A strong invariance principle for positively or negatively associated random fields

Statistics and Probability Letters 78 (2008) 2121–2129www.elsevier.com/locate/stapro

A strong invariance principle for positively or negativelyassociated random fieldsI

Alexey Shashkin∗

Moscow State University, Faculty of Mathematics and Mechanics, Moscow 119991, Russia

Received 20 October 2007; received in revised form 24 January 2008; accepted 30 January 2008Available online 3 February 2008

Abstract

We prove a new coupling theorem stating the proximity of positively or negatively associated random vectors to vectors withindependent increments. It is applied to the proof of strong invariance principle under weakest known conditions.c© 2008 Elsevier B.V. All rights reserved.

The coupling or reconstruction techniques is one of the most powerful methods of working with dependent systemsof random variables. It allows to construct an independent random system which is in a specified sense close to thegiven one; thus one can obtain some properties of the first system from well-known results for the independent one.Examples of such technology were given, e.g. by Strassen (1965), Berkes and Philipp (1979), Berbee (1987), Peligrad(2002) and Dedecker and Prieur (2004) and others. In this paper we prove a reconstruction result for associated randomsystem (or more general in a specified sense) using the connection between ideal ζ -metric and Prohorov metric. Thisseems to be the first theorem of this type applicable directly to positively or negatively associated random vectors.As an application we derive strong invariance principle for a positively or negatively associated random field withcovariance function vanishing at infinity at power rate. Previously known results require the exponential decrease ofcovariance function.

Association of random variables is an important concept in the study of dependent random systems, which appearedin the paper by Esary et al. (1967). Positive association was introduced by Newman (1984), and negative oneby Joag-Dev and Proschan (1983). Association and its generalizations find important applications in reliability theory,statistics, mathematical physics. Since Newman’s proof of central limit theorem for strictly stationary associatedrandom fields (Newman, 1980), there has been a considerable progress in the study of limit behaviour of associatedsystems, see Bulinski and Shashkin (2007). The strong invariance principle, however, was not established untilthe paper by Yu (1996). That result was generalized to random fields by Balan (2005) and later to random fieldswith more general dependence conditions than positive or negative association (Bulinski and Shashkin, 2006). Thestrong invariance principle proved in present paper comprises the results of Yu and Balan and is proved via a newmethodology based on coupling. This technique also allows us to simplify the constructions used in the proof.

In what follows, |U | denotes the cardinality of a finite set U . For a system of random variables X j , j ∈ T

and finite set I = i1, . . . , in ⊂ T , we write X I for the random vector (X i1 , . . . , X in ), when the ordering of I is

I The work has been supported by RFBR, grant 07-01-00373.∗ Tel.: +7 495 939 14 03; fax: +7 495 939 20 90.

E-mail addresses: [email protected], [email protected].

0167-7152/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2008.01.078

http://www.elsevier.com/locate/stapro

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.spl.2008.01.078

2122 A. Shashkin / Statistics and Probability Letters 78 (2008) 2121–2129

inessential. By ‖ · ‖ we denote the Euclidean norm of a vector, by dist(·, ·) the corresponding metric, and by ‖ · ‖∞ —the usual maximal norm of a vector or of a function which is essentially bounded. The notation Law(ξ) stands for thedistribution of the random element ξ.

Definition 1. A random system X = X t , t ∈ T is called associated, if for any pair of finite sets I, J ⊂ T and allbounded coordinatewise nondecreasing Borel functions f : R|I |

→ R, g : R|J |→ R one has

cov( f (X I ), g(X J )) ≥ 0. (1)

System X is positively associated (or PA) if the above requirement holds under additional assumption that I ∩ J = ∅.Finally, X is negatively associated (or NA), if (1) holds with opposite sign, when I ∩ J = ∅.

Definition 2. If X is PA or NA random field, then its Cox–Grimmett coefficients (ur )r∈N are defined by the relation

ur = supj∈Zd

∑k∈Zd :‖k− j‖∞≥r

|cov(X j , Xk)|, r ∈ N. (2)

After these preliminary notes we pass to the coupling results.

Lemma 3. Let Y = (Y1, . . . , Ym) be a PA or NA random vector such that E‖Y‖2 < ∞, and let W = (Y1, . . . , Ym−1).

Suppose that random vector Z = (Z1, . . . , Zm) is independent of Y , and consists of independent random variablessuch that Law(Zk) = Law(Yk), k = 1, . . . ,m. Then for any function f : Rm

→ R, having bounded second orderpartial derivatives, the following estimates hold:

|E f (W, Ym)− E f (W, Zm)| ≤

m−1∑k=1

∥∥∥∥ ∂2 f

∂xk∂xm

∥∥∥∥∞

|cov(Yk, Ym)|,

|E f (Y )− E f (Z)| ≤

∑1≤k<v≤m

∥∥∥∥ ∂2 f

∂xk∂xv

∥∥∥∥∞

|cov(Yk, Yv)|.

Proof. At first we consider the PA case. Introduce functions f± letting

f±(x1, . . . , xm) = f (x1, . . . , xm)±

∑1≤i< j≤m

∥∥∥∥ ∂2 f

∂xi∂x j

∥∥∥∥∞

xi x j , x ∈ Rm .

Here the sign + or − is the same on both sides. Then the functions f+ and (− f−) are supermodular (see the definitionin Christofides and Vaggelatou (2004)). Consequently, by the first statement obtained in the proof of Theorem 1in Christofides and Vaggelatou (2004) (formula (4) of that paper), we have

E f+(Y1, . . . , Ym−1, Ym) ≥ E f+(Y1, . . . , Ym−1, Zm), (3)

E f−(Y1, . . . , Ym−1, Ym) ≤ E f−(Y1, . . . , Ym−1, Zm). (4)

These inequalities rewritten in terms of f give respectively the lower and upper bounds needed for the first desiredestimate. The second estimate ensues from the first one, the Fubini theorem and induction in m.

As for the NA case, the arguments remains the same with the reverse inequality signs in (3) and (4).

Lemma 3 contains some well-known inequalities applied to study the behaviour of associated random systems. Forexample, taking a function h ∈ C2

b(R) and f (x) = h(x1 +· · ·+ xm) we obtain the inequality of Lewis (1998). Takingf (x) = expi

∑mk=1 tk xk, where t1 . . . , tm ∈ R and applying the Lemma 3 separately to the real and imaginary parts

of f , we obtain the Newman’s inequality for characteristic functions (Newman, 1980) (up to a positive multiplicativeconstant which is inessential for applications). Taking Lipschitz functions h : Rm−1

→ R and g : R → R andapplying the Lemma to the function f (x) = h(x1 . . . , xm−1)g(xm), we obtain a partial important case of the inequalityfrom Bulinski and Shabanovich (1998) (first for differentiable h and g, then by appropriate approximation to Lipschitzones).

A. Shashkin / Statistics and Probability Letters 78 (2008) 2121–2129 2123

Lemma 4 (Berkes and Philipp, 1979). Let Si , i = 1, 2, 3, be Polish spaces. Suppose that (ξ1, ξ2) and (η1, η2) are tworandom elements with values respectively in (S1 × S2,B(S1)⊗ B(S2)) and (S2 × S3,B(S2)⊗ B(S3)) and such thatLaw(η1) = Law(ξ2). Then there exist a probability space and random elements ζ 1, ζ 2 and ζ 3 defined on it such thatζ i takes values in (Si ,B(Si )) (for i = 1, 2, 3), Law(ζ 1, ζ 2) = Law(ξ1, ξ2) and Law(ζ 2, ζ 3) = Law(η1, η2).

Lemma 5. Let Y be the same random vector as in Lemma 3. Then one can redefine it on a new probabilityspace (without changing its distribution) supporting also random vector Z = (Z1, . . . , Zm), such thatLaw(Y1, . . . , Ym−1) = Law(Z1, . . . , Zm−1), random variable Zm is independent of (Z1, . . . , Zm−1) and

P(‖Y − Z‖ > Am1/6 R1/3

m

)≤ Am1/6 R1/3

m , (5)

where Rk =∑k−1

j=1 |cov(Y j , Yk)| for k > 1, and A ∈ (0, 5) is an absolute constant.

Proof. Recall that the ideal ζ -metric of order 2 between two distributions of random vectors Y and Z (taking values inRm) is the quantity ζ2(Y, Z) = supg∈G |Eg(Y )−Eg(Z)|,where G = g : Rm

→ [−1, 1], ‖∇g(x)−∇g(y)‖ ≤ ‖x−y‖

for all x, y ∈ Rm. Here ∇g(z) is the gradient of the function g taken at point z ∈ Rm . Using convolutions it is

easy to show that the supremum in the definition of ζ2 can be taken only over twice differentiable functions fromG. But then ζ2(Y, Z) ≤ Rm by the first assertion of Lemma 3. Thus by the inequality of Senatov (1984), we haveπ(Y, Z) ≤ Am1/6 R1/3

m where π(·, ·) is the Levy–Prokhorov distance between the distributions of Y and Z in Rm

endowed with Euclidean metric. Hence the theorem of Strassen (1965) implies the lemma.

Lemma 6. Let Y be the same random vector as in Lemma 3. Then one can redefine it on a new probability space(without changing its distribution) together with independent random variables Z1, . . . , Zm , such that Law(Yk) =

Law(Zk) and P(|Yk − Zk | > ak,m) ≤ ak,m for k = 1, . . . ,m, where

ak,m = Am∑v=k

v1/6 R1/3v , (6)

and the quantities Rm and A are the same as in Lemma 5 (R1 := 0).

Proof. We proceed by induction in m. If m = 2 then the claim follows from Lemma 5. So, suppose that it istrue for all (m − 1)-dimensional random vectors satisfying lemma’s condition (here m > 2). By Lemma 5 thereexists a probability space (Ω1,F1,P1) on which there are defined random vectors Y and Z satisfying (5). By theinduction hypothesis applied to the random vector (Z1, . . . , Zm−1) there exists a probability space (Ω2,F2,P2)

supporting random vector U = (U1, . . . ,Um−1) and independent random variables V = (V1, . . . , Vm−1) such thatLaw(U ) = Law(Z1, . . . , Zm−1) and

P2

(|Uk − Vk | > A

m−1∑v=k

v1/6 R1/3v

)≤ A

m−1∑v=k

v1/6 R1/3v for k = 1, . . . ,m − 1.

Clearly, enlarging the probability space we can think that on (Ω2,F2,P2) there is a random variable T independentfrom pair (U, V ) and such that Law(T ) = Law(Zm). Apply Lemma 4 with ξ1 = (Y1, . . . , Ym), ξ2 =

(Z1, . . . , Zm), η1 = (U1, . . . ,Um−1, T ), η2 = (V1, . . . , Vm−1). By that Lemma we can construct three randomvectors ζ i (i = 1, 2, 3) on one probability space, with values respectively in Rm,Rm and Rm−1, such thatLaw(ζ 1) = Law(Y ), the variables (ζ 3

1 , . . . , ζ3m−1, ζ

2m) are independent, and

P(|ζ 1

k − ζ 2k | ≥ Am1/6 R1/3

m

)≤ Am1/6 R1/3

m , k = 1, . . . ,m, (7)

P

(|ζ 2

k − ζ 3k | ≥ A

m−1∑v=k

v1/6 R1/3v

)≤ A

m−1∑v=k

v1/6 R1/3v , k = 1, . . . ,m − 1. (8)

Lemma follows from (7) and (8), if we re-denote Yk := ζ 1k , k = 1, . . . ,m, Zm := ζ 2

m and Zk := ζ 3k , k = 1, . . . ,m −1.

Theorem 7. Let Y = (Yn)n∈N be a PA or NA sequence of random variables having standard Gaussian distribution.Suppose that there exist such C > 0 and γ > 9/2 that for any n > 1 and all k < n one has |cov(Yk, Yn)| ≤ Cn−γ .


Then one can redefine Y without changing its distribution on a new probability space, supporting also independentN (0, 1) random variables (Zn)n∈N, and such that for any n > 1 one has P(|Yn − Zn| > bn) ≤ bn , where

bn = 6AC1/3(2γ − 9)−1(n − 1)3/2−γ /3.

Proof. Fix n > 1. According to Lemma 6 there exists a probability measureµn on (R2n,B(R2n)) having the followingproperties: the projection of µn onto the first n coordinates is Law(Y1, . . . , Yn), the projection of µn onto the last ncoordinates is the direct product of measures Law(Y1)⊗ · · · ⊗ Law(Yn), and

µnx ∈ R2n: |xk − xn+k | > ak,n ≤ ak,n (9)

where ak,n is defined in (6), k = 1, . . . , n. By the conditions of theorem one has ak,n ≤ A∑nv=k v

1/6 R1/3v ≤

AC1/3∑∞

v=k v1/2−γ /3

≤ bk . Hence, in (9) one can replace ak,n with bn . Define random variables U (n)k k,n∈N and

V (n)k k,n∈N letting

µn = Law(U (n)1 , . . . ,U (n)

n , V (n)1 , . . . , V (n)

n )

and U (n)m = T (n)m = 0 for m > n. Further, for any n ∈ N introduce (2k)-dimensional random vectors

T (n)k = (U (n)1 , . . . ,U (n)

k , V (n)1 , . . . , V (n)

k ),

and notation ν(n)k = Law(T (n)k ). Note that for any k ∈ N the sequence of probability measures ν(n)k is tight.Consequently, via the Cantor diagonal procedure we can find a sequence (nm)m∈N such that nm → ∞ as m → ∞

and (ν(nm )k ) → Qk for any k ∈ N, where Qk is some probability measure on R2k . These measures (Qs)s∈N form a

consistent family, as is checked directly. Hence by the Kolmogorov theorem there exists a probability space supportinga sequence of random vectors (ξk, ηk)k∈N such that

Law(ξ1, . . . , ξk, η1, . . . , ηk) = Qk, k ∈ N.

For n > k and Borel sets B1, . . . , Bk ∈ B(R) we clearly have

ν(n)k (B1 × . . .× Bk × Rk) = P(Y1 ∈ B1, . . . , Yk ∈ Bk),

ν(n)k (Rk

× B1 × . . .× Bk) = P(Y1 ∈ B1) . . .P(Yk ∈ Bk).

Furthermore, if l ≤ k then the set Dk,l = x ∈ R2k: |xl − xk+l | > bl is open in R2k . Thus by the portmanteau

theorem

P(|ξl − ηl | > bl) = Qk(Dk,l) ≤ lim infm→∞

ν(nm )k (Dk,l) ≤ bl .

It remains to assign Yk := ξk and Zk := ηk, k ∈ N.

The main application of the last theorem is the strong invariance principle. We start with some more notations.For n ∈ Nd set 〈n〉 = n1 . . . nd and n − 1 = (n1 − 1, . . . , nd − 1). We write m ≤ n (m, n ∈ Zd ) tomean that mi ≤ ni , i = 1, . . . , d , and n → ∞ to mean that n1 → ∞, . . . , nd → ∞. Call a block the set(a, b] := ((a1, b1] × . . .× (ad , bd ]) ∩ Zd where a, b ∈ Zd and a ≤ b. For τ > 0 and d > 1 denote

Gτ =

n ∈ Nd

: nk ≥

(∏j 6=k

n j

)τ, k = 1, . . . , d

.

If d = 1, then set Gτ := N. For a finite V ⊂ Zd let S(V ) =∑

j∈V X j , and for n ∈ Nd write Sn = S((0, n]).

We formulate the next Theorem for random fields indexed by Zd . However it can be easily restated for the fieldsdefined on Nd , since only points from this set appear in the statement and proof. Our approach is slightly moreconvenient to work with, namely when we calculate the asymptotic variances of normalized partial sums (Lemma 12).


Theorem 8. Let X = X j , j ∈ Zd be a PA or NA wide-sense stationary centred random field such that

sup j∈Zd E|X j |2+δ < ∞ for some δ > 0. Assume that the Cox–Grimmett coefficients (ur )r∈N of X exist and there are

such c0, λ > 0 that

ur ≤ c0r−λ (10)

for all r ∈ N. Suppose also that σ 2=∑

j∈Zd cov(X0, X j ) > 0. Then for any τ > 0 one can redefine X on a new

probability space supporting also a d-parametric Brownian motion Wt , t ∈ Rd+, such that for some ε > 0 one

almost surely has

〈n〉ε−1/2(Sn − σWn) → 0 as n → ∞, n ∈ Gτ .

Theorem of Balan (2005) is a consequence of Theorem 8. Indeed, if X is associated, then σ 2 > 0 unless X j = 0 a.s.for all j ∈ Zd . Note that if d = 1, then there is no any restriction on the argument n ∈ N. So, the strong invarianceprinciple of Yu (1996) also appears to be a special case of Theorem 8. As is well known, the strong invarianceprinciple implies lots of other limit theorems, in particular, the functional central limit theorem and the law of theiterated logarithm. See Bulinski and Shashkin (2007) for these and other limit theorems.

Proof. Let α ∈ N and β ∈ (0, α) be some numbers to be specified later. Take also ρ := τ/8. Without loss of generalitywe may assume that ρ < 1/d . For m ∈ Z+ write D(m) =

∑ml=0 lα. The key idea of the proof is to divide the set

Gτ into integer blocks of increasing volume (as n goes to infinity) and to approximate the sum Sn with the sums oversuch blocks. For k ∈ Zd

+ define the point Nk = (D(k1), . . . , D(kd)) and set Bk = (Nk−1,Nk], k ∈ Nd . Also letL = i ∈ Nd

: Bi ⊂ Gρ, H =⋃

i∈L Bi . For each N = (N1, . . . , Nd) ∈ H , let N (1), . . . , N (d) be the points in Nd

defined as follows:

N (s)i := Ni , i 6= s and N (s)

s := min ns : n ∈ H, ni = Ni , i 6= s .

Finally, let Mk =

((N(1)k )1, . . . , (N

(d)k )d

), Rk = (Mk,Nk], so that Rk = (Mk,Nk] is the largest block of the type

(a,Nk] contained in H , and Lk = i : Bi ⊂ Rk. These notations are kept from Balan (2005).If U = (a, b] is a block in Zd

+ for some a, b ∈ Zd+, and W = W (t), t ∈ Rd

+ is the d-parametric Brownian motion,then let

W (U ) =

∑I⊂1,...,d

(−1)|I |W (b − (b − a, eI )),

where (·, ·) is the usual inner product in Rd and eI ∈ Rd is a vector such that (eI )i = Ii ∈ I , i = 1, . . . , d.To estimate |Sn − σWn| for n ∈ Gτ , pick k ∈ Nd such that Nk ≤ n < Nk+1. Note that

|Sn − σWn| ≤ |S((0, n])− S((0,Nk])| + |S((0,Nk])− S(Rk)| + |S(Rk)− σW (Rk)|

+ |σW (Rk)− σW ((0,Nk])| + |σW ((0,Nk])− σW ((0, n])|. (11)

For any k ∈ Nd we have Wk =∑

t∈Nd ,t≤k Tk, where T = T j , j ∈ Zd are independent N (0, 1) random variables.

The random field T satisfies theorem’s conditions; therefore we proceed only with first three summands at the right-hand side of (11). The first two of them are estimated via

Lemma 9 (Balan, 2005; Berkes & Morrow, 1981). Let conditions of Theorem 8 be satisfied and α be large enough.Then

〈Nk〉ε1−1/2 (|S((0, n])− S((0,Nk])| + |S((0,Nk])− S(Rk)|) → 0 a.s.

when n → ∞, n ∈ Gτ , for some nonrandom ε1 > 0. Here k = k(n) is defined as before (11).

Introduce independent (both mutually and also from X ) random variables wk, k ∈ Nd such that wk ∼ N (0, 〈k〉

β).

For such k ∈ Nd that Nk ∈ Gρ write

S(Rk) =

∑t∈Lk

S(Bk) =

∑t∈Lk

√σ 2

t + τ 2t (ξt − ηt )+

∑t∈Lk

√σ 2t + τ 2

t

|Bt |− σ

√|Bt |ηt


−

∑t∈Lk

wk + σ∑t∈Lk

√|Bt |ηt , (12)

σ 2t = VarS(Bt ), τt = Varwt , ξt = (S(Bt )+ wt )/

√σ 2

t + τ 2t , ηt = Φ−1(Ft (ξt )),

here Ft (x) = P(ξt ≤ x) and Φ is the standard Gaussian distribution function.In what follows C stands for generic positive constants determined by σ 2, D2+δ, τ and (ur )r∈N. We need seven

auxiliary lemmas estimating all terms at the right-hand side of (12) except for the last one. Note that if n ∈ Gτ , thatfor all n except for a finite number we have Nk(n) ∈ Gρ , which we will use without further mentioning. A standardargument comparing sums and integrals yields an elementary

Lemma 10. If k ∈ L then for any i = 1, . . . , d we have ki ≥ e−dρ/e〈k〉

ρ .

Lemma 11. Suppose that a random field X satisfies the conditions of Theorem 8. Then there exist such positive Cand µ, determined by σ 2, D2+δ and (ur )r∈N, that for any block U = (a, b] ⊂ Zd

supx∈R

|P(S(U ) ≤ x√

VarS(U ))− Φ(x)| ≤ C |U |−µ and E|S(U )|2+µ

≤ C |U |1+µ/2,

Proof. The first estimate is proved in Bulinski and Suquet (2001), the moment inequality in PA case in Bulinski (1993)(the proof is given for associated fields but remains valid for PA), and for NA case in Christofides and Vaggelatou(2004).

Lemma 12. If U = (a, b] ⊂ Zd is a block, then∣∣∣σ 2− |U |

−1VarS(U )∣∣∣ ≤ (c0 + 2dσ 2

0 )l0(U )−λ/(1+λ), (13)

where l0(U ) is the minimal edge length of U, and σ 20 =

∑j∈Zd |cov(X0, X j )|. If U1,U2 ⊂ Zd are two disjoint

blocks and l0(U1,U2) = l0(U1) ∧ l0(U2), then

|cov(S(U1), S(U2))|√

|U1||U2|≤(c0dist(U1,U2)

−λ)∧

(2c0 (l0(U1,U2))

−λ/(1+λ)). (14)

Proof. Clearly

σ 2|U | − VarS(U ) =

∑j∈U,k 6∈U

|cov(X j , Xk)| ≤ c0

∑j∈U,k 6∈U

| j − i |−λ.

Take the sets T = j ∈ U : dist( j,Zd\ U ) ≥ l0(U )1/(1+λ)

, R = U \ T . Then by (10)∑j∈T,k 6∈U

|cov(X j , Xk)| ≤ c0|U |l0(U )−λ/(1+λ),

∑j 6∈T,k 6∈U

|cov(X j , Xk)| ≤ σ 2|U \ T | ≤ σ 22d |U |

l0(U )l0(U )

1/(1+λ).

These two bounds imply (13). The same argument allows to obtain the second bound in (14). For example, if|U1| ≤ |U2|, one should take

T = j ∈ U1 : dist( j,Zd\ U1) ≥ l0(U1,U2)

1/(1+λ),

and again estimate two sums. The first estimate in (14) is obvious due to (10).

Lemma 13. If α > β + 3 then there exists such ε2 > 0 that∑t∈L ,t≤k

wt = O(〈N1/2−ε2

k 〉

)a.s., k → ∞.


Proof. The bound P(|wt | > 〈t〉α/2−1) ≤ 〈t〉2−α+β is elementary. Hence the series∑

t∈L P(|wt | > 〈t〉α/2−1) isconvergent.Thus by the Borel–Cantelli lemma, with probability one we have |wt | ≤ 〈t〉α/2−1 except for a finitenumber of t’s. Thus almost surely∑

t∈L ,t≤k

wt = O(∑t≤k

〈t〉α/2−1) = O(Nα/(2α+2)

k

)as k → ∞. (15)

Lemma 14. Suppose that α > β + 3 and α > 3(1 + λ)/ρλ. Then there exists such ε3 > 0 that with probability 1∑t∈Lk

(√|Bt |

−1(σ 2t + τ 2

t )− σ

)√|Bt |ηt = O(〈N1/2−ε3

k 〉) as k → ∞,Nk ∈ Gρ .

Proof. As in the previous lemma, it suffices to prove that

P((√

|Bt |−1(σ 2

t + τ 2t )− σ

)√|Bt |ηt > 〈t〉α/2−1

)≤ C〈t〉2−α+β (16)

(and then argue as in (15)). If t ∈ Gρ , then l0(Bt ) ≥ eρdα/e〈t〉ρα by the Lemma 10. Therefore, by the Lemma 12 the

left-hand side of (16) can be bounded by

〈t〉2−α|Bt |σ

−1∣∣∣∣σ 2

−σ 2

t + τ 2t

|Bt |

∣∣∣∣ ≤ (c0 + 2dσ 2)σ−1e−dν/e〈t〉2−ν

+ σ−1〈t〉2−α+β ,

where ν = ραλ/(1 + λ).

Lemma 15. For any K ∈ (0,√

2αµ) and for all k ∈ L large enough, one has

supx∈R:|x |≤K

√log〈k〉

|Φ−1(Fk(x))− x | ≤ C〈k〉−αµ+K 2/2.

Proof. Let U j , j ∈ Nd be independent (mutually and from X ) centered Gaussian random variables having variance

〈t〉β−α if j ∈ Bt . If j 6∈ Zd\ Nd then U j := 0. Consider the random field X = X j , j ∈ Zd

, whereX j := X j + U j , j ∈ Zd . The first assertion of the Lemma 11 applies to X , from which the lemma is derived asin Csorgo and Revesz (1975).

Denote et =

√σ 2

t + τ 2t (ξt −ηt ), t ∈ Nd . The next lemma is proved in the same way as Lemmas 3.3 and 3.4 in Bulinski

and Shashkin (2006).

Lemma 16. For all k ∈ Nd large enough one has Ee2k ≤ C〈k〉

α−~ with ~ = 2αµδ/(4 + 3δ). If ~ > 2 then thereexists ε4 > 0 such that∑

t∈Lk

|et | ≤ C(ω)〈Nk〉1/2−ε4 a.s., k ∈ Nd .

Now we estimate the main term in the difference between the partial sums of X and the Brownian motion. SetYm = ηψ(m),m ∈ N.

Lemma 17 (Balan, 2005). Suppose that L 6= ∅ and γ0 > (1 + 1/ρ)(1 − 1/d). Then there exists such a bijectionψ : N → L that if l < m (for l,m ∈ N), then ψ(l)s∗ ≤ ψ(m)s∗ for some s∗

= s∗(l,m) ∈ 1, . . . , d. This bijectioncan be built in such a way that m ≤ 〈ψ(m)〉γ0 for all m except for a finite number.

Lemma 18. The sequence (Ym)m∈N is either PA or NA. If 2(α − β) < λρα/(1 + λ), then for any γ < (λρα/(1 +

λ) − 2(α − β))/(3γ0) one can find such C > 0 that for all j,m ∈ N with j < m the bound |cov(Y j , Ym)| ≤ Cm−γ

is satisfied.


Proof. The functions Fk and Φ−1 are nondecreasing, hence the first assertion is immediate. As for the second one,take M > 0 to be specified later. Define the function HM (x) := (|x | ∧ M)sign(x), x ∈ R. Then by the bi-linearity ofcovariance and Markov inequality we have, for all k, l ∈ L ,

|cov(ηk, ηl)| ≤ |cov(HM (ηk), HM (ηl))| + 4MEZI|Z | > M + 2EZ2I|Z | > M

≤ |cov(HM (ηk), HM (ηl))| + 6Me−M2/2, (17)

where Z ∼ N (0, 1). If g(x) = HM (Φ−1(x)), then Lip (g) ≤√

2πeM2/2. The random variable ξk has a density notexceeding 2〈k〉

α−β/√

2π. Thus

Lip (HM Φ−1 Fk) ≤ 2eM2/2

〈k〉α−β , k ∈ Nd . (18)

Now let j,m ∈ N be such numbers that j < m, and at first assume that dist(Bψ( j), Bψ(m)) = 1. Then, bythe construction of the blocks Bk, k ∈ L, for all i ∈ 1, . . . , d one has ψ(m)i ≤ ψ( j)i + 1. Consequently,〈ψ(m)〉 ≤ 2d

〈ψ( j)〉. By the Lemmas 10 and 12, setting ν := ρλα/(1 + λ), for all m large enough we have

|cov(ξψ(m), ξψ( j))| = |Bψ(m)||Bψ( j)|−1/2

|cov(S(Bψ(m)), S(Bψ( j)))|

≤ 2c0(l0(Bψ( j), Bψ(m)))−λ/(1+λ)

≤ c0e−dν/e(〈ψ(m)〉 ∧ 〈ψ( j)〉)−ν

≤ c0e−dν/e2ν〈ψ(m)〉−ν ≤ c0e−dν/e2νm−ν/γ0 . (19)

Now assume that dist(Bψ( j), Bψ(m)) > 1. Then, again by construction and Lemma 10, this distance is not less than

mini=1,...,d

(ψ(m)i − 1)α ≥ 2−α mini=1,...,d

ψ(m)αi ≥ 2−αe−dρα/e〈ψ(m)〉ρα.

Hence, again by Lemma 12, the quantity cov(ξψ(m), ξψ( j)) equals

(|Bψ(m)||Bψ( j)|)−1/2

|cov(S(Bψ(m)), S(Bψ( j)))| ≤ c02αλedραλ/em−λρα/γ0 (20)

for m large enough. Now the lemma follows from (17)–(20) after the appropriate selection of M .

The following lemma is the principal result stating the closeness of the last term in (12) to the corresponding sumbuilt for multiparameter Brownian motion. Thus it yields the bound for the third term in (11). It relies on Theorem 7in the same way as the corresponding Theorem 4.4 in Balan (2005) is based on the strong approximation techniquesof Berkes and Philipp (1979).

Lemma 19. Suppose that in the conditions of Lemma 18 we can take γ > 15/2. Then we can redefine the randomfield X, without changing its distribution, on a new probability space together with a d-parametric Wiener processW = Wt , t ∈ Rd

+, such that for some ε5 > 0∑t∈Lk

σ√

|Bt |

∣∣∣ηt − |Bt |−1/2W (Bt )

∣∣∣ = O(〈Nk〉

1/2−ε5)

a.s.

Proof. Theorem 7 allows us to redefine the sequence (Ym)m∈N on a new probability space together with a sequenceof independent N (0, 1) random variables (Zm)m∈N, in such a way that P(|Ym − Zm | > bm) ≤ bm . Since γ > 15/2,we have bm = O(m3/2−γ /3),m → ∞. Therefore, by the Borel–Cantelli lemma one has, with probability 1,∑

t∈Lk

√|Bt |

∣∣ηt − Zψ−1(t)

∣∣ ≤ C〈k〉α/2

∑t∈Lk

(ψ−1(t)

)3/2−γ /3≤ C〈Nk〉

α/2(α+1)

(the last sum over t is uniformly bounded in k). Note that a random field on Zd can be viewed as a random elementin a Polish space. The same is true about a random field on Rd

+ with continuous realizations. Thus by the Lemma 4,arguing as in Berkes and Philipp (1979), we can redefine all the random variables mentioned in the current prooftogether with a Brownian motion, so that Zm = W (Bψ(m))/

√|Bψ(m)|,m ∈ N. Finally, employing Lemma 4 once

more we redefine all this construction on a new space supporting the field X in such a way that ηt (t ∈ L) are definedin accordance with their definition.

Theorem follows from (11) and (12) and the lemmas above, if we select γ0 greater than (1 + 1/ρ)(1 − 1/d), after thatα ∈ N large enough and, finally, let β = α − 4.


Acknowledgements

The author is grateful to Professors A.V. Bulinski and V.V. Senatov for useful discussions. He also thanks theReferee for suggestions on improving the paper.

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A strong invariance principle for positively or negatively associated random fields