By Propostons 4.1 and 4.7, our work s reduced to studyng the dstrbuton of a random mxture of exponental random varables, where the random coeffcents are the average crossng tmes β = E ν 1 ω T ν n 20. The followng proposton, whch s the man result of ths secton, establshes a Posson lmt of pont processes arsng from the random coeffcents β. Proposton 5.1. For n 1 let N n,ω be a pont process defned by 39 N n,ω = δ β /n 1/κ. Then, under the measure Q, N n,ω converges weakly n the space M p to a non-homogeneous Posson pont process wth ntensty λx κ 1, where λ = C 0 κ and C 0 s the constant n 16. Q That s, N n,ω = Nλ,κ. Proof. For a pont process ζ = 1 δ x M p and a functon f : 0, ] R +, defne the Laplace functonal ζf = 1 fx. Snce the weak convergence n the space M p s equvalent to convergence of the Laplace functonals evaluated at all contnuous functons wth compact support of the type [δ, ] for some δ > 0 see Proposton 3.19 n [Res08], the statement of the proposton wll follow once we check that for any such f [ 40 lm E Q e Nn,ωf] } = exp 1 e fx λx κ 1 dx. Remark 5.2. An nspecton of the argument of Propostons 3.16 and 3.19 n [Res08] reveals that the convergence n 40 for all contnuous functons wth compact support as above wll follow once t s checked for such functons that are, n addton, Lpschtz contnuous on 0,. 0

17 WEAK QUENCHED LIMITS 17 Recall from 16 that Qβ 1 > x C 0 x κ. Thus, f the β were..d., the concluson of the proposton would follow mmedately; see e.g. Proposton 3.21 n [Res08]. Snce the sequence β s only statonary under Q, our strategy s to show that the dependence between the β s weak enough so that the pont process N n,ω converges weakly to the same lmt as f the β were..d. Recallng the notaton n 9 and 10 and the formula for quenched expectatons of httng tmes n 12, we may wrte Thus, β = A Z + Y, where β = E ν 1 ω T ν = ν ν = ν ν ν 1 j=ν 1 W j ν 1 j=ν 1 W ν 1,j + 2W ν 1 1R ν 1,ν 1. A = W ν 1 1, Z = 2R ν 1,ν 1, and Y = ν ν ν 1 j=ν 1 W ν 1,j. Note that Y and Z only depend on the envronment from ν 1 to ν 1, and therefore Y, Z } 1 s an..d. sequence of random varables wth the same dstrbuton as Y 1, Z 1 = ν + 2 ν j=0 W 0,j, 2R 0,ν 1. Also, note that the sequence A } 1 s statonary under the measure Q. From ths decomposton of β we can see that the reason β s not an..d. sequence s that the sequence A s not..d. The random varables A all have the same dstrbuton under Q as A 1 = W 1. Furthermore, W 1 has exponental tals under Q. That s, there exst constants C, C > 0 such that 41 QW 1 > x C e Cx ; see Lemma n [Pet08]. In addton, W 1 can be very well approxmated by W j, 1 for large j. That s, there exst constants C 1, C 2, C 3 > 0 such that for every j = 1, 2,..., 42 QW 1 W j, 1 > e C 1j C 2 e C 3j. To see ths, defnng the ladder locatons ν k to the left of the orgn n the natural way see [PZ09], observe that for any c > 0, QW 1 W ν k, 1 > e ck e ck E Q [W 1 W ν k, 1] = e ck E Q [Π ν k, 1W ν k 1] = e ck E Q [Π 0,ν 1 ] k E Q [W 1 ]. Snce E Q [Π 0,ν 1 ] < 1 by the defnton of the ladder locatons, choosng c small enough gves us an exponental bound QW 1 W ν k, 1 > e ck C e Ck, k = 1, 2,... for some postve C, C. The bound 42 now follows by wrtng, for a > 0, QW 1 W j, 1 > e cj QW 1 W νaj, 1 > e cj + Qν aj > j, and notcng that, by 15, for a > 0 small enough, the latter probablty s exponentally small as a functon of j. Keepng the exponental bounds 41 and 42 n mnd, we modfy the sequence of the crossng tmes n order to reduce the dependence. For n 1 we set A n and β n = A n Z + Y, = 1, 2,.... Notce that β n and β n Next, we gve a comparson of β n the random varables β n. = W ν 1 n,ν 1 1 j are ndependent f j > n. wth β that wll allow us to analyze the tal behavour of

18 18 JONATHON PETERSON AND GENNADY SAMORODNITSKY Lemma 5.3. There exst constants, C, C > 0 such that Q β 1 β n 1 > e n1/4 Ce C n, n = 1, 2,.... Proof. From the decompostons of β and β n we obtan that β β n = A A n Z. Note that Z 1 = 2R 0,ν 1 2R 0. By 16 there exsts a constant C such that QZ 1 > x Cx κ for all x > 0. Therefore, for any x > 0 Q β 1 β n 1 > x Q A 1 A n 1 > e C 1 n + Q Z 1 > e C 1 n x C 2 e C 3 n + Ce C 1κ n x κ. Choosng x = e n1/4 completes the proof. Based on the truncated crossng tmes β n we defne a sequence of pont processes by N n,ω n = δ n β, n = 1, 2,.... /n 1/κ 1 Lemma 5.4. N n,ω n Q = N λ,κ as n for λ = C 0, the constant n 16. Proof. Let f : 0, ] R + be a contnuous functon vanshng for all 0 < x < δ for some δ > 0, and Lpshtz on the nterval δ,. We wll prove the followng analogue of 40: 43 lm E Q [e ] n } N n,ωf = exp 1 e fx λx κ 1 dx. Accordng to Remark 5.2, ths wll gve us the clam of the lemma. For 0 < τ < 1 we defne a sequence of random random varables We clam that K n τ = card = 1,..., n : both β n for some + 1 j + τn, j n. }. 44 lm lm sup QK n τ > 0 = 0. τ 0 To see ths, let 0 < ε < 1, and consder a sequence of events 0 > δn 1/κ and β n j > δn 1/κ B n ε = for some = 1,..., n, β n > δn 1/κ but maxy, Z εn 1/κ}. Snce by 16 there exsts a constant C such that QmaxY 1, Z 1 > x Cx κ for all x > 0, whle by 41 the random varable A 1 has an exponentally fast decayng tal, we see that QB n ε nq maxy 1, Z 1 εn 1/κ, β n 1 > δn 1/κ nq maxy 1, Z 1 εn 1/κ, A maxy 1, Z 1 > δn 1/κ = O nqmaxy 1, Z 1 > δn 1/κ E Q A1 + 1 κ 1A > δ/ε = O δ κ E Q A1 + 1 κ 1A > δ/ε as n, for example, Breman s lemma [Bre65]. Therefore, 45 lm lm sup QB n ε = 0. ε 0

19 For τ, ε > 0 WEAK QUENCHED LIMITS 19 QK n τ > 0 QB n ε + Q for some = 1,..., n, some + 1 j + τn, maxy, Z > εn 1/κ and maxy j, Z j > εn 1/κ We conclude that QB n ε + τn 2 QmaxY 1, Z 1 > εn 1/κ 2 QB n ε + C 2 ε 2κ τ. lm τ 0 lm sup QK n τ > 0 lm sup QB n ε, and so 44 follows from 45. Fx, for a moment, ε > 0 and take τ > 0 such that for some n 0 we have QK n τ > 0 ε for all n n 0 ; ths s possble by 44. Consder the random sets Snce fx = 0 f x δ, we can wrte [ ] n N 46 E Q e n,ωf = E Q exp By the choce of τ, D n = = 1,..., n : β n > δn 1/κ }. = E Q [exp + E Q [exp := H 1 n + H 2 n. 47 lm sup H n 2 D n f D n f } n β /n 1/κ } ] n β /n 1/κ 1K n τ = 0 D n f } ] n β /n 1/κ 1K n τ > 0 lm sup QK n τ > 0 ε. Moreover, gven the event K n τ = 0}, the ponts n the random set D n are separated, for large n, by more than n and, hence, gven also the random set D n, the random varables β n, D n are ndependent, each one wth the correspondng condtonal dstrbuton. That s, [ H n 1 = QK n τ = 0E Q E Q exp f β n 1 /n 1/κ} n β 1 > δn 1/κ] } cardd n K n τ = 0. The power law 16 and Lemma 5.3 show the weak convergence to the Pareto dstrbuton Q β n 1 /n 1/κ > t n β 1 > δn 1/κ t/δ κ for t δ, and so by the bounded convergence theorem, E Q exp f β n 1 /n 1/κ} n β 1 > δn 1/κ 1 e fδt κt κ+1 dt. Now the clam 43 follows from 46, 47 and the followng lmtng statement: for the constant C 0 n 16, exp C 0 1 αδ κ} lm lm nf E Q α carddn Kn 48 τ = 0 τ 0 = lm lm sup E Q α carddn Kn τ = 0 exp C 0 1 αδ κ} τ 0

20 20 JONATHON PETERSON AND GENNADY SAMORODNITSKY for all 0 < α < 1. In order to complete the proof of the lemma t, therefore, remans to prove 48. We splt the set 1, 2..., n} nto a unon of the followng sets. Let I 1.n = 1,..., [n 3/4 ]}, J 1.n = [n 3/4 ] + 1,..., [n 3/4 ] + [n 2/3 ], I 2.n = [n 3/4 ] + [n 2/3 ] + 1,..., 2[n 3/4 ] + [n 2/3 ]}, J 2.n = 2[n 3/4 ] + [n 2/3 ] + 1,..., 2[n 3/4 ] + 2[n 2/3 ]}, etc. the last nterval can be a bt shorter than the rest. Clearly, the cardnalty m n of the unon of all ntervals J k.n satsfes m n /n 0 as n. We wrte D n = D n I D n J, where D n I resp. D n J contans all the ponts of D n that are n one of the ntervals I k.n resp. J k.n. Observe that the ntervals I k.n are separated by more that n, so for and j n two dfferent of ths type, β n and β n j are ndependent. We have E Q α carddn 1 K n τ = 0 E Q = α carddi n E Q α CardDn I 1.n [n/[n 3/4 ]+[n 2/3 ]]. Repeatng the argument leadng to 44 that shows that β n and β n j can both exceed δn 1/κ for 0 < j n 3/4 only on an event of a vanshng probablty tells us that Therefore, Q CardD n I 1.n = 1 n 3/4 Q β n 1 > δn 1/κ n 3/4 C 0 δ κ n 1 = C 0 δ κ n 1/4, Q CardD n I 1.n > 1 = on 1/4, E Q α CardDn I 1.n = 1 1 αc 0 δ κ n 1/4 + on 1/4, mplyng that lm sup E Q α carddn Kn τ = 0 and the upper lmt part n 48 follows from 44. Smlarly, E Q α carddn 1 K n τ = 0 E Q E Q α carddi n 1 QK n τ = 0 exp C 0 1 αδ κ}, α carddi n QK n τ > 0 QD J n > 0. 1 K n τ = 0, D J n = 0 The last term vanshes n the lmt snce m n /n 0. Therefore, lm nf E Q α carddn Kn τ = 0 exp C 0 1 αδ κ} QK n τ > 0, and the lower lmt part n 48 follows from 44 as well. Now we are ready to fnsh the proof of Proposton 5.1, whch we accomplsh by checkng 40 for nonnegatve contnuous functons f on 0, ] wth compact support that are Lpschtz contnuous on 0,. For any such functon f, [ }] [ E e Nn,ωf] = E exp fβ /n 1/κ = E [ n N e n,ωf exp fβ /n 1/κ fβ n /n 1/κ }].

21 Now, let Ω n := WEAK QUENCHED LIMITS 21 } ω Ω : β β n e n1/4, = 1, 2,... n Lemma 5.3 mples that QΩ c n 0 as n. Snce f s Lpschtz wth some constant c, on the event Ω n we have fβ /n 1/κ fβ n /n 1/κ c n 1/κ β β n cn 1 1/κ e n1/4, and so by Lemma 5.4 [ lm E e Nn,ωf] [ ] [ ] = lm E n N e n,ωf 1 Ωn = E e N λ,κf, provng 40. In addton to the already establshed convergence of the pont processes N n,ω, n the sequel we wll also need the followng tal bound on the sums of the average crossng tmes β that are not extremely large. Lemma 5.5. Let κ [1, 2. Then for any δ > 0, 1 lm lm sup Q ε 0 + n 1/κ β 1 β εn 1/κ } E Q[β 1 1 β1 εn 1/κ } ] δ = 0. Proof. Clearly, β 1 β εn 1/κ } = β εn 1/κ εn 1/κ 1 β >εn 1/κ }. Therefore, 1 Q n 1/κ β 1 β εn 1/κ } E Q[β 1 1 β1 εn 1/κ } ] δ 1 Q n 1/κ β εn 1/κ E Q [β 1 εn ] 1/κ 49 δ/2 + Q ε 1 β >εn 1/κ } nqβ 1 > εn 1/κ 50 δ/2. We wll frst handle the term n 50. For ε > 0, let G ε : M p Z + be defned by G ε ζ = 1 1 x >ε} when ζ = 1 δ x. Then, snce G ε s contnuous on the set M p ε = ζε} = 0}, we conclude by Proposton 5.1 and the contnuous mappng theorem that n 1 β >εn 1/κ } = G ε N n,ω = G ε N λ,κ. Further, t follows from 16 that nqβ 1 > εn 1/κ C 0 ε κ = E[G ε N λ,κ ] as n. Now, snce G ε N λ,κ has Posson dstrbuton wth mean λε κ /κ, we see that lm ε 0 lm sup Q lm ε 0 P ε 1 β >εn 1/κ } nqβ 1 > εn 1/κ δ/2 G ε N λ,κ E[G ε N λ,κ ] δ 2ε 4ε 2 lm ε 0 δ 2 VarG 4ε 2 κ λ εn λ,κ = lm ε 0 δ 2 = 0. κ

22 22 JONATHON PETERSON AND GENNADY SAMORODNITSKY Next, we estmate the probablty n 49. By Chebychev s nequalty and the fact that the β are statonary under Q, ths probablty s bounded above by 4 δ 2 n 2/κ Var Q β εn 1/κ 51 = 4 δ 2 n 2/κ n Var Qβ 1 εn 1/κ + 8 δ 2 n 2/κ n k Cov Q β 1 εn 1/κ, β k+1 εn 1/κ. Now, the tal decay 16 of β 1 and Karamata s theorem see p. 17 n [Res08] mply that lm sup n 2/κ 1 Var Q β 1 εn 1/κ lm n 2/κ 1 E Q [β1 2 ε 2 n 2/κ 4 κ ] = C 0 2 κ ε2 κ. Snce κ < 2 ths vanshes as ε 0 and so to fnsh the proof of the lemma t s enough to show that 1 52 lm lm sup ε 0 n 2/κ n k Cov Q β 1 εn 1/κ, β k+1 εn 1/κ = 0. k=1 k=1 To bound the covarance terms, we use 12 to wrte β k+1 = ν k+1 1 j=ν k 1 + 2W j ν k+1 1 = ν k+1 ν k + 2 W ν1,j + 2W ν1 1Π ν1,ν k 1R νk,ν k+1 1 j=ν k =: β k+1 + 2W ν1 1Π ν1,ν k 1R νk,ν k+1 1. Note that β k+1 s ndependent of β 1, so that for some constant C Cov Q β 1 εn 1/κ, β k+1 εn 1/κ = Cov Q β 1 εn 1/κ, β k+1 εn 1/κ β k+1 εn 1/κ Var Q β 1 εn 1/κ Var Q β k+1 εn 1/κ β k+1 εn 1/κ 53 C ε 1 κ/2 n 1/κ 1/2 E Q [β k+1 β k βk+1 εn 1/κ } ] for n large enough. An examnaton of the formula for β k+1 shows that R νk,ν k+1 1 β k+1. Therefore, E Q [β k+1 β [ ] k βk+1 εn 1/κ } ] = 4E Q Wν 2 1 1Π 2 ν 1,ν k 1Rν 2 k,ν k+1 11 βk+1 εn 1/κ } [ [ ] [ ] 4E Q W 2 ν1 1] EQ Π 2 ν1,ν k 1 EQ Rν 2 k,ν k+1 11 Rνk,ν k+1 1 εn 1/κ } [ ] [ ] ] = 4E Q W 2 1 EQ Π 2 k ,ν 1 EQ [R0,ν 11 2 R0,ν 1 εn 1/κ }, where n the last step we used the nvarance of the dstrbuton Q under shfts by the ladder locatons ν. Further, E Q [W 1 2 ] < by 41, and E Q[Π 0,ν 1 ] < ] 1 by the defnton of the ladder locatons. Also, snce R 0,ν 1 β 1, E Q [R0,ν R 0,ν 1 εn 1/κ } C ε 2 κ n 2/κ 1 for large n. Combnng ths wth 53 and 54 we see that for some 0 < ρ < 1, Cov Q β 1 εn 1/κ, β k+1 εn 1/κ C 2 ε 2 κ n 2/κ 1 ρ k, and ths bound on the covarance s suffcent to prove 52. lemma. Ths fnshes the proof of the

23 WEAK QUENCHED LIMITS 23 We conclude ths secton by gvng a corollary of Lemma 5.5 that s of ndependent nterest. In [PZ09] t was shown that, f 0 < κ < 1, then n 1/κ E ω T νn = n 1/κ n β converges n dstrbuton to a κ-stable random varable. The followng corollary shows that E ω T νn has a stable lmt law when κ [1, 2 as well. Corollary 5.6. If κ = 1, then there exsts a b > 0 and a sequence D n = E[β 1 1 β1 n}] C 0 log n such that lm Q Eω T νn nd n x = L 1,b x, x R. n If κ 1, 2, then lm Q Eω T νn ne Q [E ω T ν1 ] In both cases b κ = λ/κ. n 1/κ x = L κ,b x, x R. Proof. Ths s a drect applcaton of Proposton 5.1 and Lemma 5.5 to Theorem 3.1 n [DH95]. 6. Weak quenched lmts of httng tmes - quenched centerng Havng done the necessary preperatory work n Sectons 4 and 5 we are now ready to prove Q Theorem 1.3. Recall, that by Proposton 4.1 t s enough to show that σ n,ω = HNλ,κ for some λ > 0, where σ n,ω = HN n,ω s gven n 21, whle H and N n,ω are defned by 3 and Q 39, respectvely. Snce N n,ω = Nλ,κ by Proposton 5.1, f the mappng H : M p M 1 were contnuous the statement of Theorem 1.3 would follow by the contnuous mappng theorem. Unfortunately, H s not a contnuous mappng. To overcome ths, we employ a truncaton technque. For ε > 0 defne the a mappng H ε : M p M 1 by modfyng the defnton 21 as follows: 55 Hε ζ = P x τ 11 x >ε}, 1 when ζ = 1 δ x. It turns out that ths mappng s contnuous on the relevant subset of M p. Lemma 6.1. Hε s contnuous on the set M ε p := ζ M p : ζε} = 0}. v ε Proof. Let ζ n ζ M p. Then, by [Res08, Proposton 3.13] there exsts an nteger M and a labellng of the ponts of ζ and ζ n for n suffcently large such that M M ζ ε, = δ x, and ζ n ε, = δ n x, lm H ε ζ n = lm P wth x n 1, xn 2,... xn M x 1, x 2,... x M as n. Consequently, M M x n τ 1 = P x τ 1 = H ε ζ n the space M 1. Proof of Theorem 1.3. Snce PN λ,κ / M ε p = 0, Proposton 5.1, Lemma 6.1 and the contnuous mappng theorem [Bl99, Theorem 2.7] mply that for every ε > 0, 56 Hε N n,ω Q = H ε N λ,κ, as n.