The moderate deviation principle

Note further that, for any δ 0 and T 0, there exists ǫ 0 such that Jx ¶ ǫ implies sup 0¶t¶T |x − y| δ, where y t = t − a + for some a ∈ [0, T ]... Then, for θ f 0, P sup 0¶t¶T | 1 κ Z κt − y| ¶ δ ¶ P Z κa ¶ δκ = P T [0, κδ] ¾ κa ¶ e −κaθ Y Φ j¶κδ E exp θ S j = e −κaθ exp X Φ j¶κδ log 1 1 − θ f j , and the result follows because the sum on the right is bounded by a constant multiple of κδ. ƒ

4.3 The moderate deviation principle

Recall from the beginning of Section 4.2 that it is sufficient to show Theorem 1.15 for the approx- imating process Z defined in 17. We initially include the case c = ∞ in our consideration, and abbreviate b κ := a κ κ 2 α−1 1 −α ¯ ℓκ ≪ κ α 1 −α ¯ ℓκ, so that we are looking for a moderate deviation principle with speed a κ b κ . Lemma 4.12. Let 0 ¶ u v, suppose that f and a κ are as in Theorem 1.15 and define I [u,v = Z v u s − α 1 −α ds = 1 −α 1 −2α v 1 −2α 1 −α − u 1 −2α 1 −α . Then the family T[κu, κv − κv − u a κ κ0 satisfies a large deviation principle with speed a κ b κ and rate function I [u,v t =    1 2 I [u,v t 2 if u 0 or t ¶ 1 c I [0,v f 0, 1 c f 0 t − 1 2 I [0,v 1 c f 0 2 if u = 0 and t ¾ 1 c I [0,v f 0. Proof. Denoting by Λ κ the logarithmic moment generating function of b κ T [κu, κv − κv − u, observe that Λ κ θ = log E exp θ b κ T [ κu, κv − κv − u = X w ∈S∩[κu,κv log E exp θ b κ T [w] − θ κb κ v − u = X w ∈S∩[κu,κv ξ θ b κ ¯ f w − θ κb κ v − u = Z I κ ¯ f w ξ θ b κ ¯ f w d w − θ κb κ v − u, 22 where I κ = {w ¾ 0 : ιw ∈ [κu, κv} and ιw = max S∩[0, w]. Since κu ¶ inf I κ κu+ f 0 −1 and κv ¶ sup I κ κv + f 0 −1 we get Λ κ θ − Z I κ ¯ f w ξ θ b κ ¯ f w − θ b κ d w ¶ 2 θ b κ f 0 . 23 1253 Now focus on the case u 0. A Taylor approximation gives ξw = w + 1 2 1 + o1w 2 , as w ↓ 0. By dominated convergence, Z I κ ¯ f w ξ θ b κ ¯ f w − θ b κ d w ∼ 1 2 Z I κ 1 ¯ f w d w × θ 2 b 2 κ ∼ 1 2 κ 1 −2α 1 −α ¯ ℓκ Z v u w − α 1 −α d w × θ 2 b 2 κ = a κ b κ 1 2 I [u,v θ 2 . Together with 23 we arrive at Λ κ θ ∼ a κ b κ 1 2 I [u,v θ 2 . Now the Gärtner-Ellis theorem implies that the family T [ κu, κv − κv − ua κ satisfies a large deviation principle with speed a κ b κ having as rate function the Fenchel-Legendre transform of 1 2 I [u,v θ 2 which is I [u,v . Next, we look at the case u = 0. If θ ¾ 1 c f 0 then Λ κ θ = ∞ for all κ 0, so assume the contrary. The same Taylor expansion as above now gives ¯ f w ξ θ b κ ¯ f w − θ b κ ∼ 1 2 θ 2 b 2 κ ¯ f w as w ↑ ∞. In particular, the integrand in 22 is regularly varying with index − α 1 −α −1 and we get from Karamata’s theorem, see e.g. [Bingham et al., 1987, Theorem 1.5.11], that Λ κ θ ∼ 1 2 θ 2 b 2 κ κ 1 −2α 1 −α ¯ ℓκ Z v s −α1−α ds = a κ b κ 1 2 I [0,v θ 2 . 24 Consequently, lim κ→∞ 1 a κ b κ Λ κ θ = 1 2 I [0,v θ 2 if θ 1 c f 0, ∞ otherwise. The Legendre transform of the right hand side is I [0,v t = 1 2 I [0,v t 2 if t ¶ 1 c I [0,v f 0, 1 c f 0 t − 1 2 I [0,v 1 c f 0 2 if t ¾ 1 c I [0,v f 0. Since I [0,v is not strictly convex the Gärtner-Ellis Theorem does not imply the full large deviation principle. It remains to prove the lower bound for open sets t, ∞ with t ¾ 1 c I [0,v f 0. Fix ǫ ∈ 0, u and note that, for sufficiently large κ, P T [κu, κv − κva κ t ¾ P T [κǫ, κv − κv − ǫa κ 1 c I [0,v f 0 × P T 0, κǫ − κǫa κ −ǫ | {z } →1 P T [0] a κ t − 1 c I [0,v f 0 + ǫ . 1254 so that by the large deviation principle for T [κǫ,κv −κv −ǫa κ and the exponential distribution it follows that lim inf κ→∞ 1 a κ b κ log P T [0, κv − κva κ t ¾ − 1 c I [0,v f 0 2 2 I [ǫ,v − t − 1 c I [0,v f 0 + ǫ 1 c f 0. Note that the right hand side converges to −I [0,v t when letting ǫ tend to zero. This establishes the full large deviation principle for T [0, κv − κva κ . ƒ We continue the proof of Theorem 1.15 with a finite-dimensional moderate deviation principle, which can be derived from Lemma 4.12. Lemma 4.13. Fix 0 = t t 1 · · · t p . Then the vector 1 a κ Z κt j − κt j : j ∈ {1, . . . , p} satisfies a large deviation principle in R p with speed a κ b κ and rate function I a 1 , . . . , a p = p X j=1 I [t j −1 ,t j a j −1 − a j , with a := 0 . Proof. We note that, for −∞ ¶ a j b j ¶ ∞, we have interpreting conditions on the right as void, if they involve infinity P a j a κ ¶ Z κt j − κt j b j a κ for all j = P T [0, κt j + a κ a j ¶ κ t j , T [0, κt j + a κ b j κ t j for all j . To continue from here we need to show that the random variables T [0, κt + a κ b and T [0, κt+ a κ b are exponentially equivalent in the sense that lim κ→∞ a −1 κ b −1 κ log P T [0, κt + a κ b − T [0, κt − a κ b a κ ǫ = −∞ . 25 Indeed, first let b 0. As in Lemma 4.12, we see that for any t ¾ 0 and θ ∈ R, a −1 κ b −1 κ log E exp θ b κ T [ κt, κt + a κ b − a κ b −→ 0, 26 Chebyshev’s inequality gives, for any A 0, P T [0, κt + a κ b − T [0, κt − a κ b a κ ǫ ¶ e −Aa κ b κ E exp A ǫ b κ T [ κt, κt + a κ b − a κ b . A similar estimate can be performed for PT [0, κt + a κ b − T [0, κt − a κ b −a κ ǫ, and the argu- ment also extends to the case b 0. From this 25 readily follows. Using Lemma 1.15 and independence, we obtain a large deviation principle for the vector 1 a κ T [ κt j −1 , κt j − κ t j − t j −1 : j ∈ {1, . . . , p} , 1255 with rate function I 1 a 1 , . . . , a p = p X j=1 I [t j −1 ,t j a j . Using the contraction principle, we infer from this a large deviation principle for the vector 1 a κ T [0, κt j − κ t j : j ∈ {1, . . . , p} with rate function I 2 a 1 , . . . , a p = p X j=1 I [t j −1 ,t j a j − a j −1 . Combining this with 25 we obtain that a −1 κ b −1 κ log P T [0, κt j + a κ a j ] κ t j , T [0, κt j + a κ b j ] κ t j for all j ∼ a −1 κ b −1 κ log P − a κ b j T [0, κt j ] − κ t j −a κ a j for all j , and observing the signs the required large deviation principle. ƒ We may now take a projective limit and arrive at a large deviation principle in the space P 0, ∞ of functions x : 0, ∞ → R equipped with the topology of pointwise convergence. Lemma 4.14. The family of functions 1 a κ Z κt − κt : t κ0 satisfies a large deviation principle in the space P 0, ∞, with speed a κ b κ and rate function I x = ¨ 1 2 R ∞ ˙ x t 2 t α 1 −α d t − 1 c f 0 x if x is absolutely continuous and x ¶ 0. ∞ otherwise. Proof. Observe that the space of functions equipped with the topology of pointwise convergence can be interpreted as the projective limit of R p with the canonical projections given by πx = xt 1 , . . . , xt p for 0 t 1 . . . t p . By the Dawson-Gärtner theorem, we obtain a large deviation principle with rate function ˜Ix = sup t 1 ...t p p X j=2 I [t j −1 ,t j x t j −1 − x t j + I [0,t 1 − x t 1 . Note that the value of the variational expression is nondecreasing, if additional points are added to the partition. We first fix t 1 0 and optimize the first summand independently. Observe that sup t 1 ...t p p X j=2 I [t j −1 ,t j x t j −1 − x t j = 1 2 sup t 1 t 2 ...t p p X j=2 x t j − x t j −1 2 1 −α 1 −2α t 1 −2α 1 −α j − t 1 −2α 1 −α j −1 . Recall that t j − t j −1 t −α 1 −α j ¶ 1 −α 1 −2α t 1 −2α 1 −α j − t 1 −2α 1 −α j −1 ¶ t j − t j −1 t −α 1 −α j −1 . 1256 Hence we obtain an upper and [in brackets] lower bound of 1 2 sup t 1 t 2 ...t p p X j=2 x t j − x t j −1 t j − t j −1 2 t α 1 −α j[ −1] t j − t j −1 . It is easy to see that using arguments analogous to those given in the last step in the proof of the first large deviation principle that this is + ∞ if x fails to be absolutely continuous, and otherwise it equals 1 2 Z ∞ t 1 ˙ x t 2 t α 1 −α d t. In the latter case we have ˜Ix = lim t 1 ↓0 1 2 Z ∞ t 1 ˙ x t 2 t α 1 −α d t + I [0,t 1 − x t 1 . If x 0 the last summand diverges to infinity. If x = 0 and the limit of the integral is finite, then using Cauchy-Schwarz, I [0,t 1 − x t 1 ¶ 1 2 I −1 [0,t 1 Z t 1 ˙ x t d t 2 ¶ 1 2 Z ǫ ˙ x t 2 t α 1 −α d t, hence it converges to zero. If x 0, lim t 1 ↓0 I [0,t 1 − x t 1 = lim t 1 ↓0 − 1 c f 0 x t 1 + 1 −α 1 −2α t α 1 −α 1 1 c f 0 2 = − 1 c f 0 x , as required to complete the proof. ƒ Lemma 4.15. If c ∞, the function I is a good rate function on L 0, ∞. Proof. Recall that, by the Arzelà-Ascoli theorem, it suffices to show that for any η 0 the family {x : Ix ¶ η} is bounded and equicontinuous on every compact subset of 0, ∞. Suppose that Ix ¶ η and 0 s t. Then, using Cauchy-Schwarz in the second step, |x t − x s | = Z t s ˙ x u du ¶ Z t s |˙x u | u α 21 −α u − α 21 −α du ¶ Z t s ˙ x u 2 u α 1 −α du 1 2 Z t s u − α 1 −α du 1 2 ¶ q η 1 −α 1 −2α t 1 −2α 1 −α − s 1 −2α 1 −α 1 2 , which proves equicontinuity. The boundedness condition follows from this, together with the obser- vation that 0 ¾ x ¾ −cη f 0. ƒ To move our moderate deviation principle to the topology of uniform convergence on compact sets, recall the definition of the mappings f m from 21. We abbreviate ¯ Z κ := 1 a κ Z κt − κt : t . 1257 Lemma 4.16. f m ¯ Z κ m ∈N are exponentially good approximations of ¯ Z κ on L 0, ∞. Proof. We need to verify that, denoting by k · k the supremum norm on any compact subset of 0, ∞, for every δ 0, lim m →∞ lim sup κ→∞ a −1 κ b −1 κ log P k ¯Z κ − f m ¯ Z κ k δ = −∞. The crucial step is to establish that, for sufficiently large κ, for all j ¾ 2, P sup t j −1 ¶ t t j ¯ Z κ t − ¯Z κ t j −1 δ ¶ 2 P ¯ Z κ t j − ¯Z κ t j −1 δ 2 . 27 To verify 27 we use the stopping time τ := inf{t ¾ t j −1 : | ¯Z κ t − ¯Z κ t j −1 | δ}. Note that P ¯ Z κ t j − ¯Z κ t j −1 δ 2 ¾ P sup t j −1 ¶t t j ¯ Z κ t − ¯Z κ t j −1 δ P ¯ Z κ t j − ¯Z κ t j −1 δ 2 τ ¶ t j , and, using Chebyshev’s inequality in the last step, P ¯ Z κ t j − ¯Z κ t j −1 δ 2 τ ¶ t j ¾ P ¯ Z κ t j − ¯Z κ τ ¶ δ 2 τ ¶ t j ¾ 1 − 4 δ 2 Var ¯ Z κ t j −t j −1 . As this variance is of order a −2 κ κ 1 −2α 1 −α ¯ ℓκ −1 → 0, the right hand side exceeds 1 2 for sufficiently large κ, thus proving 27. With 27 at our disposal, we observe that, for some integers n 1 ¾ n ¾ 2 depending only on m and the chosen compact subset of 0, ∞, P k ¯Z κ − f m ¯ Z κ k δ ¶ n 1 X j=n P sup t j −1 ¶t t j ¯ Z κ t − ¯Z κ t j −1 δ ¶ 2 n 1 X j=n P ¯ Z κ t j − ¯Z κ t j −1 δ 2 . Hence, we get lim sup κ→∞ a −1 κ b −1 κ log P k ¯Z κ − f m ¯ Z κ k δ ¶ − n 1 inf j=n I [t j −1 ,t j δ 2 , and the right hand side can be made arbitrarily small by making m = 1 t j −t j −1 large. ƒ Proof of Theorem 1.15. We apply [Dembo and Zeitouni, 1998, Theorem 4.2.23] to transfer the large deviation principle from the topological Hausdorff space P 0, ∞ to the metrizable space L 0, ∞ using the sequence f m of continuous functions. There are two conditions to be checked for this, on the one hand that f m ¯ Z κ m ∈N are exponentially good approximations of ¯ Z κ , as verified in Lemma 4.16, on the other hand that lim sup m →∞ sup I x¶ η d f m x, x = 0, for every η 0, where d denotes a suitable metric on L 0, ∞. This follows easily from the equicontinuity of the set {Ix ¶ η} established in Lemma 4.15. Hence the proof is complete. ƒ 1258 5 The vertex with maximal indegree In this section we prove Theorem 1.5 and Proposition 1.18.

5.1 Strong and weak preference: Proof of Theorem 1.5