LIMIT THEOREMS FOR WEIGHTED BERNOULLI RANDOM FIELDS UNDER HANNAN'S CONDITION

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LIMIT THEOREMS FOR WEIGHTED BERNOULLI RANDOM FIELDS UNDER HANNAN’S CONDITION

Jana Klicnarov´a, Dalibor Volny, Yizao Wang

To cite this version:

Jana Klicnarov´a, Dalibor Volny, Yizao Wang. LIMIT THEOREMS FOR WEIGHTED BERNOULLI RANDOM FIELDS UNDER HANNAN’S CONDITION. Stochastic Processes and their Applications, Elsevier, 2016. �hal-01878199�

RANDOM FIELDS UNDER HANNAN’S CONDITION

JANA KLICNAROV ´A, DALIBOR VOLN ´Y, AND YIZAO WANG

Abstract. Recently, invariance principles for partial sums of Bernoulli random fields over rectangular index sets have been proved under Han- nan’s condition. In this note we complement previous results by estab- lishing limit theorems for weighted Bernoulli random fields, including central limit theorems for partial sums over arbitrary index sets and invariance principles for Gaussian random fields. Most results improve earlier ones on Bernoulli random fields under Wu’s condition, which is stronger than Hannan’s condition.

1. Introduction

We are interested in limit theorems for partial sums of weighted stationary random fields {X_j}_j∈

Z^d, in form of

(1.1) Sn= X

j∈Z^d

bn,jXj, n∈N, where {b_n,j}_j∈Zd are coefficients such that P

jb²_n,j < ∞. We will impose further conditions on the dependence of {X_j}_j∈Zd so thatSnis well defined in theL² sense.

Limit theorems for partial sums of dependent random variables have a long history. In the cased= 1, limit theorems for stationary sequences have been extensively developed. The most considered case is the unweighted case with bn,j =1{j∈{1,...,n}}, which yields

Sn=X1+· · ·+Xn.

In this case, various conditions on the weak dependence of stationary se- quence {X_j}_j∈Z have been developed. Heuristically, when the sequence of {X_j}j∈Z is said to have weak dependence,Sn should behave asymptotically as partial sums of i.i.d. random variables; in particular one expects

(1.2)

S_bntc

√n

t∈[0,1]

⇒σ{Bt}_t∈[0,1]

for a standard Brownian motionBand some constantσ >0. Results of this type are referred to as (weak) invariance principles or functional central limit theorems. See for examples [14, 24, 25, 28, 29, 36] and references therein for recent developments. In case thatb_n,j is a more general`² sequence, the same rational applies: S_n in (1.1) with weakly dependent {X_j}j∈Z should

1

behave as if {X_j}_j∈Z are independent and identically distributed (i.i.d.) random variables. Different types of invariance principles may arise, see for example [12] where the limiting processes are fractional Brownian motions.

A few conditions have been known that lead to the above invariance principle (1.2). See for example [15, 16] on comparisons between different conditions. In this paper we focus on Hannan’s condition [18] and Wu’s condition [36], and particular their extensions to random fields. In order to compare with our results for random fields, assume in addition that the stationary random variables {X_j}_j∈Z have the form

X_j =f ◦T^j({_k}_k∈Z)

for some measurable function f : R^Z → R and the shift operator T on R^Z defined by [Tj(w)]_k =w_j+k for w ={w_k}_k∈Z ∈ R^Z, and a sequence of stationary random variables {_k}_k∈Z. We assumeEXj = 0 and EX_j² <∞.

Furthermore, introduce F_j := σ(k : k ≤ j), F−∞ := T

jF_j and F∞ :=

W

jF_j, and assume f to be regular in the sense thatE(f | F−∞) = 0,E(f | F∞) =f. With these notations, Hannan’s condition reads as

(1.3) X

j∈Z

kE(f◦T^j | F₀)−E(f◦T^j | F₋₁)k₂ <∞.

To introduce Wu’s condition, furthermore set first ^∗_j =j for j6= 0 and ^∗₀ to be an independent copy of 0 and independent of all the other random variables, and then set X_j^∗ = f◦T^j({^∗_k}_k∈Z). Then Wu’s condition reads as

(1.4) X

j∈Z

kX_j−X_j^∗k₂ <∞.

Both conditions have been extensively investigated (e.g. [9, 13, 38]). It has been shown in [34] that Wu’s condition is strictly stronger than Hannan’s condition, in the sense that one can find an example such that the invariance principle as in (1.2) holds for Hannan’s condition, but Wu’s condition is violated. On the other hand, Wu’s condition is very practical in proving limit theorems [38], and conditions of similar types have lead to strong invariance principles [3, 37].

Limit theorems for stationary random fields (d ≥ 2) have also been in- vestigated since long time ago. There is a vast literature on limit theorems for general stationary random fields, and we refer to [7, 8, 10, 11] and the references therein. A main motivation of the recent developments was to extend the well-investigated dependence conditions for stationary sequences to random fields. However, the success so far has been mostly limited to Bernoulli random fields {X_j}_j∈

Z^d:

(1.5) Xj =f ◦Tj({_k}_k∈

Z^d) wheref :R^Z

d →Ris a measurable function,{T_j}_j∈

Z^d are the shift operators onR^Z

d such that forw={w_k}_k∈Zd ∈R^Z

d, [T_j(w)]_k=w_j+kfor allk, j ∈Z^d,

and {_k}_k∈Zd are i.i.d. random variables. We assume EXj = 0 and EX_j² <

∞.

It is important to point out that in the cased= 1,{_j}_j∈Zis only assumed to be stationary. At the technical level, the reason for restricting {_j}_j∈

Z^d

to be i.i.d. whend≥2 is the following. Conditional expectations of random variables, in the form of E(Y | F_j) with F_j = σ(_k : k ≤ j, k ∈ Z^d), are involved in the proof. Such conditional expectations are extensively used to express random variables of interest in terms of sums of orthogonal random variables. For example in one dimension, one can write

(1.6) Y =X

j∈Z

[E(Y | F_j)−E(Y | F_j−1)] =:X

j

P_jY

for any regular random variable Y with finite moment, and for all j 6= j⁰, P_jY and P_j⁰Y are uncorrelated. To extend this key decomposition to the random-field setup (Lemma 2.1 below) and maintain the orthogonality of the terms, one needs the following commuting property of the filtration:

(1.7) E[E(Y | F_j)| F_k] =E[E(Y | F_k)| F_j] =E(Y | F_j∧k)

with (j∧k)q= min(jq, kq), q= 1, . . . , d. This identity, unfortunately, is not true for filtrations generated by general stationary {_j}_j∈

Z^d except when they are i.i.d. Therefore, many arguments based on the telescoping (1.6) fail to be generalized to high dimensions for arbitrary stationary{_j}_j∈

Z^d. A deeper reason for working under the assumption of {_j}_j∈

Z^d being i.i.d. is that the so-constructed random fields as in (1.5) can be approxi- mated by m-dependent random fields, but it is not clear how to construct an m-dependent approximation for general stationary random fields. For general stationary sequences in one dimension, instead ofm-dependent ap- proximation one can proceed alternatively by establishing martingale ap- proximation, using the projections based on conditional expectations, and then apply the martingale central limit theorem. However, the martingale central limit theorem has been well known (e.g. [7]) to be much less practical in high dimensions than in one dimension, and limit theorems for stationary random fields established via martingale approximations have very strin- gent conditions [2, 26, 27, 30]. So for general stationary random fields in form of (1.5) with stationary{_j}_j∈Zd, it is still an open question that what should be the general approach to establish central limit theorems. This is the limitation of most of the results so far on weighted Bernoulli random fields. It might be true that the situation with stationary {_j}_j∈

Z^d is actu- ally much more complicated than in one dimension. An exceptional result for general stationary random fields is due to Dedecker [11] via a very in- volved Lindeberg method, and the condition on weak dependence is much more complicated than the one-dimensional case. Another very recent result that goes beyond the i.i.d. regime is in [33], although it is still not yet as satisfactory as in one dimension.

As discussed above, latest advances on limit theorems for stationary ran- dom fields have been mostly limited to Bernoulli random fields in form of (1.1) and (1.5). In particular, progress has been made on Wu’s condition in the past few years. Observe that Wu’s condition in one dimension (1.4) can be naturally extended to high dimensions, formally as

(1.8) X

j∈Z^d

kX_j −X_j^∗k₂ <∞, withX_j^∗ =f◦Tj({^∗_k}_k∈

Z^d) and{^∗_k}_k∈

Z^dsimilarly as in the cased= 1. This extension is first considered by El Machkouri et al. [17], where invariance principles for unweighted partial sums are established. Limit theorems have been also established for fractional Brownian sheets [35] and set-indexed random fields [4, 17]. All these results can be formulated as limit theorems for weighted Bernoulli random fields as in (1.1), under Wu’s condition and certain assumptions onbn,j. Recent results on Bernoulli random fields under other conditions include also [23].

In this paper, we consider limit theorems for Bernoulli random fields under Hannan’s condition ((2.1) below), and continue the development in [34]. It is proved in [34] that for stationary Bernoulli random field under Hannan’s condition,





 1 n^d/2

X

1≤i≤nt

X_i







t∈[0,1]^d

⇒σ{Bt}_t∈[0,1]d

as n → ∞ in D([0,1]^d), where the limiting random field is the Brownian sheet B up to a multiplicative constant σ, as in the case for {X_j}_j∈Zd be- ing i.i.d. Here, we complement results in [34] by considering more general weights b_n,j: we extend a few of aforementioned results [4, 17, 35] on limit theorems for weighted Bernoulli random fields under Wu’s conditions to the strictly weaker Hannan’s condition (except in one case where the results are not comparable; see Remark 4.4).

There are two key ingredients in the proofs here. One is the assumption on the i.i.d. random variables discussed above: in particular, this assumption allows the approximation of the stationary random fields by m-dependent ones. The other is a moment inequality for weighted partial sums, in form of

X

j∈Z^d

b_n,jX_j p

≤C



 X

j∈Z^d

b²_n,j





1/2

for some p≥2. We establish such an inequality in Lemma 2.2 under Han- nan’s condition. It plays the key role in bounding the error term in the m-dependent approximation of the random fields. Such an inequality has been known under Wu’s condition (1.8) [17, Proposition 1]. (See also for a different extension of Wu’s condition proposed by [32], where a similar mo- ment inequality was established for the unweighted partial sums.) Our proof

of the main result, Theorem 2.5, makes essential use of the two keys, and the proof is inspired by Bierm´e and Durieu [4] (see also Remark 2.9). Here we present a variation of the same idea, using m-dependent approximation instead ofmn-dependent approximation.

The paper is organized as follows. The main result, Theorem 2.5, is established in Section 2. As consequences, we present two applications.

First, central limit theorems for partial sums over arbitrary index sets are investigated in Section 3. Second, invariance principles in [4, 17, 35] are established under the Hannan’s condition in Section 4.

2. A central limit theorem Consider i.i.d. random variables {_i}_i∈

Z^d defined in a probability space (Ω,B,P). Set F_i = σ(_j : j ∈ Z^d, j ≤ i), i ∈ Z^d and F_i^(q)

q = σ(_j : j ∈ Z^d, jq ≤iq), q = 1, . . . , d, iq ∈ Z. Because of the independence of {_j}_j∈

Z^d, {F_j}_j∈

Z^d are commuting in the sense of (1.7). Next, as in [34], introduce the projection operator

P_i=

d

Y

q=1

P_i^(q)

q with P_i^(q)

q (·) =E(· | F_i^(q)

q )−E(· | F_i^(q)

q−1).

In this way, P_i^(q)_q and Pi are commuting operators from L²(Ω,B,P) to L²(Ω,B,P), due to the commuting property of the filtration. For more properties of these filtrations and operators, see [20, 35]. The following decomposition based on these projection operators is useful.

Lemma 2.1. Let Y be a random variable measurable with respect to the σ-algebra F_∞ =σ(_j :j ∈ Z^d), with EY = 0,E|Y|^p < ∞, for some p≥ 2.

Then,

Y = X

j∈Z^d

P_jY := lim

m→∞

X

j∈{−m,...,m}^d

P_jY in L^p.

Proof. By definition ofPj, X

j∈{−m+1,...,m}^d

P_jY =E(Y | F_m1) + X

δ∈{−1,1}^d\{1}

(−1)^hδiE(Y | F_mδ) with hδi = Pd

q=11_{δq=−1}, 1 = (1, . . . ,1) ∈ Z^d and mδ, m1 ∈ Z^d. By martingale convergence theorem, E(Y | F_m1)→Y almost surely and inL^p. The other 2^d−1 terms all converge to zero in L^p. Indeed, observe that for each q= 1, . . . , d,

m→∞lim E(Y | F_−m^(q)) =E Y

\

m∈N

F_−m^(q)

!

almost surely and inL^p, by backwards martingale convergence theorem. By Kolmogorov’s zero-one law, the limit is a constant and hence necessarily zero since EY = 0. To complete the proof, it suffices to observe for each

δ ∈ {−1,1}^d\ {1} with δq =−1 for some q ∈ {1, . . . , d}, kE(Y | F_mδ)k_p ≤

kE(Y | F_m^(q))k_p for j∈Z^d.

In view of Lemma 2.1, throughout, an infinite sum of random variables is understood as the limit of partial sums over sequence of finite sets in the L^p sense.

With projection operators defined above, the Hannan’s condition states as

(2.1) ∆_p(X) := X

i∈Z^d

kP₀X_ik_p <∞,

for somep≥2. We first give two lemmas on Bernoulli random fields under Hannan’s condition.

Lemma 2.2. Suppose ∆p(X)<∞ for somep≥2. Then for all{a_i}_i∈

Z^d ∈

`²(Z^d), (2.2)

X

i∈Z^d

a_iX_i p

≤C_p,d



 X

i∈Z^d

a²_i





1/2

∆_p(X) with Cp,d = (p−1)^d/2.

Proof. Observe that it suffices to show

X

i∈Λ

aiXi

p

≤Cp,d

X

i∈Λ

a²_i

!1/2

∆p(X) for all finite Λ⊂Z^d. Then, by Lemma 2.1,

X

i∈Λ

a_iX_i=X

i∈Λ

a_i X

j∈Z^d

P_jX_i= X

j∈Z^d

P_j X

i∈Λ

a_iX_i

! .

Now letY be any random variable withE|Y|^p <∞in the same probability space, and recall thatP_j =Q_d

q=1P_j^(q)

q . Then,

X

j∈{−m,...,m}^d

PjY

2

p

=

m

X

j1=−m

Pj1





X

j2,...,jd∈{−m,...,m}

d

Y

q=2

PjqY





2

p

≤(p−1)

m

X

j1=−m

Pj1





X

j2,...,j_d∈{−m,...,m}

d

Y

q=2

PjqY





2

p

(2.3)

= (p−1)

m

X

j1=−m

X

j2,...,jd∈{−m,...,m}

d

Y

q=2

P_jq(P_j1Y)

2

p

,

where in the first and last equalities above we used the commuting property of the projection operators, and the inequality is a Marcinkiewicz–Zygmund type inequality for one-dimensional martingales due to Rio [31, Theorem 2.1]. Iterating the same argument we arrive at, for allm∈N,

(2.4)

X

j∈{−m,...,m}^d

PjY

2

p

≤C_p,d² X

j∈{−m,...,m}^d

kP_jYk²_p. Taking Y =P

i∈ΛaiXi, we obtain for all m∈N,

X

j∈{−m,...,m}^d

P_j X

i∈Λ

a_iX_i

! p

≤C_p,d² X

j∈{−m,...,m}^d

P_j X

i∈Λ

a_iX_i

!

2

p

≤C_p,d² X

j∈Z^d

X

i∈Λ

|a_i|kP₀Xi−jk_p

!2

≤C_p,d² X

j∈Z^d

X

i∈Λ

a²_ikP₀Xi−jk_pX

`∈Λ

kP₀X`−jk_p

=C_p,d² ∆²_p(X)X

i∈Λ

a²_i,

where we applied triangle inequality and Cauchy–Schwarz inequality in the second and third steps, respectively. Thus, we have shown (2.2).

Remark 2.3. It is not clear to us whether the constant C_p,d = (p−1)^d/2 is optimal for d≥2. It is proved by Rio [31] that it is optimal when d= 1.

The constant C_p,d will play a role when establishing tightness with entropy conditions for invariance principles. See Remark 4.4 below.

Lemma 2.4. Suppose∆₂(X)<∞. ThenP

j∈Z^d|Cov(X₀, X_j)| ≤∆²₂(X)<

∞.

Proof. Hannan’s condition enables to write X_i =P

jP_jX_i. Since {P_j}_j∈

Z^d

are orthogonal in the sense that E[(PjX)(P_kY)] = 0 for all j, k∈Z^d, j 6=k and X, Y ∈L²(Ω,B,P), it follows that

X

k∈Z^d

|E(X0X_k)| ≤ X

k∈Z^d

X

i∈Z^d

E|(P_iX0)(PiX_k)|

≤ X

k∈Z^d

X

i∈Z^d

kP_iX₀k₂kP_iX_kk₂ = ∆²₂(X).

As a consequence, we introduce

(2.5) σ² := X

j∈Z^d

Cov(X₀, X_j)

which is finite under Hannan’s condition.

To state the main result, introduce ~b_n = {b_n,j}_j∈

Z^d ∈ `²(Z^d), b_n :=

(P

j∈Z^db²_n,j)^1/2. For{~bn}_n∈N⊂`²(Z^d), we are interested in S_n= X

j∈Z^d

b_n,jX_j,

which by Lemma 2.2 is defined in the L² sense under ∆₂(X) < ∞, and moreover for σ_n² := Var(Sn),

(2.6) σ²_n≤Cb²_n∆²₂(X)<∞, n∈N for some constantC >0. Our main result is the following.

Theorem 2.5. Let {X_i}_i∈

Z^d be a stationary Bernoulli random field as in (1.5) satisfying Hannan’s condition (2.1). If

(2.7) lim

n→∞sup

j∈Z^d

|b_n,j| b_n = 0 and

(2.8) lim inf

n→∞

σ_n² b²_n >0 hold, then

(2.9) S_n

σn

⇒ N(0,1).

The condition (2.8) is subtle as it involves both the coefficients and the dependence of underlying random fields (viaσ_n). The following corollary is more convenient, as it imposes only conditions on coefficients. However, we see later in Example 3.2 that there are examples that satisfy the conditions in Theorem 2.5, but the conclusion of Corollary 2.6 does not hold. Recall that fork∈Z^d, the shift operator yieldsT_k~b_n={b_n,j+k}_j∈

Z^d. Lete₁, . . . , e_d be thedcanonical unit vector in R^d.

Corollary 2.6. Let {X_i}_i∈

Z^d be a stationary Bernoulli random field as in (1.5) satisfying Hannan’s condition (2.1). Under the notations as in Theorem 2.5, if

(2.10) lim

n→∞

kT_eq~b_n−~b_nk_`2

b_n = 0, for all q= 1, . . . , d hold, then

(2.11) lim

n→∞

σ_n bn

=σ with σ defined as in (2.5), and

Sn

b_n ⇒ N(0, σ²).

Proof of Corollary 2.6. We first show (2.11). Recall (2.6). Observe that

−2 X

k∈Z^d

bn,kbn,k+j =kT_j~b_n−~b_nk²_{`</sub>2 − k~b<sub>n</sub>k<sup>2</sup><sub>`}2 − kT_j~b_nk²_`2,

and kT_j~b_n−~b_nk_`2 = o(b_n) for all fixed j ∈ Z^d, a consequence of (2.10).

Therefore,

(2.12) lim

n→∞

1 b²_n

X

k∈Z^d

b_n,kb_n,k+j = 1 for allj ∈Z^d.

Thus, by the dominated convergence theorem, (2.6) and (2.12) imply (2.11).

Ifσ = 0, thenσ_n²/b²_n→0, and the central limit theorem is degenerate and trivially holds. If σ > 0, then (2.8) holds. By Cauchy–Schwarz inequality, (2.10) implies

(2.13) lim

n→∞

1 b²_n

X

j∈Z^d

b²_n,j+eq−b²_n,j

= 0, for allq = 1, . . . , d.

It has been shown in [5, Lemma 8], using an idea from [28], that (2.13) implies (2.7). The desired result now follows from Theorem 2.5.

Remark 2.7. Condition (2.10) was introduced in Bierm´e and Durieu [4, Theorem 3.1]. Condition (2.7) was also assumed there. It has been pointed out in [5, Remark 3] that (2.7) was redundant.

Proof of Theorem 2.5. We proceed an m-dependent approximation argu- ment. For each m ∈ N, set G_j^(m) = σ(_i : i ∈ Z^d,|j − i|_∞ ≤ m), X_j^(m) = E(X_j | G_j^(m)), j ∈ Z^d. In this way, {X_j^(m)}_j∈

Z^d is a (2m + 1)- dependent stationary random field. Write

S_n^(m) = X

j∈Z^d

bn,jX_j^(m) and σ_m,n² = Var(S_n^(m)).

Observe that P₀X_j^(m)= X

δ∈{0,1}^d

(−1)^{δ1+···+δd}E h

E(X_j | G_j^(m)) F_−δi

= X

δ∈{0,1}^d

(−1)^{δ1+···+δd}E h

E(X_j | F_−δ) G_j^(m)i

=E

P₀X_j G_j^(m)

,

where in the second equality we used the fact that the σ-algebrasG_j^(m) and F_` are conditionally independent and hence commuting, because they are generated by independent random variables{_j}_j∈

Z^d. Thus, (2.14) ∆p(X^(m))≤∆p(X),

and Sn^(m) is well defined in theL^p sense if ∆_p(X)<∞, p≥2.

We will approximateS_nbySn^(m). To establish a central limit theorem for m-dependent random variables, we will apply a result due to Heinrich [19],

which requires each partial sum to be of finite number of random variables.

Therefore, we introduce a finite setV_n⊂Z^d for each nsuch that|V_n| → ∞ and limn→∞b⁻²_n P

j∈Vnb²_n,j = 1. Set S_V^(m)_n = X

j∈Vn

bn,jX_j^(m) and σ²_m,Vn = Var(S_V^(m)_n ).

We first summarize a few estimates in the following lemma.

Lemma 2.8. With the construction described above,

(2.15) lim

m→∞lim sup

n∈N

Var(S_n−S_n^(m))

σ_n² = 0, lim

m→∞lim sup

n∈N

|σ_m,n² −σ²_n| σ_n² = 0, and with the choice of Vn described above, for everym large enough,

(2.16) lim

n→∞

Var(Sn^(m)−S_V^(m)_n )

σ_m,n² = 0, lim

n→∞

σ_m,V²

n

σ_m,n² = 1.

Proof of Lemma 2.8. In the sequel, we let C denote constant number inde- pendent fromnand m, but may change from line to line. We first show the first part of (2.15). Indeed, by Lemma 2.2,

Var(Sn−S_n^(m))≤Cb²_n



 X

j∈Z^d

P0(Xj−X_j^(m)) 2





2

.

Observe that for eachj,kP₀(Xj−X_j^(m))k₂ ≤ kX_j−X_j^(m)k₂→0 asm→ ∞, and that

X

j∈Z^d

P0(Xj−X_j^(m))

2 ≤ X

j∈Z^d

kP₀(Xj)k₂+kP₀(X_j^(m))k₂

≤∆₂(X^(m)) + ∆₂(X)≤2∆₂(X), which is finite under Hannan’s condition. By the dominated convergence theorem, limm→∞sup_n∈NVar(Sn^(m)−S_n)/b²_n= 0, and the first part of (2.15) follows from the assumption (2.8). To see the second part, it suffices to observe

σ²_m,n−σ_n²

≤Var^1/2(S_n^(m)−S_n)Var^1/2(S_n^(m)+S_n).

We have seen that σ_n² ≤Cb²_n in (2.6). Again by Lemma 2.2 and (2.14), σ_m,n² ≤Cb²_n∆²₂(X^(m))≤Cb²_n∆²₂(X).

Therefore, Var(Sn^(m)+S_n) ≤2(σ_m,n² +σ_n²) ≤Cb²_n, for allm, n∈N. It then follows

lim sup

n→∞

|σ_m,n² −σ²_n|

σ_n² ≤Clim sup

n→∞

bn

σn

Var^1/2(Sn^(m)−Sn) σn

. The second part of (2.15) now follows from the first part and (2.8).

For (2.16), to show the first part, using the same argument as above it suffices to observe

Var(S_n^(m)−S_V^(m)

n ) σ²_m,n ≤C

P

j /∈Vnb²_n,j

σ_m,n² ∆²₂(X^(m))≤C P

j /∈Vnb²_n,j b²_n

b²_n σ_n²

σ²_n

σ_m,n² ∆²₂(X), again by Lemma 2.2 and (2.14). By the second part of (2.15), for m large enough, saym≥m₀, lim sup_n→∞|σ_m,n² −σ_n²|/σ²_n≤1/2, whence

(2.17) lim sup

n→∞

σ_n²

σ²_m,n ≤2, m≥m₀.

Therefore the first part of (2.16) follows, for m≥m0. For the second part, observe that

|σ_m,n² −σ²_m,Vn| ≤Var^1/2(S_n^(m)−S_V^(m)

n )Var^1/2(S_n^(m)+S_V^(m)

n ), and by Lemma 2.2 and (2.14),

σ_m,V² _n ≤C



 X

j∈Vn

b²_n,j



∆²₂(X^(m))≤Cb²_n∆²₂(X).

Thus,

(2.18) |σ_m,n² −σ²_m,V

n|

σ_m,n² ≤C Var(Sn^(m)−S_V^(m)

n ) σ_m,n²

!^1/2 bn

σ_n σn

σ_m,n.

By (2.8), (2.17) and the first part of (2.16), for m ≥ m₀ the second part

of (2.16) follows.

Now we prove the desired central limit theorem (2.9) in three steps.

1) We first show, form large enough,

(2.19) lim

n→∞

S_V^(m)

n

σ_m,Vn ⇒ N(0,1).

For this purpose, we apply the central limit theorem for m-dependent ran- dom variables due to Heinrich [19]. We need also

(2.20) lim sup

n→∞

bn

σ_m,Vn <∞,

which follows from (2.8) and (2.16), for m large enough. For (2.19), the required conditions in Heinrich’s theorem can be easily verified: for any m∈N large enough fixed,

1 σ²_m,V

n

X

j∈Vn

E

b²_n,jX_j^(m)2

≤ b²_n σ_m,V²

n

Var(X₀^(m))≤C <∞

for some constantC and nlarge enough, and for all >0, and m^2d

σ_m,V²

n

X

j∈Vn

E

b²_n,jX_j^(m)21_{|X(m)

j |≥m^−2dσm,Vn_|

bn,j|}

≤ m^2db²_n σ_m,V²

n

E X₀^(m)21

{|X₀^(m)|≥m^−2d/sup_j ^|bn,j|

σm,Vn}

!

→0 asn→ ∞ where the last step is due to (2.20) and the assumption (2.7).

2) Observe that

Sn^(m)

σm,n

= S^(m)_n −S_V^(m)

n

σm,n

+ S_V^(m)

n

σm,Vn

σ_m,Vn σm,n

.

From (2.16) and (2.19), it follows that for mlarge enough,

(2.21) S^(m)n

σ_m,n ⇒ N(0,1).

3) At last, to show (2.9), observe that Sn

σ_n −Sn^(m)

σ_m,n = 1

σ_n(Sn−S_n^(m)) +σm,n−σn

σ_nσ_m,n S_n^(m). By Lemma 2.8, it follows that

(2.22) lim

m→∞lim sup

n→∞ Var Sn

σn

− Sn^(m)

σm,n

!

= 0.

Therefore, applying [6, Theorem 4.2] to (2.21) and (2.22), we have thus

proved (2.9).

Remark 2.9. The same mn-dependent approximation as in [4, Theorem 3.1] can be applied here, once one notices that

(2.23) lim

n→∞

Var(S_n−S_n^(mn)) b²_n = 0

holds (in the same way as in the proof of the first part of (2.15)) in place of [4, Eq. (3.4)] for an appropriately chosen increasing sequence {m_n}_n∈N, and the rest of the proof therein can be carried out with minor changes. In order not to introduce too much duplication, we chose to present a different proof. Our result is more general also in the sense that we consider the normalization ofσ_n instead ofb_n.

3. Central limit theorems for set-indexed partial sums In this section, we consider the case

S_n≡S_Γn = X

i∈Γ_n

X_i

for a sequence of subsets {Γ_n}_n∈N of Z^d with the cardinality of subsets

|Γ_n| → ∞ as n → ∞. This corresponds to the case bn,j = 1{j∈Γ_n} and b_n=|Γ_n|^1/2. Then, in view of Corollary 2.6, it is easy to notice that (2.10) is equivalent to

(3.1) lim

n→∞

|∂Γ_n|

|Γ_n| = 0,

where ∂Γn = {i ∈ Γn : ∃j /∈ Γn,|i−j|∞ = 1} is the boundary set of Γn. Indeed, if we identify Γ_n with an element in `<sup>2</sup>(Z<sup>d</sup>) via b<sub>n,j</sub> =1<sub>{j∈Γn}</sub>, then for each q = 1, . . . , d we have kT<sub>eq</sub>Γ<sub>n</sub>−Γ<sub>n</sub>k<sup>2</sup><sub>`2</sub> ≤2|∂Γ_n| ≤Pd

m=1kT_emΓ_n− Γ_nk²_`2. We have thus obtained the following.

Corollary 3.1. For a Bernoulli random field with ∆₂(X) < ∞, and a sequence of subsets {Γ_n}_n∈N of Z^d satisfying|Γ_n| → ∞ and (3.1),

(3.2) S_Γn

|Γ_n|^1/2 ⇒ N(0, σ²) with σ² given in (2.5).

In the rest of this section, we discuss what happens if we are interested in the convergence of

(3.3) Sn

σn

⇒ N(0,1).

This follows from (2.8), by Theorem 2.5. To see the role of the condi- tion (2.8), we provide two examples. First, by Example 3.2, we show that condition (2.8) cannot be removed: otherwise (3.3) may no longer hold under Hannan’s condition. Second, by Example 3.3, we show that the assumption in Corollary 2.6 is strictly stronger than (2.8), in the sense that there are examples satisfying (2.8), but the conclusion of Corollary 2.6 does not hold.

Note also that Example 3.2 also shows that when S_n/b_n ⇒ N(0, σ²) with σ² = 0, one should not expectSn/σn to converge, without further assump- tions.

For the sake of simplicity, both examples are given in one dimension. Let {_i}_i∈Z be the i.i.d. random variables that generate the Bernoulli random field (1.5).

Example 3.2. Consider Γ_n ={0,1, . . . , n−1}. We construct an example such thatSn/σn converges to different limits along different subsequences.

Suppose that there exists a collection of mutually independent random variables {ζ_n^(k)}_n∈Z,k∈N such that for each k ∈ N, {ζ_n^(k)}n∈Z are i.i.d., and

for each n, ζn^(k) is σ(_n)-measurable. We further assume that Eζn^(k) = 0,Var(ζ_n^(k)) = 1. A detailed construction is given at the end.

For coefficients {α_k}_k∈N satisfying P

k|α_k| < ∞ and a sequence of in- creasing positive integers{n_k}_k∈N, set

W_n^(k)=α_k(ζ_n^(k)−ζ_n−n^(k)

k), k∈N and X_n=

∞

X

k=1

W_n^(k).

Observe that P₀X_n = P∞

k=1P₀Wn^(k), which equals −α<sub>`</sub>ξ<sub>0</sub><sup>(`)</sup> if n = n<sub>`</sub> for some`∈N, and 0 otherwise. Thus, ∆2(X) =P∞

`=1|α<sub>`</sub>|<∞.

WriteS_n=S_Γn =Pn−1

i=0 X_i and S_n(W^(k)) =Pn−1

i=0 W_i^(k). So S_n=

∞

X

k=1

S_n(W^(k)).

By independence,

E(S_nk(W^(`)))² =

2n_{`</sub>α<sup>2</sup><sub>`} `≤k 2n<sub>k</sub>α<sup>2</sup><sub>`</sub> ` > k , and

Var(Snk) =

∞

X

`=1

Var(Snk(W^(`))) =

k−1

X

`=1

2n`α<sup>2</sup><sub>`</sub> +

∞

X

`=k+1

2nkα_`²+ 2nkα²_k. One can chooseαk and nk so that

(3.4) Var(Snk)∼Var(Snk(W^(k))) = 2nkα²_k ask→ ∞.

For example, taking αk = 2^−k2 and nk = 2^3k2 k∈ N, it yields Var(Snk) ∼ Var(Snk(W^(k))) = 2^k2+1.

Now in view of (3.4), for our purpose it suffices to chooseζ_kappropriately such that

(3.5) Z_k:= S_nk(W^(k))

α_k√ n_k

converge to different limits along even and odd sequences.

To do so, we now give an explicit construction of {ζ_n^(k)}_n∈Z,k∈N. For the sake of simplicity, consider (Ω,B,P) = ([0,1]^Z,B([0,1])^Z,Leb^Z), and n(ω) =ωn, ω ∈Ω. Here, T is the shift operator although we do not use it explicitly. In this way, for any sequence{d_k}_k∈N with d_k∈[0,1], we choose a family of sets{A^±_k}_k∈N⊂ B([0,1]) such thatA⁺_k ∩A⁻_k =∅,µ(A^±_k) =dk/2, and set

ζ_n^(k)(ω) = 1

√dk

(1_A⁺

k −1_A⁻

k)(ωn), n∈Z, k∈N.

So for n 6= n⁰, ζn^(k) and ζ_n^(k0⁰⁾ are independent. In order that {ζ_n^(k)}_n∈Z,k∈N satisfy the conditions that we assumed at the beginning, it remains to choose {A⁺_k, A⁻_k}_k∈Nsuch that for each fixedn,{ζ_n^(k)}_k∈Nare mutually independent.

This can be done via a variation of dyadic expansion as follows. First, pick A⁺₁ := (0, d₁/2], A⁻₁ := (d₁/2, d₁]. Suppose A⁺_k, A⁻_k have been selected for k ∈N. Then each of A⁺_k, A⁻_k and (0,1]\(A⁺_k ∪A⁻_k) can be expressed as a disjoint union of left-open-right-closed intervals, and together these intervals form a partition of (0,1], say (0,1] =Sjk

j=1(a_j,k, b_j,k]. Now, set A⁺_k+1 :=

j_k

[

j=1

aj,k, aj,k+ (bj,k−aj,k)dk+1

2

#

A⁻_k+1 :=

jk

[

j=1

aj,k+ (bj,k−aj,k)d_k+1

2 , aj,k+ (bj,k−aj,k)dk+1

# .

The so-constructed{ζ_n^(k)}_n∈Z,k∈N are then mutually independent.

Now set d_k = 1 for k even and d_k = 1/n_k for k odd. For {A^±_k}_k∈N and {ζ_n^(k)}_n,k∈N described above, whenkis even, Z_k in (3.5) becomes

√1 nk

n_k−1

X

i=0

(ζ_i^(k)−ζ_i−n^(k)

k)

which is the normalized sum of 2n_kRademacher random variables, and thus Z_2k⇒ N(0,2) ask→ ∞. At the same time, forkodd,Z_kin (3.5) becomes

n_k−1

X

i=0

(ζ_i^(k)−ζ_i−n^(k)

k),

which is the sum of 2n_ki.i.d. random variables withP(ζ₁^(k)=±1) = 1/(2n_k) and P(ζ₁^(k) = 0) = 1−1/nk. Clearly as k→ ∞ Z2k−1 has a non-degenerate limiting distribution which is not Gaussian. So S_n/σ_n does not converge.

Example 3.3. Consider Xi = i −i−1. Observe that Xi and Xj are uncorrelated if |i− j| ≥ 2. Therefore, this stochastic process satisfies

∆2(X) < ∞. We now construct a sequence of subsets {Γ_n}_n∈N such that lim infn→∞σ_n/b_n>0 but limn→∞σ_n/b_n does not exist.

We construct Γn iteratively. Set Γ1 = {0,1}. For n ∈ N, set Γn+1 = Γn∪Bn with

Bn=

{a_n+ 2, an+ 3, . . . , an+ 2ⁿ+ 1} neven {a_n+ 2, a_n+ 4, . . . , a_n+ 2·2ⁿ} nodd

with a_n = max{j : j ∈ Γ_n}. By construction, Var(S_Γn+1) = Var(S_Γn) + Var(S_Bn), and Var(Bn) = 2E²₀ for n even, and 2ⁿ⁺¹E²₀ forn odd. At the same time, |Γ_n|= 2ⁿ. It is clear that the desired result follows.

4. Invariance principles for Gaussian random fields In this section, we present two invariance principles for weighted Bernoulli random fields. LetT be an index set equipped with a pseudo-metric. Con- sider random fields in form of

(4.1) S_n(t) = X

j∈Z^d

b_n,j(t)X_j, t∈T.

Under Hannan’s condition on{X_j}_j∈Zd and appropriate assumptions on the coefficients bn,j(t), we shall establish, for an increasing sequence of positive numbers{b_n}_n∈N,

(4.2)

Sn(t) b_n

t∈T

⇒ {Gt}_t∈

T

where G is a zero-mean Gaussian process. The space of weak convergence will be specified below. Most results improve earlier ones [4, 17, 35], in the sense that Wu’s condition is replaced by Hannan’s condition.

We first provide an overview on how to establish (4.2), illustrating how previous proofs can be adapted without much changes. To establish such an invariance principle, we proceed as in the standard two-step proof: we first show convergence of finite-dimensional distributions and then tightness.

To show the convergence of finite-dimensional distributions, we first remark that marginally, forbn(t) := (P

j∈Z^db²_n,j(t))^1/2, one should expect

(4.3) Sn(t)

bn(t) ⇒ N(0, σ²), for allt∈T

with σ² as in (2.5) as a consequence of Theorem 2.5. Comparing this with (4.2), it suggests that limn→∞b²_n(t)/b²_n = Var(Gt)/σ². Moreover, by Cramer–Wold’s device, for the weak convergence to hold, we need to show, for all λ∈R^m, t∈T^m, m∈N,

(4.4) 1 bn

m

X

r=1

λ_rS_n(t_r)⇒ N(0,Σ²_λ,t) with Σ²_λ,t= Var

m

X

r=1

λ_rGtr

! . The linear combinations of finite-dimensional distributions can again be rep- resented as a linear random field via

m

X

r=1

λrSn(tr) = X

j∈Z^d

ebn,jXj with ebn,j =

m

X

r=1

λrbn,j(tr),

to which one can apply Theorem 2.5 again. This is the standard procedure to establish finite-dimensional convergence of linear random fields. In our setup we have thus proved the following as a consequence of Theorem 2.5.

Writeeb_n= (P

jeb²_n,j)^1/2.

Proposition 4.1. Consider random fields in form of (4.1) with {X_j}_j∈

Z^d

satisfying Hannan’s condition∆₂(X)<∞. Suppose there exists a sequence of real numbers{b_n}n∈N such that