Markov chain approximation of pure jump processes - arXiv

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES ´ AND RENE ´ L. SCHILLING ANTE MIMICA (†), NIKOLA SANDRIC,

arXiv:1611.07236v1 [math.PR] 22 Nov 2016

Abstract. In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains.

1. Introduction Let Xn , n ∈ N be a sequence of continuous-time Markov chains where Xn takes values on the lattice n−1 Zd , and let X be a Markov process on Rd . We are interested in the following two questions: (i) Under which conditions does {Xn }n∈N converge weakly to some (non-symmetric) Markov process? (ii) Can a given Markov process X be approximated (in the sense of weak convergence) by a sequence of Markov chains? These questions have a long history. If X is a diffusion process determined by a generator in non-divergence form these problems have been studied in [SV06] using martingale problems. The key ingredient in this approach is that the domain of the corresponding generator is rich enough, i.e. containing the test functions Cc∞ (Rd ). On the other hand, if the generator of X is given in divergence form, it is a delicate matter to find nontrivial functions in its domain. In order to overcome this problem, one resorts to an L2 -setting and the theory of Dirichlet forms; for example, [SZ97] solve these problems for symmetric diffusion processes X using Dirichlet forms. The main assumptions are certain uniform regularity conditions and the boundedness of the range of the conductances of the approximating Markov chains. These results are further extended in [BK08], where the uniform regularity condition is relaxed and the conductances may have unbounded range. Very recently, [DK13] discusses these questions for a non-symmetric diffusion process X. Let us also mention that the problem of approximation of a reflected Brownian motion on a bounded domain in Rd is studied in [BC08]. As far as we know, the paper [HK07] is among the first papers studying the approximation of a jump process X. In this work the authors investigate convergence to and approximation of a symmetric jump process X whose jump kernel is comparable to the jump kernel of a symmetric stable L´evy process. These results have been extended in [BKK10], where the comparability assumption is imposed on the small jumps only, whereas the big jumps are controlled by a certain integrability condition. The case where X is a symmetric process which has both a continuous and a jump part is dealt with in [BKU10]. Let us point out that all of these approaches require some kind of “stable-like” property (or control) of the jump kernel, and the main step in the proofs is to obtain heat kernel estimates of the chains {Xn }n∈N . This is possible due to the uniform ellipticity 2010 Mathematics Subject Classification. 60J25, 60J27, 60J75. Key words and phrases. non-symmetric Dirichlet form, non-symmetric Hunt process, Markov chain, Mosco convergence, semimartingale, semimartingale characteristics, weak convergence. 1

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´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

assumption in the continuous case and the “stable-like” assumption in the jump case. In general, this is very difficult to verify, and in many cases it is even impossible. Using a completely different approach, [CKK13] study the convergence and approximation problems for pure jump processes X on a metric measure space satisfying the volume doubling condition. The proof of tightness is based on methods developed in [BC08, Lemma 2.1] and only works if the approximating Markov chains Xn are symmetric. In order to prove the convergence of the finite-dimensional distributions of {Xn }n∈N to those of X, Mosco convergence of the corresponding symmetric Dirichlet forms is used. This type of convergence is equivalent to strong convergence of the corresponding semigroups. It was first obtained in [Mos94] in the case when all the forms are defined on the same Hilbert space, and then it was generalized in [Kim06] (see also [CKK13] and [KS03]) to the case where the forms are defined on different spaces. We are interested in the convergence and approximation problems for non-symmetric pure jump processes. We will use two approaches: (i) via Dirichlet forms, and (ii) via semimartingale convergence results. The first approach (Section 2.1–2.3) follows the roadmap laid out in [CKK13]: To obtain tightness of {Xn }n∈N we use semimartingale convergence results developed in [JS03]. More precisely, we first ensure that the processes Xn , n ∈ N, are regular Markov chains (in particular, they are semimartingales), then we compute their semimartingale characteristics, and finally we provide conditions for the tightness of {Xn }n∈N in terms of the corresponding conductances. This is based on a result from [JS03] which states that a sequence of semimartingales is tight if the corresponding characteristics are C-tight (i.e. tight and all accumulation points are processes with continuous paths). To get the convergence of the finite-dimensional distributions of {Xn }n∈N in the non-symmetric case we can still use Mosco convergence, but for non-symmetric Dirichlet forms. Just as in the symmetric case, this type of convergence is equivalent to the strong convergence of the corresponding semigroups. It was first obtained in [Hin98] for forms defined on the same Hilbert space, and and then it was generalized in [T¨06] to forms living on different spaces. Our second approach (Section 3.1), is based on a result from [JS03] which provides general conditions under which a sequence of semimartingales converges weakly to a semimartingale. If the processes Xn , n ∈ N, are regular Markov chains and if the limiting process X is a so-called (pure jump) homogeneous diffusion with jumps, we obtain conditions (in terms of conductances and characteristics of X) which imply the desired convergence. As an application, we can now answer question (ii) and provide conditions for the approximation of a given Markov process, both in the Dirichlet form set-up (Section 2.4) and the semimartingale setting (Section 3.2).

Notation. Most of our notation is standard or self-explanatory. Throughout this paper, we write Zdn := n1 Zd = { n1 m : m ∈ Zd } for the d-dimensional lattice with grid size n1 . For p ≥ 1 we use Lpn as a shorthand for LpP (Zdn ), the standard Lp -space on Zdn . If p = 2, the scalar product is given by hf, giL2n := a∈Zdn f (a)g(a). We write Cck (Rd ) for the k-times continuously differentiable, compactly supported functions, and CcLip (Rd ) is the space of Lipschitz continuous functions with compact support. Br (x) is the open ball with radius r > 0 and centre x, diag = {(x, x) : x ∈ Rd } denotes the diagonal in Rd . Finally, the sum A + B of subsets A, B ⊆ Rd is defined as A + B = {a + b : a ∈ A, b ∈ B}.

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES

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2. Convergence of Markov chains using Dirichlet forms Our starting point is a sequence of continuous-time Markov chains {Xtn }t≥0 with state space Zdn and infinitesimal generator X An f (a) = (f (b) − f (a))C n (a, b), f ∈ DAn , b∈Zdn

where the domain is given by n o X DAn := f : Zdn → R : |f (b)|C n (a, b) < ∞ for all a ∈ Zdn . b∈Zdn

A sufficient condition for the existence of Xn is that the kernel C n : Zdn × Zdn → [0, ∞) satisfies the following two properties ∀n ∈ N, ∀a ∈ Zdn : C n (a, a) = 0; X ∀n ∈ N : sup C n (a, b) < ∞,

(T1) (T2)

a∈Zdn

b∈Zdn

see e.g. [Nor98]; in this case, the chain {Xtn }t≥0 is regular, i.e. it has only finitely many jumps on finite time-intervals. If the chain is in state a ∈ Zd , it jumps to state b ∈ Zd P with probability C n (a, b)/ c∈Zdn C n (a, c) after an exponential waiting time with paramP eter c∈Zdn C n (a, c). Moreover, {Xtn }t≥0 is conservative and defines a semimartingale. 1 2 Observe that (T1) implies L∞ n ∪ Ln ∪ Ln ⊆ DAn . Indeed, we have X X |f (b)|C n (a, b) ≤kf k∞ sup C n (a, b), a∈Zdn

b∈Zdn

X

b∈Zdn

|f (b)|C n (a, b) ≤ sup

a∈Zdn

X

C n (a, b)

b∈Zdn

X

b∈Zdn

b∈Zdn

|f (b)| =kf kL1n sup

a∈Zdn

X

b∈Zdn

and, using the Cauchy–Schwarz inequality, X X p p |f (b)|C n (a, b) = |f (b)| C n (a, b) C n (a, b) b∈Zdn

C n (a, b)

b∈Zdn

≤

X

b∈Zdn

2

n

|f (b)| C (a, b)

≤kf kL2n sup

a∈Zdn

X

1/2  X

b∈Zdn

n

C (a, b)

1/2

C n (a, b).

b∈Zdn

We are interested in conditions which ensure the convergence of the family {Xn }n∈N as n → ∞. 2.1. Tightness. The proof of convergence relies on convergence criteria for semimartingales; our standard reference will be the monograph [JS03]. Let {St }t≥0 be a d-dimensional semimartingale on the stochastic basis (Ω, F , {Ft}t≥0 , P), and denote by h : Rd → Rd a truncation function, i.e. a bounded and continuous function which such that h(x) = x in a neighbourhood of the origin. Since a semimartingale has c`adl`ag (right-continuous, finite left limits) paths, we can write ∆St := St − St− , t > 0, and ∆S0 := S0 , for the jumps of S and set X ¯ t := ¯ t. S(h) (∆Ss − h(∆Ss )) and S(h)t := St − S(h) s≤t

´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

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The process {S(h)t }t≥0 is a special semimartingale, i.e. it admits a unique decomposition

(2.1)

S(h)t = S0 + M(h)t + B(h)t ,

where {M(h)t }t≥0 is a local martingale and {B(h)t }t≥0 is a predictable process of bounded variation on compact time-intervals. Definition 2.1. Let {St }t≥0 be a semimartingale and h : Rd → Rd be a truncation function. The characteristics of the semimartingale (relative to the truncation h) is a triplet (B, A, N) consisting of the bounded variation process B = {B(h)t }t≥0 appearing in (2.1), the compensator N = N(ω, ds, dy) of the jump measure X µ(ω, ds, dy) := δ(s,∆Ss (ω)) (ds, dy) s:∆Ss (ω)6=0

of the semimartingale {St }t≥0 and the quadratic co-variation process i,c k,c Aik t = hSt , St i,

i, k = 1, . . . , d, t ≥ 0,

of the continuous part {Stc }t≥0 of the semimartingale. ˜ N) where A(h) ˜ ik := hM(h)i , M(h)k iL2 , The modified characteristics is the triplet (B, A, t t t i, k = 1, . . . , d, with {M(h)t }t≥0 being the local martingale appearing in (2.1). Using [JS03, Proposition II.2.17 and Theorem II.2.42] we can easily obtain the (modified) characteristics of {Xtn }t≥0 from the infinitesimal generator; as before, we write h : Rd → Rd for the truncation function: Z tX n B (h)t = h(b)C n (Xsn , Xsn + b) ds, 0

A˜n (h)ik t = N n (ds, b) =

Z

b∈Zdn

t

0

X

hi (b)hk (b)C n (Xsn , Xsn + b) ds,

b∈Zdn C n (Xsn , Xsn

+ b) ds

and, since {Xtn }t≥0 is purely discontinuous, An ≡ 0. In order to show the tightness of the family {Xtn }t≥0 , n ∈ N, we need further conditions: X ∀ρ > 0 : lim sup sup (T3) C n (a, a + b) < ∞; n→∞

lim lim sup sup

(T4) (T5)

a∈Zdn |b|>ρ

r→∞

n→∞

a∈Zdn

X

C n (a, a + b) = 0;

|b|>r

X n ∃ρ > 0 ∀i = 1, . . . , d : lim sup sup bi C (a, a + b) < ∞; d n→∞ a∈Zn

(T6)

|b| 0 ∀i, k = 1, . . . , d : lim sup sup bi bk C n (a, a + b) < ∞. d n→∞ a∈Zn

|b| 0 and all ε > 0,
(2.2) lim lim sup Pn N n ([0, T ], {a ∈ Zdn : |a| > r}) > ε = 0, r→∞

n→∞

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES

and the families of processes {B n (h)t }t≥0 , {A˜n (h)t }t≥0 and

nR P t 0

a∈Zdn

g(a)N n (ds, a)

5

o

t≥0

,

n ∈ N, are tight for every bounded function g : Rd → R which vanishes in a neighbourhood of the origin. and it is enough to show the tightness Clearly, (2.2) is a direct consequence of (T4), nR P o t n n n of the families {B (h)t }t≥0 , {A˜ (h)t }t≥0 and 0 a∈Zdn g(a)N (ds, a) , n ∈ N. t≥0

According to [JS03, Theorem VI.3.21] tightness of {B n (h)t }t≥0 , n ∈ N, follows if we can show that (i) for every T > 0 there exists some r > 0 such that   n n sup |B (h)t | > r = 0; lim P n→∞

t∈[0,T ]

(ii) for every T > 0 and r > 0 there exists some τ > 0 such that   n n n sup |B (h)u − B (h)v | > r = 0. lim P n→∞

u,v∈[0,T ],|u−v|≤τ

Fix T > 0; without loss of generality we may assume that h(x) = x for all x ∈ Bρ (0) ⊂ Rd where ρ > 0 is given in (T5). For i = 1, . . . , d, we have Z t X n i n n n sup |B (h)t | = sup hi (b)C (Xs , Xs + b) t∈[0,T ]

t∈[0,T ]

0

b∈Zdn

Z tX Z t X n n n ≤ sup bi C (Xs , Xs + b) ds + sup |hi (b)|C n (Xsn , Xsn + b) ds t∈[0,T ]

0

t∈[0,T ]

|b| 0 set Rd χε (x) := ε−d χ(x/ε), x ∈ Rd . R ¯ε (0) and d χε (x) dx = 1. The Friedrichs By definition, χε ∈ Cc∞ (Rd ), supp χε ⊆ B R mollifier J1/n of g ∈ CcLip (Rd ) is defined as Z Z d χ1/n (x − y)g(y) dy = J1/n g(x) = χ1/n ∗ g(x) = n χ(y)g(x − y/n) dy, ¯1 (0) B

Rd

x ∈ Rd ; for brevity, we use gn := J1/n g. It is easy to see that {gn }n≥1 converges uniformly ¯1 (0) for all n ∈ N; in particular, {gn }n≥1 to g(x), kgn k∞ ≤ kgk∞ and supp gn ⊆ supp g + B 2 d converges to g(x) in L (R , dx). Hence, it remains to prove that lim E(gn − g, gn − g) = 0.

n→∞

First, observe that for all n ∈ N and x, y ∈ Rd Z |gn (x) − gn (y)| ≤ χ(z)|g(x − z/n) − g(y − z/n)| dz ≤ Lg |x − y|, ¯1 (0) B

¯R (0) where Lg > 0 is the Lipschitz constant of g(x). Pick R > 0 such that supp gn ⊆ B for all n ∈ N. Then, we have that (gn (y) − gn (x))2 1Rd ×Rd \diag (x, y)


c (0) + 1B c (0)×B + 1 = (gn (y) − gn (x))2 1B2R (0)×B2R (x, y) (0) B (0)×B (0)\diag 2R 2R 2R 2R

2 c (0) (x, y) + g (y)1B c (0)×B (x, y) + L2g |x − y|21B2R (0)×B2R (0) (x, y) ≤ gn2 (x)1B2R (0)×B2R n 2R (0) 2R

2 c (0) (x, y) + kgk 1B c (0)×B (x, y) + L2g |x − y|21B2R (0)×B2R (0) (x, y). ≤ kgk2∞1B2R (0)×B2R ∞ 2R (0) 2R

Since

E(gn − g, gn − g) = ≤

Z

Rd ×Rd \diag

Z

Rd ×Rd \diag

(gn (y) − g(y) − gn (x) + g(x))2ks (x, y) dx dy
2(gn (y) − gn (x))2 + 2(g(y) − g(x))2 ks (x, y) dx dy,

the assertion follows directly from (C1) and the dominated convergence theorem.



Let us now describe the Dirichlet form related to the Markov chains {Xtn }t≥0 , n ∈ N, introduced in the previous Section 2.1. first, recall that for n ∈ N, we denote by L2n the standard Hilbert space on Zdn with scalar product X hf, giL2n := n−d f (a)g(a), f, g ∈ L2n . a∈Zdn

The following result is a direct consequence of [FU12] and [SW15]. Proposition 2.4. Assume that C n , n ∈ N, satisfy (T1), (T2) and (C1)1. For every n ∈ N we define the bilinear forms X X E n (f, g) := n−d (f (b) − f (a))(g(b) − g(a))Csn (a, b) a∈Zdn b∈Zdn

X X 1 (f (b) − f (a))g(b)Can (a, b). H n (f, g) := E n (f, g) − n−d 2 d d a∈Zn b∈Zn

1(C1)

is assumed to hold for each chain {Xtn }t≥0 by replacing the kernel k(x, y) by C n (a, b)

´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

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where Csn (a, b) := 12 (C n (a, b) + C n (b, a)), resp., Can (a, b) := 12 (C n (a, b) − C n (b, a)) are the symmetric and antisymmetric parts of C n (a, b). (i) H n (f, g) is a well defined non-symmetric bilinear form on F n := {f ∈ L2n : E n (f, f ) < ∞}. (ii) (H n , F n ) is a regular lower bounded semi-Dirichlet form (in the sense of [MR92]); (iii) H n (f, g) = h−An f, giL2n for all f ∈ L2n and g ∈ F n . In particular, the associated Hunt process is {Xtn }t≥0 . Recall that X An f (a) = (f (b) − f (a))C n (a, b) and L2n ⊆ DAn . b∈Zdn

(iv) The estimates (2.3) and (2.4) hold for E = E n , H = H n and X C n (a, b)2 a . α0 = α0n := sup n C d a∈Zn s (a, b) d b∈Zn Csn (a,b)6=0

Corollary 2.5. Assume that C n , n ∈ N, satisfy (T1), (T2) as well as X sup (T2∗ ) C n (a, b) < ∞. b∈Zdn

a∈Zdn

Then F n = L2n for every n ∈ N.

Proof. Clearly, (T2) and (T2∗ ) imply (C1). The claim follows from Jensen’s and H¨older’s inequalities.  In order to study the convergence of the forms H n as n → ∞ we need a few further notions. Denote by a ¯ := [a1 − 1/2n, a1 + 1/2n) × · · · × [ad − 1/2n, ad + 1/2n) the half-open cube with centre a = (a1 , . . . , ad ) ∈ Zdn and side-length n−1 , and for x = (x1 , . . . , xd ) ∈ Rd we set [x]n := ([nx1 + 1/2] /n, . . . , [nxd + 1/2] /n) , where [u] is the integer part of u ∈ R. Note that for a ∈ Zdn and x ∈ a ¯ we have [x]n = a.

a¯

d 1) a ∈ Z(1, n n 2 / √ d 1/2n

Figure 1. Definition of the discretization By rn : L2 (Rd , dx) → L2n and en : L2n → L2 (Rd , dx) we denote the restriction and extension operators which are defined by Z d f (x)dx, a ∈ Zdn rn f (a) = n a ¯

en f (x) = f (a),

x∈a ¯.

These operators have the following properties: for f ∈ L2 (Rd , dx) and fn ∈ L2n , n ∈ N: (i) sup krn kL2n ≤ kf kL2 , lim krn f kL2n = kf kL2 and ken fn k2L2 = ken fn2 kL1 = kfn k2L2n ; n∈N

n→∞

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES

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(ii) rn en fn = fn and hrn f, fn iL2n = hf, en fn iL2 ; (iii) lim ken rn f − f kL2 = 0; n→∞

see [CKK13, Lemma 4.1]. Let us recall from [KS03] the notions of strong and weak convergence. Let C ⊆ L2 (Rd , dx) be dense in (L2 (Rd , dx), k · kL2 ). A sequence fn ∈ L2n , n ∈ N, converges strongly to f ∈ L2 (Rd , dx) if for every {gm }m≥1 ⊆ C satisfying lim kgm − f kL2 = 0,

m→∞

we have that lim lim sup krn gm − fn kL2n = 0.

m→∞

The sequence fn ∈

L2n ,

n→∞

n ∈ N, converges weakly to f ∈ L2 (Rd , dx), if lim hfn , gn iL2n = hf, giL2

n→∞

for every sequence {gn }n≥1 , gn ∈ L2n , converging strongly to g ∈ L2 (Rd , dx). In the following lemma we give an equivalent characterization of the strong and weak convergence, which simplifies the use of these types of convergence. Lemma 2.6. (i) A sequence {fn }n≥1 , fn ∈ L2n , converges strongly to f ∈ L2 (Rd , dx) if, and only if, lim ken fn − f kL2 = 0. n→∞

(ii) A sequence {fn }n≥1 , fn ∈ L2n , converges weakly to f ∈ L2 (Rd , dx) if, and only if, lim hen fn , gi = hf, gi,

n→∞

g ∈ L2 (Rd , dx).

Proof. (i) Let fn ∈ L2n , n ∈ N, and f ∈ L2 (Rd , dx). Assume that {fn }n≥1 converges strongly to f , and let {gm }m≥1 ⊆ C be an approximating sequence of f satisfying the conditions from the definition of strong convergence. Then, by the properties of the operators rn and en , n ∈ N, we find ken fn − f kL2 ≤ ken fn − en rn f kL2 + ken rn f − f kL2 = kfn − rn f kL2n + ken rn f − f kL2

≤ kfn − rn gm kL2n + krn gm − rn f kL2n + ken rn f − f kL2 ≤ kfn − rn gm kL2n + kgm − f kL2 + ken rn f − f kL2 .

Letting first n → ∞ and then m → ∞, the necessity of the claim follows. For the ‘if’ part, we proceed as follows. Let {gm }m≥1 ⊆ C be any approximating sequence of f . Using the strong convergence en fn → f and the fact that rn en fn = fn , we see krn gm − fn kL2n = krn gm − rn en fn kL2n ≤ kgm − en fn kL2

≤ kgm − f kL2 + kf − en fn kL2 .

Letting first n → ∞ and then m → ∞ proves the assertion.

(ii) Let fn ∈ L2n , n ∈ N, and f ∈ L2 (Rd , dx). Assume that {fn }n≥1 converges weakly to f . Since for every g ∈ L2 (Rd ) the sequence {rn g}n≥1 converges strongly to g, it follows immediately that lim hen fn , giL2 = hf, giL2 , g ∈ L2 (Rd , dx). n→∞

For the sufficiency part we pick g ∈ L2 (Rd , dx) and any sequence {gn }n≥1 , gn ∈ L2n such that {gn }n≥1 converges strongly to g. By our assumption, lim hfn , rn giL2n = lim hen fn , giL2n = hf, giL2 ;

n→∞

n→∞

´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

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this shows, in particular, that hen fn , giL2 ≤ kf kL2 for all g with kgkL2 = 1, and so supn∈N ken fn kL2 < ∞. In order to prove the claim we have to show that lim hfn , gn − rn giL2n = 0.

n→∞

Using the properties of the operators rn and en along with the Cauchy-Schwarz inequality yields |hfn , gn − rn giL2n | = |hfn , rn en gn − rn giL2n | = |hen fn , en gn − giL2 |

≤ ken fn kL2 ken gn − gkL2 ,

proving the assertion.



For further details on strong and weak convergence we refer to [KS03] and [T¨06]. 2.3. Convergence of the Finite-Dimensional Distributions. We can now combine the relative compactness from Section 2.1 and the convergence results from Section 2.2 to show the convergence of the finite-dimensional distributions of the chains {Xtn }t≥0 , n ∈ N, to those of a non-symmetric pure jump process {Xt }t≥0 . The latter will be determined by a kernel k : Rd × Rd \ diag → R satisfying (C1). Theorem 2.7. Assume that the chains {Xtn }t≥0 , n ∈ N, satisfy (T1), (T2) and (C1). Let {Xt }t≥0 be a non-symmetric process determined by a kernel k : Rd × Rd \ diag → R satisfying (C1). Denote by {Ptn }t≥0 , n ∈ N, and {Pt }t≥0 the transition semigroups of {Xtn }t≥0 , n ∈ N, and {Xt }t≥0 , respectively, If {Ptn rn f }n≥1 converges strongly to Pt f for all t ≥ 0 and f ∈ L2 (Rd , dx), then there exists a Lebesgue null set B such that the finite-dimensional distributions of {Xtn }t≥0 , n ∈ N, converge along Q on B c to those of {Xt }t≥0 . Proof. Choose an arbitrary countable family C ⊆ CcLip (Rd ) which is dense in CcLip (Rd ) with respect to k · k∞ . By the Markov property, the properties of the operators rn and en , and a standard diagonal argument, we can extract a subsequence {ni }i≥1 ⊆ N such that for all m ≥ 1, all t1 , . . . , tm ∈ Q, 0 ≤ t1 ≤ · · · ≤ tm < ∞, and all f1 , . . . fm ∈ C, {E·n [rn f1 (Xtn1 ) · · · rn fm (Xtnm )]}n≥1 converges strongly to E· [f1 (Xt1 ) · · · fm (Xtm )]. Again by a diagonal argument, we conclude that there is a further subsequence {n′i }i≥1 ⊆ {ni }i≥1 and a Lebesgue null set B ⊆ Rd , such that the above convergence holds pointwise  on B c . The assertion now follows from [EK86, Proposition 3.4.4]. Denote by D(Rd ) the space of all c`adl`ag functions f : [0, ∞) → Rd . The space D(Rd ) equipped with Skorokhod’s J1 topology becomes a Polish space, the so-called Skorokhod space, cf. [EK86]. Corollary 2.8. Assume that the conditions of Theorems 2.2 and 2.7 hold, and let B be the Lebesgue null set from Theorem 2.7. Denote by µn and µ the initial distributions of {Xt }t≥0 and {Xtn }t≥0 , n ∈ N, respectively. If µ(B) = 0 and if µn → µ weakly, then the following convergence holds in Skorokhod space: (2.5)

d

{Xtn }t≥0 −−−→ {Xt }t≥0 . n→∞

Proof. The assertion follows by combining Theorems 2.2, 2.7, [JS03, Lemma VI.3.19] and the remark following that lemma.  Theorem 2.7 states that (2.5) follows if we can prove “strong convergence” of Ptn → Pt , t > 0. A sufficient condition for this convergence is given in the following theorem.

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES

11

Theorem 2.9. Assume that (C1) holds for both {Xtn }t≥0 , n ∈ N, and {Xt }t≥0 . The semigroups {Ptn rn f }n≥1 converge strongly to Pt f for all t ≥ 0 and f ∈ L2 (Rd , dx) if the following conditions are satisfied: 0 < lim inf α0n ≤ lim sup α0n < ∞; n→∞ n→∞ Z ∀ρ > 0 : sup (1 ∧ |y|2)ks (x, x + y) dy < ∞;

(C2) (C3)

x∈Bρ (0)

∀ρ > 0 : lim sup sup

(C4)

n→∞

a∈Bρ (0)

X

b∈Zdn

(1 ∧ |b|2 )Csn (a, a + b) < ∞;

∀ε > 0 ∃n0 ∈ N ∀n0 ≤ m ≤ n, f ∈ L2m :

(C5) (C6)

Rd

E n (rn em f, rn em f )

1/2

≤ E m (f, f )1/2 + ε;

for all sufficiently small ε > 0 and large m ∈ N n lim E¯m,ε (f, f ) = Em,ε (f, f ), f ∈ CcLip (Rd ) n→∞

where for all f ∈ CcLip (Rd ) ZZ 1 Em,ε (f, f ) := (f (y) − f (x))2 ks (x, y) dx dy, 2 {(x,y)∈Bm (0)×Bm (0): |x−y|>ε} d ZZ n n E¯m,ε (f, f ) := (f (y) − f (x))2 Csn (a, b)1a¯×¯b (x, y) dx dy, 2 {(x,y)∈Bm (0)×Bm (0): |x−y|>ε} (C7)

(i)

x 7→

(ii)

x 7→

(iii)

(C8)

(iv) Z

Rd

(C9)

x 7→ x 7→

Z

B1 (0)

Z

B1c (0)

Z

B1 (0)

Z

Rd

B1 (0)

|y||ks(x, x + y) − ks (x, x − y)| dy ∈ L2loc (Rd , dx),

L2 (Rd ,dx)

loc −−−−→ 0; (1 ∧ |y|2)|ks (x, x + y) − nd Csn ([x]n , [x]n + [y]n )| dy −−

n→∞

for all sufficiently large R > 1 Z Z

BR (−x)

ks (x, x + y) dy

2

dx < ∞;

for all sufficiently large R > 1 Z 2 Z n→∞ d n |ks (x, x + y) − n Cs ([x]n , [x]n + [y]n )| dy dx −−−→ 0; c (0) B2R

(C11) Z

ks (x, x + y) dy ∈ L2 (Rd , dx) ∪ L∞ (Rd , dx),

(1 ∧ |y|)|ka(x, x + y)| dy ∈ L2loc (Rd , dx);

c (0) B2R

(C10)

|y|2ks (x, x + y) dy ∈ L2loc (Rd , dx),

BR (−x)

|y||ks(x, x + y) − ks (x, x − y) L2 (Rd ,dx)

loc − nd Csn ([x]n , [x]n + [y]n ) + nd Csn ([x]n , [x]n − [y]n )| dy −− −−−−→ 0;

n→∞

´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

12

(C12)

Z

Rd

(C13)

L2 (Rd ,dx)

loc −−−−→ 0; (1 ∧ |y|)|ka(x, x + y) − nd Can ([x]n , [x]n + [y]n )| dy −−

n→∞

for all sufficiently large R > 1 2 Z Z n→∞ d n |ka (x, x + y) − n Ca ([x]n , [x]n + [y]n )| dy dx −−−→ 0. c (0) B2R

BR (−x)

Proof. According to the properties of the operators rn and en , n ∈ N, Propositions 2.3 and 2.4, and [T¨06, Theorem 2.41 and Remark 2.44] the assertion follows if (i) for every sequence {fn }n≥1 , fn ∈ F n , converging weakly to some f ∈ L2 (Rd , dx) and satisfying lim inf H1n (fn , fn ) < ∞, we have that f ∈ F ; n→∞

(ii) for every g ∈ Cc2 (Rd ) and every sequence {fn }n≥1 , fn ∈ F n , converging weakly to f ∈ F, lim H n (rn g, fn ) = H(g, f ). n→∞

Indeed, (i) and (ii) imply that for any sequence {fn }n≥1 , fn ∈ L2n , converging strongly to f ∈ L2 (Rd , dx), the sequence {Ptn fn }n≥1 converges strongly to Pt f for every t ≥ 0, cf. [T¨06, Theorem 2.41 and Remark 2.44]. For any fixed f ∈ L2 (Rd , dx) we set fn = rn f . Since rn f → f strongly we conclude that Ptn rn f → Pt f strongly as claimed. Let us now prove that (C1)–(C13) imply (i) and (ii). We begin with (i). According to Proposition 2.4, we have E1n (f ) ≤

4(1 ∨ α0n ) n H1 (f ), 1 ∧ α0n

f ∈ F n.

Let {fn }n≥1 , fn ∈ F n , be an arbitrary sequence converging weakly to some f ∈ L2 (Rd , dx) such that lim inf n→∞ H1n (fn ) < ∞; the condition (C2) ensures that lim inf n→∞ E1n (fn ) < ∞. Finally, [CKK13, Theorem 4.6] states that under (C3)–(C6) E(f, f ) ≤ lim inf E n (fn , fn ), n→∞

which proves (i). In order to prove (ii) we proceed as follows. According to Proposition 2.4 we have for g ∈ Cc2 (Rd ) and fn ∈ F n H n (rn g, fn ) = h−An rn g, fn iL2n ,

n ∈ N.

Using (C7) it is shown in [SW15, Theorem 2.2] that the generator (A, DA ) of {Pt }t≥0 (or, equivalently, of (H, F )) has the following properties: (i) Cc2 (Rd ) ⊆ DA ; (ii) for every g ∈ Cc2 (Rd ), Z Ag(x) = (g(x + y) − g(x) − h∇g(x), yi1B1(0) (y))ks(x, x + y) dy Rd Z 1 + h∇g(x), yi(ks(x, x + y) − ks (x, x − y)) dy 2 B1 (0) Z + (g(x + y) − g(x))ka (x, x + y) dy; Rd

(iii) for all g ∈ Cc2 (Rd ) and all f ∈ F , H(g, f ) = h−Ag, f iL2 .

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES

13

Therefore, it suffices to prove that {An rn g}n≥1 converges strongly to Ag for every g ∈ Cc2 (Rd ). Observe that for any g ∈ Cc2 (Rd ) and n ∈ N, X An rn g(a) = (rn g(a + b) − rn g(a))C n (a, a + b) b∈Zdn

=

X

b∈Zdn

(rn g(a + b) − rn g(a) − hrn ∇g(a), bi1{|b|≤1} (b))Csn (a, a + b)

1X hrn ∇g(a), bi(Csn (a, a + b) − Csn (a, a − b)) 2 |b|≤1 X + (rn g(a + b) − rn g(a))Can (a, a + b). +

b∈Zdn

Using the triangle inequality we get n

ken A rn g − AgkL2 ≤ √ −1

5 X i=1

kAni − Ai kL2 ,

d the Ai and Ani are given by where for ρ := 1 + (2n) Z
A1 := g(x + y) − g(x) − h∇g(x), yi1B1(0) (y) ks (x, x + y) dy B (0) Z ρ
An1 := rn g([x]n + [y]n ) − rn g([x]n ) − hrn ∇g([x]n ), [y]n i1B1 (0) ([y]n ) × Bρ (0)

A2 :=

Z

Bρc (0)

An2 :=

Z

Bρc (0)

× nd Csn ([x]n , [x]n + [y]n ) dy


g(x + y) − g(x) ks (x, x + y) dy


rn g([x]n + [y]n ) − rn g([x]n ) nd Csn ([x]n , [x]n + [y]n ) dy

Z 1 A3 := h∇g(x), yi1B1(0) (y)(ks (x, x + y) − ks (x, x − y)) dy 2 Bρ (0) Z 1 n hrn ∇g([x]n ), [y]n i1B1 (0) ([y]n )× A3 := 2 Bρ (0) A4 :=

Z

Bρ (0)

An4 := A5 :=

Z

Bρ (0)

Z

Bρc (0)

An5 :=

Z

Bρc (0)

× nd (Csn ([x]n , [x]n + [y]n ) − Csn ([x]n , [x]n − [y]n )) dy
g(x + y) − g(x) ka (x, x + y) dy
rn g([x]n + [y]n ) − rn g([x]n ) nd Can ([x]n , [x]n + [y]n ) dy


g(x + y) − g(x) ka (x, x + y) dy


rn g([x]n + [y]n ) − rn g([x]n ) nd Can ([x]n , [x]n + [y]n ) dy.

√ In the remaining part of the √ proof we assume R > 1√ + d/2 such that supp g ⊆ BR (0) and we write ρ := 1 + (2n)−1 d and σ := 1 − (2n)−1 d. kAn1 − A1 kL2

´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

14

≤

Z

B2R (0)

Z

 g(x + y) − g(x) − h∇g(x), yi1B1(0) (y)

Bρ (0)

2  1 2 − rn g([x]n + [y]n ) + rn g([x]n ) + hrn ∇g([x]n ), [y]n i1B1 (0) ([y]n ) ks (x, x + y) dy dx Z Z   + r g([x] + [y] ) − r g([x] ) − hr ∇g([x] ), [y] i1 ([y] ) × n n n n n n n n B (0) n 1 

B2R (0)

Bρ (0)

d

× ks (x, x + y) − n

≤

Z

B2R (0)

Z

Bσ (0)



Csn ([x]n , [x]n

g(x + y) − g(x) − h∇g(x), yi

2  1 2 + [y]n ) dy dx


2  1 2 − rn g([x]n + [y]n ) + rn g([x]n ) + hrn ∇g([x]n ), [y]n i ks (x, x + y) dy dx Z Z 2  21
+ 4kgk∞ + 2ρk∇gk∞ ks (x, x + y) dy dx 

B2R (0)

2

+ k∇ gk∞

Z

B2R (0)

Z

Bσ (0)




+ 2kgk∞ + ρk∇gk∞ × Z Z × B2R (0)

Bρ (0)\Bσ (0)

2

|y| ks (x, x + y) − nd Csn ([x]n , [x]n + [y]n ) dy

Bρ (0)\Bσ (0)

ks (x, x + y) − nd Csn ([x]n , [x]n + [y]n ) dy

2

2

 12 dx

 21 dx .

By monotone and dominated convergence theorem, Taylor’s theorem, (C7) (i) and (ii), and (C8), we conclude that kAn1 − A1 kL2 → 0. Next, kAn2 − A2 kL2 Z Z ≤ B2R (0)

Bρc (0)

2  1 2   g(x + y) − g(x) − rn g([x]n + [y]n ) + rn g([x]n ) ks (x, x + y) dy dx

2  1 Z 2   dx g(x + y) − r g([x] + [y] ) k (x, x + y) dy + n n n s c Bρ (0) B c (0)  Z 2R Z   r g([x] + [y] ) − r g([x] ) × + n n n n n c Z

Bρ (0)

B2R (0)

d

× ks (x, x + y) − n

+ ≤

Z

Z

c (0) B2R

B2R (0)

+

Z

Z

Z

Bρc (0)

c (0) B2R

Bρc (0)

Csn ([x]n , [x]n

2  1 2 + [y]n ) dy dx


2  1 2 rn g([x]n + [y]n ) ks (x, x + y) − nd Csn ([x]n , [x]n + [y]n ) dy dx


2  1 2   g(x + y) − g(x) − rn g([x]n + [y]n ) + rn g([x]n ) ks (x, x + y) dy dx

Z

BR (−x)

g(x + y) − rn g([x]n + [y]n ) ks (x, x + y) dy

2

 21 dx

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES

+ 2kgk∞ + kgk∞

Z

B2R (0)

Z

c (0) B2R

Z

Bρc (0)

Z

ks (x, x + y) − nd Csn ([x]n , [x]n + [y]n ) dy

BR (−x)

2

ks (x, x + y) − nd Csn ([x]n , [x]n + [y]n ) dy

15

 21 dx

2

 21 dx .

Again, by monotone and dominated convergence theorem, Taylor’s theorem, (C7) (ii), (C8), (C9) and (C10), we have that kAn2 − A2 kL2 → 0. Further, kAn3 − A3 kL2 Z Z   1 h∇g(x), yi1B1 (0) (y) − hrn ∇g([x]n ), [y]n i1B1 (0) ([y]n ) × ≤ 2 Bρ (0) B2R (0) 1 2
2 × ks (x, x + y) − ks (x, x − y) dy dx Z Z 1 + hrn ∇g([x]n ), [y]n i1B1 (0) ([y]n )× 2 Bρ (0) B2R (0) × ks (x, x + y) − ks (x, x − y) − n

1 ≤ 2

Z

B2R (0)

+ ρk∇gk∞ + k∇gk∞

Z

Bσ (0)

Z

Z



B2R (0)

B2R (0)

d

Csn ([x]n , [x]n

+ [y]n ) + n

d

Csn ([x]n , [x]n

1 2
2 − [y]n ) dy dx

1 2 
2 h∇g(x), yi − hrn ∇g([x]n ), [y]n i ks (x, x + y) − ks (x, x − y) dy dx Z

Z

Bρ (0)\Bσ (0)

Bσ (0)

|y|×

ks (x, x + y) − ks (x, x − y) dy

2

1 2

dx

× ks (x, x + y) − ks (x, x − y) − nd Csn ([x]n , [x]n + [y]n ) + nd Csn ([x]n , [x]n − [y]n ) dy Z Z 1 + ρk∇gk∞ 2 B2R (0) Bρ (0)\Bσ (0) × ks (x, x + y) − ks (x, x − y) − nd Csn ([x]n , [x]n + [y]n ) + nd Csn ([x]n , [x]n − [y]n ) dy

2

2

dx

1

dx

1

2

2

,

which, by monotone and dominated convergence, Taylor’s theorem, (C7) (iii), (C8) and (C11), implies that kAn3 − A3 kL2 → 0. Next, kAn4 − A4 kL2 2  1 Z Z 2   dx g(x + y) − g(x) − r g([x] + [y] ) + r g([x] ) k (x, x + y) dy ≤ n n n n n a Bρ (0) B2R (0) Z Z   + r g([x] + [y] ) − r g([x] ) × n n n n n B2R (0)

Bρ (0)

d

× ka (x, x + y) − n

≤

Z

B2R (0)

Z

Bρ (0)



Can ([x]n , [x]n

2  1 2 + [y]n ) dy dx


2  1 2  g(x + y) − g(x) − rn g([x]n + [y]n ) + rn g([x]n ) ka (x, x + y) dy dx

´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

16

+ 2k∇gk∞

Z

B2R (0)

Z

Bρ (0)

|y| ka (x, x + y) − nd Can ([x]n , [x]n + [y]n ) dy

2

 21 dx .

Now, by monotone and dominated convergence, Taylor’s theorem, (C7) (iv) and (C12), we see kAn4 − A4 kL2 → 0. Finally, kAn5 − A5 kL2 Z Z ≤

Bρc )(0)

B2R (0)

2  1 2   g(x + y) − g(x) − rn g([x]n + [y]n ) + rn g([x]n ) ka (x, x + y) dy dx

2  1 Z 2   dx g(x + y) − r g([x] + [y] ) k (x, x + y) dy + n n n a c Bρ (0) B c (0)  Z 2R Z   r g([x] + [y] ) − r g([x] ) × + n n n n n c Z

Bρ (0)

B2R (0)

d

× ka (x, x + y) − n

+ ≤

Z

Z

c (0) B2R

B2R (0)

+

Z

Z

Z

Bρc (0)

+ 2kgk∞ + kgk∞

Bρc (0)



rn g([x]n + [y]n ) ka (x, x + y) − n

Can ([x]n , [x]n

2  1 2 + [y]n ) dy dx


2  1 2  g(x + y) − g(x) − rn g([x]n + [y]n ) + rn g([x]n ) ka (x, x + y) dy dx

BR (−x)

Z

B2R (0)

Z

2  1 2 + [y]n ) dy dx


d

Z

c (0) B2R

Can ([x]n , [x]n

c (0) B2R

|g(x + y) − rn g([x]n + [y]n )||ka (x, x + y)| dy

Z

Bρc (0)

Z

2

ka (x, x + y) − nd Can ([x]n , [x]n + [y]n ) dy

BR (−x)

 21 dx

2

ka (x, x + y) − nd Can ([x]n , [x]n + [y]n ) dy

 21 dx

2

 21 dx .

By monotone and dominated convergence, Taylor’s theorem, (C7) (iv), (C9), (C12) and (C13), we get that kAn5 − A5 kL2 → 0, which concludes the proof.  The conditions of Theorem 2.9 can be slightly changed to give a further set of sufficient conditions of the convergence of {Ptn rn f }n≥1 ; the advantage is that we can state these conditions only using ks and k resp. Csn and C n , which makes them sometimes easier to check. Corollary 2.10. Assume that (C1)–(C6) and (C7)(i), (ii) hold, that Z (2.6) x 7→ |y| k(x, x + y) − k(x, x − y) dy ∈ L2loc (Rd , dx) B1 (0)

and that (C8)–(C11) hold with ks and Csn replaced by k and C n , respectively. Then {Ptn rn f }n≥1 converges strongly to Pt f for all t ≥ 0 and f ∈ L2 (Rd , dx). Proof. According to [SW15, Theorem 3.1], the above assumptions imply that the generator (A, DA ) of (H, F ) satisfies (i) Cc∞ (Rd ) ⊆ DA ;

MARKOV CHAIN APPROXIMATION OF PURE JUMP PROCESSES

17

(ii) for every g ∈ Cc∞ (Rd ), Z
Ag(x) = g(x + y) − g(x) − h∇g(x), yi1B1(0) (y) k(x, x + y) dy Rd Z
1 h∇g(x), yi k(x, x + y) − k(x, x − y) dy. + 2 B1 (0)

Note that in [SW15, Theorem 3.1] slightly stronger conditions are assumed (namely (H3) which is a symmetrized version of (2.6) and the tightness assumption (H5)), but they are exclusively used to deal with the formal adjoint A∗ ; this follows easily from an inspection of the proofs of [SW15, Theorems 2.2 and 3.1].  From this point onwards we can follow the proof of Theorem 2.9. Recall that a set C ⊆ DA is an operator core for (A, DA ) if A|C = A. If we happen to know that Cc2 (Rd ) is an operator core for (A, DA), then there is an alternative proof of Theorem 2.9 and its Corollary 2.10 based on [EK86, Theorem 1.6.1]: {Ptn rn f }n≥1 converges strongly to Pt f for all t ≥ 0 and all f ∈ L2 (Rd , dx) if (and only if) {An rn g}n≥1 converges strongly to Ag for every g ∈ Cc2 (Rd ). 2.4. Approximation of a Given Process. We will now show how we can use the results of Sections 2.1–2.3 to approximate a given non-symmetric pure-jump process by a sequence of Markov chains. We assume that {Xt }t≥0 is of the type described at the beginning of Section 2.2; in particular the kernel k : Rd ×Rd \diag → R satisfies (C1). We are going to construct a sequence of approximating (in the weak sense) Markov chains. Let 0 < p ≤ 1 and define a family of kernels C n,p : Zdn × Zdn → [0, ∞), n ∈ N, by  Z Z √ nd k(x, y) dx dy, |a − b| > 2npd n,p (2.7) C (a, b) := a ¯ ¯b √  0, |a − b| ≤ 2npd . Remark 2.11. Th family of kernels defined in (2.7) has the following properties:

(i) The kernels C n,p , n ∈ N, automatically satisfy (T1). (ii) For any increasing sequence {ni }i∈N ⊂ N such that the lattices are nested, i.e. Zdni ⊆ Zdni+1 , the conditions (C5) and (C6) hold true, cf. [CKK13, Theorem 5.4]. This is, in particular, the case for ni = 2i , i ∈ N. (iii) Due to (C1) and Lebesgue’s differentiation theorem (see [Fol84, Theorem 3.21]), we have for (Lebesgue) almost all (x, y) ∈ Rd × Rd \ diag, Z Z 2d lim n k(u, v) dv du = k(x, y) n→∞

[x]n

[y]n

Z

|k(u, v) − k(x, y)| dv du = 0.

and 2d

lim n

n→∞

Z

[x]n

[y]n

Let us check the conditions (T2)–(T6). Proposition 2.12. The conditions (T2) and (T3) hold true if Z ∀ρ > 0 : sup (T1.D) k(x, y)dy < ∞. x∈Rd

Bρc (x)

´ AND R. L. SCHILLING A. MIMICA, N. SANDRIC,

18

Proof. We will only discuss (T2) since (T3) follows in a similar way. Observe that for every d ∈ N and 0 < p ≤ 1, [ c c ¯b ⊆ B√ (a) ⊆ B√ (x), a ∈ Zdn , x ∈ a ¯. d/np d/2np √ |a−b|>2 d/np

This shows that sup a∈Zdn

X

C

n,p

X

d

(a, b) = n sup a∈Zdn

b∈Zdn d

≤ n sup

a∈Zdn

≤ sup

x∈Rd

√ |a−b|>2 d/np

Z Z

Z

c B√

a ¯

c B√

Z Z a ¯

¯b

k(x, y) dy dx

k(x, y) dy dx p (x)

d/2n

k(x, y) dy, (x) d/2np

which concludes the proof.



This means that under (T1.D), the kernels C n,p , n ∈ N, define a family of regular Markov chains {Xtn }t≥0 , n ∈ N. Using the same arguments as above, it is easy to see that (T4) holds if Z lim sup (T3.D) k(x, y)dy = 0. r→∞ x∈Rd

Brc (x)

Proposition 2.13. Assume that (T1.D) holds. Then the following statements are true. (i) (T5) will be satisfied if (T4.D.1)

there is some ρ > 0 such that for i = 1, . . . , d Z lim sup sup (yi − xi )k(x, y) dy < ∞, ε↓0 x∈Rd B (x)\Bε (x) Z ρ lim sup sup |yi − xi |k(x, y) dx < ∞, ε↓0

x∈Rd

lim sup ε sup ε↓0

x∈Rd

B√dεp (x)\B√dεp −(√d/2)ε (x)

Z

Bρ (x)\Bεp (x)

k(x, y) dy < ∞.

(ii) (T6) will be satisfied if (T5.D.1)

there is some ρ > 0 such that for i, k = 1, . . . , d Z lim sup sup (yi − xi )(yk − xk )k(x, y) dy < ∞, ε↓0 x∈Rd B (x)\Bε (x) Z ρ lim sup sup |yi − xi ||yk − xk |k(x, y) dx < ∞, ε↓0

x∈Rd

lim sup ε sup ε↓0

x∈Rd

B√dεp (x)\B√dεp −(√d/2)ε (x)

Z

Bρ (x)\Bεp (x)

|yi − xi |k(x, y) dy < ∞.

Proof. We will only discuss (T5), since (T6) follows in an analogous way. Assume (T4.D.1). We have Z Z X X d n,p bi sup bi C (a, a + b) = n sup k(x, y) dy dx d d √ a∈Zn

|b|