Tightness, moment bounds. getdoc7e8b. 765KB Jun 04 2011 12:04:34 AM

3 Intermediate results for the proof of Theorem 1.5 We use the classical strategy of proving tightness of {X N } in the space DR + , M F R 2 , and then identify the limits. The main difficulty will come from the above drift term D N ,3 t Φ. Although it will be convenient to have Theorem 1.5 for a continuous parameter N ≥ 3, it suffices to prove it for an arbitrary sequence approaching infinity and nothing will be lost by considering N ∈ N ≥3 . This condition will be in force thoughout the proof of Theorem 1.5, as will the assumption that ξ N is deterministic and all the conditions of Theorem 1.5.

3.1 Tightness, moment bounds.

Recall that a sequence of processes with sample paths in DR + , S for some Polish space S is C-tight in DR + , S iff their laws are tight in DR + , S and every limit point is continuous. Proposition 3.1. The sequence {X N , N ∈ N ≥3 } is C-tight in DR + , M F R 2 . Proposition 3.1 will follow from Jakubowski’s theorem see e.g. Theorem II.4.1 in [13] and the two following lemmas. Lemma 3.2. For any function Φ ∈ C 3 b R 2 , the sequence {X N Φ, N ∈ N ≥3 } is C-tight. Lemma 3.3. For any ε 0, any T 0 there exists A 0 such that sup N ≥3 P ‚ sup t ≤T X N t B0, A c ε Œ ε. Lemma 3.2 will be established by looking separately at each term appearing in 28. The difficulty will mainly lie in establishing tightness for the last term in 28. The proof of the two lemmas is given in Section 6. An important step in proving tightness will be the derivation of bounds on the first and second moments. It will be assumed that N ∈ N ≥3 until otherwise indicated. Proposition 3.4. There exists a c 3.4 0, and for any T 0 constants C a , C b , depending on T , such that for any t ≤ T , a E ” X N t 1 — ≤ € 1 + C a log N −16 Š X N 1 exp c 3.4 t , b E ” X N t 1 2 — ≤ C b € X N 1 + X N 1 2 Š , Part a of the above Proposition is proved in Subsection 4.4, part b is proved in Subsection 5.1. Note that a immediately implies the existence of a constant C ′ a depending on T such that for any t ≤ T , E ” X N t 1 — ≤ C ′ a X N 1. Moreover, by a, b and the Markov property, if we set C ab := C ′ a C b , we have for any s, t ∈ [0, T ], E ” X N s 1X N t 1 — ≤ C ab X N 1 + X N 1 2 . 40 For establishing tightness of some of the terms of 28, and also for proving the compact containment condition, Lemma 3.3, we will need a space-time first moment bound. Recall that t N = log N −19 . 1207 Suppose Φ : R + × R 2 → R. Define |Φ| Lip , respectively |Φ| 1 2 , to be the smallest element in R + such that |Φs, x − Φs, y| ≤ |Φ| Lip |x − y|, ∀s ≥ 0, x ∈ R 2 , y ∈ R 2 , |Φs − t N , x − Φs, x| ≤ |Φ| 1 2 p t N , ∀ s ≥ t N , x ∈ R 2 , We will write ||Φ|| Lip := ||Φ|| ∞ + |Φ| Lip , ||Φ|| 1 2 := ||Φ|| ∞ + |Φ| 1 2 and ||Φ|| := ||Φ|| ∞ + |Φ| Lip + |Φ| 1 2 . Obviously the definition of ||.|| Lip also applies to functions from R 2 into R. Define P N t , t ≥ 0 as the semigroup of the rate−N random walk on S N with jump kernel p N . Lemma 3.5. There exist δ 3.5 0, c 3.5 0 and for any T 0, there is a C 3.5 T , so that for all t ≤ T and any Ψ : R 2 → R + such that ||Ψ|| Lip ≤ T , E ” X N t Ψ — ≤ e c 3.5 t X N € P N t Ψ Š + C 3.5 log N −δ 3.5 X N 1 + X N 1 2 . This lemma requires a key second moment estimate see Proposition 3.10 below, it is proved in Subsection 5.3.

3.2 On the new drift term