Tightness getdoc28f9. 373KB Jun 04 2011 12:04:11 AM

3.6 Tightness

In what follows, we shall derive estimates on pth-moment differences of ˆ A ξ t z ≡ Aξ t z − 〈ν , ψ z t 〉. Recall the assumption A ξ → u in C from Theorem 2.10. Also note that Lemma 3.9b to come will yield that ψ z t x converges to p € t 3 , z − x Š . The estimates of Lemma 3.8 and the convergence of ψ z t taken together will be sufficient to show C-tightness of the approximate densities A ξ t z at the end of this section. Lemma 3.8. For 0 ≤ s ≤ t ≤ T, y, z ∈ N −1 Z , |t − s| ≤ 1, | y − z| ≤ 1, λ 0 and p ≥ 2 we have E ” ˆ A ξ t z − ˆ A ξ s y p — ≤ Cλ, p, T € 1 + C p Q Š e λp z € |t − s| p 24 + |z − y| p 24 + N −p24 Š . Proof. Fix s, t, T, y, z, λ, p as in the statement. We decompose the increment ˆ A ξ t z− ˆ A ξ s y into a space increment ˆ A ξ t z − ˆ A ξ t y and a time increment ˆ A ξ t y − ˆ A ξ s y. We consider first the space differences. From the Green’s function representation 46, the esti- mates obtained in Lemma 3.7b for the error terms E 1 and E 3 and the linearity of M t φ and E 1 t φ, E 3 t φ in φ, we get E ” ˆ A ξ t z − ˆ A ξ t y p — 51 ≤ Cλ, p, T 1 + C p 2 Q N −p16 e λp z + E • M t € ψ z t −· − ψ y t −· Š p ˜ + E h X k=0,1 1 − 2k X m ≥2,i, j=0,1 q k,m,N i j Z t 〈 € F j ◦ Aξ s − Š × F 1 −k ◦ Aξ s − m −1 € F i ◦ p N ∗ ξ s − Š 1 ξ s − · = k , € ψ z t −s − ψ y t −s Š 〉ds p i . Recall Definition 35 and observe that 0 ≤ € p N ∗ ξ s − Š x ≤ 1 follows from ξ s − ∈ {0, 1} Z N . Use this and 0 ≤ Aξ s − x ≤ 1 together with the definition of F k from Notation 2.9 to get E ” ˆ A ξ t z − ˆ A ξ t y p — 52 ≤ Cλ, p, T 1 + C p 2 Q N −p16 e λp z + E • M t € ψ z t −· − ψ y t −· Š p ˜ + E h X m ≥2,i, j,k=0,1 q k,m,N i j Z t 〈 F 1 −k ◦ Aξ s − 1 ξ s − · = k , ψ z t −s − ψ y t −s 〉ds p i ≤ Cλ, p, T 1 + C p 2 Q N −p16 e λp z + E • M t € ψ z t −· − ψ y t −· Š p ˜ + C p Q E h Z t 〈Aξ s − + ξ s − , ψ z t −s − ψ y t −s 〉ds p i . Note that this is the main step to see why the fixed kernel interaction does not impact our results on tightness. In what follows, we shall employ a similar strategy to the proof of Lemma 6 in [13] to obtain estimates on the above. We nevertheless give full calculations as we proceeded in a different logical 643 order to highlight the ideas for obtaining bounds. Minor changes in the exponents of our bounds ensued, both due to the different logical order and the different setup. Let us first derive a bound on E • M t € ψ z t −· − ψ y t −· Š p ˜ . Using the Burkholder-Davis-Gundy inequal- ity 49 from above and observing that the jumps of the martingales M t ψ x t −· are bounded a.s. by C N −12 we have for any 0 ≤ δ ≤ t E • M t € ψ z t −· − ψ y t −· Š p ˜ 33 ≤ Cλ, pE   Z δ kψ z t −s − ψ y t −s k 2 λ 〈1, e −2λ 〉ds 53 + Z t δ kψ z t −s − ψ y t −s k 〈ξ s − , ψ z t −s − ψ y t −s 〉ds Œ p 2    + CpN −p2 48 ≤ Cλ, pE –‚ T sup ≤s≤δ kψ z t −s − ψ y t −s k 2 λ 1 λ + Z t δ kψ z t −s − ψ y t −s k 〈ξ s − , ψ z t −s − ψ y t −s 〉ds Œ p 2    + CpN −p2 . Now observe that by Lemma 3.6a and Lemma 3.4a, 〈ξ s − , ψ z t −s − ψ y t −s 〉 ≤ 〈Aξ s − , ¯ ψ z t −s + ¯ ψ y t −s 〉 ≤ 〈1, ¯ ψ z t −s + ¯ ψ y t −s 〉 = 2. 54 We can therefore apply the estimates from Corollary 3.5b to the first term in 53 and Corollary 3.5c to the second term, assuming δ ≤ € t − N −34 Š ∨ 0 and using | y − z| ≤ 1 to obtain E • M t € ψ z t −· − ψ y t −· Š p ˜ ≤ Cλ, p, T e λp z ¦ |z − y| p 2 t − δ −p + N −p2 t − δ −3p2 + t − δ p 6 © + CpN −p2 . Now set δ = t − €€ |z − y| 1 4 ∨ N −14 Š ∧ t Š and observe that δ ≤ € t − N −34 Š ∨ 0 follows. We obtain t − δ = € |z − y| 1 4 ∨ N −14 Š ∧ t and |z − y| 1 4 ≤ N −14 ⇒ |z − y| p 2 t − δ −p + N −p2 t − δ −3p2 + t − δ p 6 55 ≤ |z − y| p 4 + N −p8 + N −p24 , |z − y| 1 4 N −14 ⇒ |z − y| p 2 t − δ −p + N −p2 t − δ −3p2 + t − δ p 6 ≤ |z − y| p 4 + N −p8 + |z − y| p 24 . Plugging this back in the above estimate we finally have E • M t € ψ z t −· − ψ y t −· Š p ˜ ≤ Cλ, p, T e λp z ¦ |z − y| p 24 + N −p24 © . 644 Next we shall get a bound on the last term of 52. Recall that 〈ξ t , φ〉 = 〈ν t , φ〉. We get E h Z t 〈Aξ s − + ξ s − , ψ z t −s − ψ y t −s 〉ds p i ≤ Cp E   Z δ 〈Aξ s − + ν s − , e −λ 〉ds sup ≤s≤δ kψ z t −s − ψ y t −s k λ p   +E   ‚Z t δ 〈Aξ s − + ν s − , e −λ 〉 kψ z t −s − ψ y t −s k λ ds Œ p   . Now use that 〈Aξ s − + ν s − , e −λ 〉 = 〈Aξ s − + ξ s − , e −λ 〉 ≤ 〈2, e −λ 〉 48 ≤ Cλ 56 to obtain that the above is bounded by Cp ‚ T C λ sup ≤s≤δ kψ z t −s − ψ y t −s k λ Œ p + ‚Z t δ C λ kψ z t −s − ψ y t −s k λ ds Œ p ≤ Cλ, p, T e λp z ¦ |z − y| p 2 t − δ −p + N −p2 t − δ −3p2 + t − δ p 3 © , where we used Corollary 3.5b,c and | y − z| ≤ 1. Here we assumed δ ≤ € t − N −34 Š ∨ 0 when we applied Corollary 3.5b. Now choose δ = t − €€ |z − y| 1 4 ∨ N −14 Š ∧ t Š ≤ € t − N −34 Š ∨ 0 as before. Reasoning as in 55, we get C λ, p, T e λp z € N −p8 + |z − y| p 12 Š as an upper bound. Now we can take all the above bounds together and plug them back into 52 to obtain recall that |z − y| ≤ 1 E ” ˆ A ξ t z − ˆ A ξ t y p — ≤ Cλ, p, T 1 + C p 2 Q + C p Q e λp z € |z − y| p 24 + N −p24 Š . Next we derive a similar bound on the time differences. We start by subtracting the two Green’s function representations again, this time for the time differences, using 46 and Lemma 3.7b for 645 the error terms. E ” ˆ A ξ t z − ˆ A ξ u z p — 57 ≤ Cλ, p, T 1 + C p 2 Q N −p16 e λp z + E • M t € ψ z t −· Š − M u € ψ z u −· Š p ˜ + E h X m ≥2,i, j,k=0,1 q k,m,N i j ¨Z t u 〈 € F j ◦ Aξ s − Š F 1 −k ◦ Aξ s − m −1 € F i ◦ p N ∗ ξ s − Š × 1 ξ s − · = k , ψ z t −s 〉ds + Z u 〈 € F j ◦ Aξ s − Š F 1 −k ◦ Aξ s − m −1 € F i ◦ p N ∗ ξ s − Š ×1 ξ s − · = k , ψ z t −s − ψ z u −s 〉ds © p i ≤ Cλ, p, T 1 + C p 2 Q N −p16 e λp z + E • M t € ψ z t −· Š − M u € ψ z u −· Š p ˜ + E h X m ≥2,i, j,k=0,1 q k,m,N i j ¨Z t u 〈 F 1 −k ◦ Aξ s − 1 ξ s − · = k , ψ z t −s 〉ds + Z u 〈 F 1 −k ◦ Aξ s − 1 ξ s − · = k , ψ z t −s − ψ z u −s 〉ds « p i ≤ Cλ, p, T 1 + C p 2 Q N −p16 e λp z + E • M t € ψ z t −· Š − M u € ψ z u −· Š p ˜ + C p Q E h Z t u 〈Aξ s − + ξ s − , ψ z t −s 〉ds + Z u 〈Aξ s − + ξ s − , ψ z t −s − ψ z u −s 〉ds p i . For the martingale term we now further get via the Burkholder-Davis-Gundy inequality 49 E • M t € ψ z t −· Š − M u € ψ z t −· Š p ˜ ≤ Cp § E • M t € ψ z t −· Š − M u € ψ z t −· Š p ˜ + E • M u € ψ z t −· Š − M u € ψ z u −· Š p ˜ª ≤ CpE M · € ψ z t −· Š t − M · € ψ z t −· Š u p 2 + CpE M · € ψ z t −· − ψ z u −· Š u p 2 + CpN −p2 ≤ Cλ, pE    ‚Z t u kψ z t −s k 〈ξ s − , ψ z t −s 〉ds Œ p 2    + C λ, p Z δ∧u kψ z t −s − ψ z u −s k 2 λ 〈1, e −2λ 〉ds p 2 + C λ, pE   ‚Z u δ∧u kψ z t −s − ψ z u −s k 〈ξ s − , ψ z t −s − ψ z u −s 〉ds Œ p 2   + CpN −p2 , where we used equation 33 to bound the first and second term. Using 48 and reasoning as in 54 the above can further be bounded by C λ, p ‚Z t u kψ z t −s k ds Œ p 2 + C λ, p, T sup ≤s≤δ∧u kψ z t −s − ψ z u −s k p λ + C λ, p ‚Z u δ∧u kψ z t −s − ψ z u −s k ds Œ p 2 + CpN −p2 . 646 Under the assumption N −34 ∧ u ≤ u − δ ∧ u we obtain from Corollary 3.5a, d, e that E • M t € ψ z t −· Š − M u € ψ z u −· Š p ˜ 58 ≤ Cλ, p, T e λp z ¦ t − u p 6 + € |t − u| p 2 + N −p2 Š u − δ ∧ u −3p2 + u − δ ∧ u p 6 + N −p2 © . Finally observe that with δ = u − €€ |t − u| 1 4 ∨ N −14 Š ∧ u Š we get N −34 ∧ u ≤ u − δ and by proceeding as in 55 we obtain E • M t € ψ z t −· Š − M u € ψ z u −· Š p ˜ ≤ Cλ, p, T e λp z ¦ t − u p 6 + |t − u| p 24 + N −p24 + N −p2 © . Finally, we can bound the last expectation of the last line of 57 by using 〈Aξ t −s + ξ s − , ψ z t −s 〉 ≤ 〈1 + 1, ψ z t −s 〉 = 2. Here the last equality followed from Lemma 3.4a. We thus obtain as an upper bound on the last expectation of the last line of 57, Cp ¨ |t − u| p + E –‚Z u 〈Aξ s − + ν s − , ψ z t −s − ψ z u −s 〉ds Œ p ™« . We further have for the second term E h Z u 〈Aξ s − + ν s − , ψ z t −s − ψ z u −s 〉ds p i ≤ Cp E   Z δ∧u 〈Aξ s − + ν s − , e −λ 〉ds sup ≤s≤δ kψ z t −s − ψ z u −s k λ p   +E –‚Z u δ∧u 〈Aξ s − + ν s − , e −λ 〉 kψ z t −s − ψ z u −s k λ ds Œ p ™« 56 ≤ Cλ, p, T ¨‚ sup ≤s≤δ∧u kψ z t −s − ψ z u −s k λ Œ p + ‚Z u δ∧u kψ z t −s − ψ z u −s k λ ds Œ p « ≤ Cλ, p, T e λp z ¦ t − u p 2 u − δ ∧ u −3p2 + N −p2 u − δ ∧ u −3p2 + u − δ ∧ u p 3 © , where we assumed N −34 ∧u ≤ u−δ∧u when we applied Corollary 3.5d together with Corollary 3.5e in the last line. Now reason as from 58 on to obtain C λ, p, T e λp z ¦ |t − u| p 24 + N −p24 © as an upper bound. Taking all bounds together we have for the time differences from 57 E ” ˆ A ξ t z − ˆ A ξ u z p — ≤ Cλ, p, T 1 + C p 2 Q + C p Q e λp z ¦ |t − u| p 24 + N −p24 © . The bounds on the space difference and the time difference taken together complete the proof. 647 We now show that these moment estimates imply C-tightness of the approximate densities. We shall start including dependence on N again to clarify the tightness argument. First define ˜ A ξ N t z = ˆ A ξ N t z on the grid z ∈ N −1 Z , t ∈ N −2 N . Linearly interpolate first in z and then in t to obtain a continuous C -valued process. Note in particular that we can use Lemma 3.8 to show that for 0 ≤ s ≤ t ≤ T, |t − s| ≤ 1 and y, z ∈ R , | y − z| ≤ 1, E ” ˜ A ξ N t z − ˜ A ξ N s y p — ≤ Cλ, p, T € 1 + C p Q Š e λp z € |t − s| p 48 + |z − y| p 24 Š for λ 0, p ≥ 2 arbitrarily fixed. The next lemma shows that ˜ A ξ N t and ˆ A ξ N t remain close. The advantage of using ˜ A ξ N t is that it is continuous. Using Kolmogorov’s continuity theorem see for instance Corollary 1.2 in Walsh [19] on compacts R i 1 ,i 2 1 ≡ {t, x ∈ R + × R : t, x ∈ i 1 , i 2 + [0, 1] 2 } for i 1 ∈ N , i 2 ∈ Z we obtain tightness of ˜ A ξ N t x in the space of continuous functions on n t, x : t, x ∈ R i 1 ,i 2 1 o . Indeed, we can use the Arzelà-Ascoli theorem. With arbitrarily high probability, part ii of Corollary 1.2 of [19] provides a uniform in N modulus of continuity for all N ≥ N . Pointwise boundedness follows from the boundedness of A ξ N t x together with Lemma 3.9b below. Now use a diagonalization argument to obtain tightness of ˜ A ξ N t x : t ∈ R + , x ∈ R N ∈N in the space of continuous functions from R + × R to R equipped with the topology of uniform convergence on compact sets. Next observe that if we consider instead the space of continuous functions from R + to the space of continuous functions from R to R, both equipped with the topology of uniform convergence on compact sets, tightness of ˜ A ξ N t x : t ∈ R + , x ∈ R N ∈N in the former space is equivalent to tightness of ˜ A ξ N t · : t ∈ R + N ∈N in the latter. Finally, tightness of A ξ N t : t ∈ R + N ∈N as cadlag C 1 -valued processes recall that 0 ≤ Aξ N t x ≤ 1 by construction and also the continuity of all weak limit points follow from the next lemma. Lemma 3.9. For any λ 0, T ∞ we have a P € sup t ≤T k ˜ A ξ N t − ˆ A ξ N t k −λ ≥ 7N −14 Š → 0 as N → ∞. b sup t ≤T k〈ν N , ψ · t 〉 − P t 3 u k −λ → 0 as N → ∞. Proof. The proof is very similar to the proof of Lemma 7 in [13]. We shall only give some additional steps for part a to complement the proof of the given reference. a For 0 ≤ s ≤ t we have k〈ν N , ψ · t 〉 − 〈ν N , ψ · s 〉k −λ = sup z 〈Aξ N , ¯ ψ z t − ¯ ψ z s 〉 e −λ|z| ≤ 2N|t − s|. Here we used Lemma 3.6a, 0 ≤ Aξ N ≤ 1 and Lemma 3.4d. Hence, this only changes by 648 ON −1 between the time-grid points in N −2 N . We obtain that P ‚ sup t ≤T k ˜ A ξ N t − ˆ A ξ N t k −λ ≥ 7N −14 Œ ≤ P € ∃t ∈ [0, T ] ∩ N −2 N , s ∈ [0, T ], |s − t| ≤ N −2 such that kAξ N t − Aξ N s k −λ + 〈ν N , ψ · t − ψ · s 〉 −λ ≥ 7N −14 Š ≤ P € ∃t ∈ [0, T ] ∩ N −2 N , s ∈ [0, T ], |s − t| ≤ N −2 such that kAξ N t − Aξ N s k −λ ≥ 6N −14 Š for N big enough. Next note that the value of A ξ N t x changes only at jump times of P t x; y or Q m,i, j.k t x; y 1 , . . . , y m ; z, i, j, k = 0, 1, m ≥ 2 for some y ∼ x respectively for some y 1 , . . . , y m ∼ x and arbitrary z ∈ N −1 Z and that each jump of A ξ N t is by definition of A ξ N t bounded by N −12 . Then, writing P a for a Poisson variable with mean a, we get as a further bound on the above X l ∈Z P € ∃z ∈ N −1 Z ∩ l, l + 1], ∃t ∈ [0, T ] ∩ N −2 N , ∃s ∈ [t, t + N −2 ] with § A ξ N t z − Aξ N s z ∧ Aξ N t+N −2 z − Aξ N s z ª ≥ N −14 e λ|l|−1 ‹ ≤ X l ∈Z N N 2 T P   C N −12   X y ∼0 P N −2 0; y + X i, j,k=0,1,m ≥2 X y 1 ,..., y m ∼0 X u Q m,i, j,k N −2 0; y 1 , . . . , y m ; u    ≥ N −14 e λ|l|−1    ≤ X l ∈Z CT N 3 P € C N −12 P € N −2 €€ N − θ N Š + C Q ŠŠ ≥ N −14 e λ|l|−1 Š ≤ X l ∈Z CT N 3 P € P € N −2 € N + C Q ŠŠŠ p ≥ C N p 4 e λp|l|−1 for some p 0. Now apply Chebyshev’s inequality. Choose p 0 such that 3 − p4 0. Then the resulting sum is finite and goes to zero for N → ∞. b The proof of part b follows as the proof of Lemma 7b of [13].

3.7 Characterizing limit points