Asymptotics of the scaling factor µ

Note that condition 4.4.94 can be met because of condition 4.1.27. Proof of Lemma 4.4.6: Combining Lemma 4.4.4, Lemma 4.4.5 and formula 4.4.94 we see that there exist constants M i , tending to zero as i → ∞, such that for all x ∈ K Ni and ω ∈ C ξ i , K [0, ∞ g ∗ x k i − E ω g ∗ X N i τ i − σ k i E ω gX N i τ i ≤ M i . 4.4.96 We may conclude from 4.4.96 that E ω gX N i τ i ≤ σ k i −1 M i + g ∗ ∞ . 4.4.97 Since σ k i → ∞ as i → ∞ and since g is continuous on K and nonzero on K ◦ , there exists constants ˜ M i , tending to zero as i → ∞, such that for all x ∈ K Ni and ω ∈ C ξ i , K [0, ∞ E ω g ∗ X N i τ i ≤ ˜ M i . 4.4.98 When we insert this into 4.4.96, then after a suitable redefinition of our constants M i we arrive at 4.4.95. We now translate the statement in Lemma 4.4.6 about the conditional law of X N i given X N i ξ i t t ≥0 = ωt t ≥0 4.4.99 into a statement about the unconditional law. Lemma 4.4.7 For i ∈ N let X N i be a solution of 4.1.5 with initial condition 4.4.22 and let λ i i ∈ N be constants satisfying 4.4.94. For i ∈ N and ξ ∈ N i let Y i ξ be given by Y i ξ : = ∞ σ k i gX N i t − g ∗ x k i λ −1 i e t λ i dt. 4.4.100 Then there exist constants M i , tending to zero as i → ∞, such that for all x ∈ K Ni E Y i Y i ξ i ≤ M i . 4.4.101 Proof of Lemma 4.4.7: Note that almost surely Y i ξ ≥ −g ∗ ∞ , and hence E |Y i ξ | = ∞ −g ∞ y P[Y i ξ ∈ dy] = −g ∞ y P[Y i ξ ∈ dy] + ∞ y P[Y i ξ ∈ dy] ≤ g ∞ + E[Y i ξ ] i ∈ N , ξ ∈ N i . 4.4.102 Lemma 4.4.6 implies that for a suitable version of the conditional expectation |E[Y i |Y i ξ i = y]| ≤ M i y ≥ −g ∗ ∞ , 4.4.103 and by symmetry the same is true for the conditional expectation of Y i ξ i given Y i . It follows that E[Y i Y i ξ i ] = ∞ −g ∗ ∞ E[Y i Y i ξ i |Y i ξ i = y]P[Y i ξ i ∈ dy] = ∞ −g ∗ ∞ y E[Y i |Y i ξ i = y]P[Y i ξ i ∈ dy] ≤ ∞ −g ∗ ∞ y E[Y i |Y i ξ i = y] P[Y i ξ i ∈ dy] ≤ M i g ∗ ∞ + E[Y i ξ i ] ≤ M i g ∞ + M i , 4.4.104 where in the last step we used that E[Y i ξ i ] = ∞ −g ∗ ∞ E[Y i ξ i |Y i = y]P[Y i ∈ dy] ≤ ∞ −g ∗ ∞ E[Y i ξ i |Y i = y] P[Y i ∈ dy] ≤ M i . 4.4.105 Lemma 4.4.8 For i ∈ N let X N i be a solution of 4.1.5 with initial condition 4.4.22, and let λ i i ∈ N be constants satisfying 4.4.94. Then there exist con- stants M i , tending to zero as i → ∞, such that for all x ∈ K Ni E ∞ g ∗ X N i , k i t − σ k i 1 N k i i ξ : ξ≤k i gX N i ξ t λ −1 i e t λ i dt 2 ≤ M i . 4.4.106 Proof of Lemma 4.4.8: Defining random variables Y i ξ as in Lemma 4.4.7 and using symmetry, we see that E ∞ g ∗ x k i − σ k i 1 N k i i ξ : ξ≤k i gX N i ξ t λ −1 i e t λ i dt 2 = 1 N 2k i i ξ : ξ≤k i η : η≤k i E Y i ξ Y i η = N k i i N k i i − N k i −1 i N 2k i i E[Y i Y i ξ i ] + 1 N 2k i i ξ : ξ≤k i η : η≤k i ξ−η≤k i −1 E Y i ξ Y i η ≤ E[Y i Y i ξ i ] + 1 N i g ∗ ∞ + σ k i g ∞ , 4.4.107 where σ k i ∼ c −k i c1 − c and hence σ k i N i → 0 by 4.4.94. But E ∞ g ∗ x k i − g ∗ X N i , k i t λ −1 i e t λ i dt 2 ≤ E ∞ g ∗ x k i − g ∗ X N i , k i t 2 λ −1 i e t λ i dt ≤ g ∗ ∞ E ∞ x k i − X N i , k i t 2 λ −1 i e t λ i dt ≤ g ∗ ∞ M λ i N k i i 4.4.108 by Corollary 4.4.2. Here λ i N k i i → 0 as i → ∞ by 4.4.94, and combining 4.4.107 and 4.4.108 and applying Lemma 4.4.7 we arrive at 4.4.106. Lemma 4.4.9 For i ∈ N let X N i be a solution of 4.1.5 with initial condition 4.1.6. Then for i ∈ N there exists positive constants γ i , M i satisfying γ i ≪ c k i and M i ≪ 1 i → ∞, such that for all t ≥ 0 E ∞ g ∗ ˆ X i t + s − ˆ G i t + s γ −1 i e −sγ i ds 2 ≤ M i . 4.4.109 Proof of Lemma 4.4.9: Let us write R i t : = g ∗ ˆ X i t − ˆ G i t = g ∗ X N i , k i β i t − σ k i 1 N k i i ξ : ξ≤k i gX N i ξ β i t. 4.4.110