RATES IN THE CLT FOR VECTOR-VALUED RANDOM FIELDS

RATES IN THE CLT FOR VECTOR-VALUED RANDOM FIELDS

A L E X A N D E R B U L I N S K I ^∗ and A L E X E Y S H A S H K I N

Dept. of Mathematics and Mechanics, Moscow State University, Moscow 119992, Russia

∗ E-mail: [email protected] [email protected]

The Lindeberg function or the Lyapunov fraction are used to establish convergence rates in the CLT for vector-valued random fields possessing dependence structure more general than positive or negative association. Thus a generalization of the classical Newman CLT is obtained. The Stein method and the Bernstein block techniques are employed. An applica- tion to kernel estimates for the density of a stationary random field is provided.

Keywords: dependence conditions, kernel estimates of density, random fields, rates in the CLT.

1. Introduction

The aim of this paper is to establish convergence rates in the CLT for sums of dependent multiindexed random vectors with values in R^s. We develop an approach to description of the dependence structure proposed by Doukhan and Louhichi (1999) for stochastic processes and by Bulinski and Suquet (2001) for random fields.

Let X = {X_t, t ∈ T } be a random field defined on a probability space (Ω, F, P) such that X_t takes values in a metric space (M, κ) for each t ∈ T . The key idea is to measure, for finite disjoint sets I, J ⊂ T (with cardinalities

|I|, |J|), the dependence between collections of random variables X_I = (X_t, t ∈ I) and XJ = (Xt, t ∈ J) in terms of a functional

F (f, g; I, J) = |cov(f (XI), g(XJ))| (1) where f : M^I → R, g : M^J → R belong to specified classes of ”test functions”

(whenever the covariance exists). Some restrictions can be imposed on I and J as well, for instance, |I| = 1.

We use here classes of bounded Lipschitz functions f, g. Recall that G : K → L (where (K, τ ) and (L, ν) are some metric spaces) is a Lipschitz function

if

Lip(G) = sup

x6=y

ν(G(x), G(y))

τ (x, y) < ∞. (2)

When K = M^I we take τ (x, y) = P

t∈Iκ(xt, yt) for x = (xt, t ∈ I), y = (yt, t ∈ I) in formula (2). Let BL(K) denote the class of bounded Lipschitz functions G : K → R (in R we use the Euclidean distance).

Often it is natural to suppose, when T is endowed with a metric ρ, that the dependence between X_I and X_J is ”rather small” if the distance ρ(I, J) = inf{ρ(t, v) : t ∈ I, v ∈ J} is ”large enough”. At the same time the dependence can increase if the distance ρ(I, J) is fixed but I and J are growing in a sense.

To give an exact formulation consider T = Z^d and introduce a set Θ consisting of functions θ(I, J) depending on finite disjoint sets I, J ⊂ Z^d such that

θ(τnI, τmJ) → 0 as |n − m| → ∞ (n, m ∈ Z^d) (3) where τ_nI = {t + n : t ∈ I} is a shift of I, |n| = max_i=1,...,d|n_i|.

For example, θ ∈ Θ if

θ(I, J) ≤ a(|I|, |J|)u(ρ(I, J)) (4) where a function a ≥ 0 is nondecreasing in each variable and u(r) & 0 as r → ∞.

Definition 1 (Bulinski and Suquet (2001)). A random field X = {X_j, j ∈ Z^d} with values in a metric space (M, κ) is called (BL, θ)-dependent if there is a function θ ∈ Θ such that

F (f, g; I, J) ≤ Lip(f )Lip(g)θ(I, J) (5) for all finite disjoint sets I, J ⊂ Z^d and any f ∈ BL(M^I), g ∈ BL(M^J).

The appearance of Lipschitz constants in the right-hand side of (5) is clear since covariance is a bilinear function and Lip(cf ) = |c|Lip(f ) for every c ∈ R.

The motivation for the concept of (BL, θ)-dependence is the following.

There are a number of interesting models described by means of families of random variables possessing properties of positive or negative association or their modifications. For definitions and examples we refer to the pioneering papers by Harris (1960), Lehmann (1966), Esary et al. (1967), Fortuin et al.

(1971), Joag-Dev and Proschan (1983); see also, e.g., Pitt (1982), Newman (1984), Lindqvist (1988), Evans (1990), Lee et al. (1990), Rachev and Xin

(1996), Ebrahimi (2002). Due to Bulinski and Shabanovich (1998) for a pos- itively or negatively associated real-valued random field X = {X_j, j ∈ Z^d} having finite second moments the inequality (5) holds with

θ(I, J) = X

i∈I

X

j∈J

|cov(X_i, X_j)|. (6)

So, for this function θ the bound (4) is valid with a(|I|, |J|) = min{|I|, |J|}

and with an analogue of the Cox–Grimmett coefficient u(r) = sup

j∈Z^d

X

q:|q−j|≥r

|cov(X_j, X_q)|, r ≥ 1.

Thus, θ appearing in (6) satisfies (3) if u(1) < ∞ and u(r) → 0 as r → ∞.

In other words Definition 1 provides a unified approach to studying both families of positively or negatively dependent random variables.

A variant of inequality (5) for smooth functions f and g in associated real-valued random variables was firstly established by Birkel (1988) (related results were proved by Newman (1984), Roussas (1994), Peligrad and Shao (1995), Bulinski (1996)). Some modifications of association for vector-valued processes and random fields leading to (5) were used by Burton et al. (1986), Bulinski (2000), Shashkin (2002).

Note that the choice of indicator functions f and g as the ”test-functions”

in (1) would lead to the Rosenblatt-type mixing coefficient (see, e.g., Doukhan (1994) and the references therein showing that the calculation or estimation of mixing coefficients is in general a difficult problem whereas using of a co- variance function is much more simple). The choice of certain power-type functions f and g in (1) was applied by Bakhtin and Bulinski (1997) to get bounds for absolute moments of partial sums of multiindexed dependent ran- dom variables. Linear functions f and g and a correlation coefficient instead of covariance in (1) was recently used by Bradley (2002) to study the bound- edness properties for spectral density of weakly stationary random field. The choice of ”complex exponential” functions is discussed by Jakubowski (1993), Doukhan and Louhichi (1999).

Remark 1. Following Doukhan and Louhichi (1999) we can define the dependence conditions for a field X = {X_t, t ∈ T } by means of specified ”test functions” f and g and inequalities

F (f, g; I, J) ≤ c(f, g; |I|, |J|)v(ρ(I, J))

where I and J are finite disjoint subsets of T , c is a nonnegative function (nondecreasing in |I| and |J|) and v(r) → 0 as r → ∞.

Remark 2. In many problems we need not consider the whole random field X on Z^d but only ”a part” X_U = (X_j, j ∈ U), U ⊂ Z^d, |U| < ∞. Then it is sufficient to use I, J ⊂ U, I ∩ J = ∅ in (5). Moreover, we can introduce θ₁ = θ₁(X_U) = sup F (f, g; {j}, U \ {j}) (7) where the supremum is taken over all j ∈ U and all f ∈ BL(M), g ∈ BL(M^{U \{j}}) with Lip(f ) ≤ 1, Lip(g) ≤ 1. Note that in (7) a set U need not be a subset of Z^d, that is we can use any finite collection of random variables X_t, t ∈ U, with values in some metric space (M, κ).

Further on let T = Z^d and M = R^s, that is we study a random field X = {X_j, j ∈ Z^d} with values in R^s. As usual EY and V ar(Y ) denote respectively the mean and covariance matrix of a random vector Y defined on a probability space (Ω, F, P).

Let U be a finite subset of Z^d. Assume that

EX_j = 0 ∈ R^s, EkX_jk² < ∞ for all j ∈ U, (8) where k · k is the Euclidean norm in R^s. We shall use kzk₁ = P_k

i=1|z_i| for z ∈ R^k as well. Note that |z| ≤ kzk ≤ kzk1 for all z ∈ R^k. These norms coinside if k = 1. Moreover, all norms are equivalent in our finitedimensional case.

Set

S = X

j∈U

X_j, V² = X

j∈U

V ar(X_j).

Suppose det V² > 0 and define

Y_j = V⁻¹X_j, W = (W₁, . . . , W_s) =X

j∈U

Y_j, R = kV⁻¹k²₁|U|θ₁ (9) where V⁻¹ is the inverse matrix to the square root of V², kAk₁ is the matrix norm corresponding to the vector norm kzk1.

Evidently, V², W , R are functions of Xj, j ∈ U, and we use also notation V²(X_U), W (X_U), R(X_U).

Consider a function h : R^s → R such that for some positive constants M₀, M₁, M₂ and for all x, x⁰ ∈ R^s, k = 1, . . . , s, one has

|h(x)| ≤ M₀,

¯¯

¯¯∂h(x)

∂x_k

¯¯

¯¯ ≤ M¹,

¯¯

¯¯∂h(x)

∂x_k − ∂h(x⁰)

∂x_k

¯¯

¯¯ ≤ M²kx − x⁰k. (10)

Using the Stein method (see Stein (1972, 1986)) we provide, for a (BL, θ)- dependent random field X = {X_j, j ∈ Z^d} with values in R^s, upper estimates of a functional

∆(h, XU) = |Eh(W ) − Eh(Z)| (11) where h is a function satisfying conditions (10), Z is a standard normal vector in R^s and U is a finite subset of Z^d.

It is worth remarking that there are various generalizations of the Stein method. We refer to Chen (1975), Tikhomirov (1980, 1983), Barbour (1990), G¨otze(1991). The approach based on diffusion approximation for positively or negatively dependent random field was used in Bulinski and Shabanovich (1998). Interesting applications of the Stein techniques (with semigroup ap- proach) in the framework of statistical models are discussed in Baddeley (2000).

Our main result (Theorem 1) gives an estimate for ∆(h, X_U) in terms of the Lindeberg function

L_ε = L_ε(X_U) = X

j∈U

EkY_jk²1{kY_jk > ε}, ε > 0, (12) of the function R appearing in (9); here random vectors Y_j are defined in (9) and 1{·} is an indicator function.

If, moreover, for some δ ∈ (0, 1]

EkX_jk^2+δ < ∞, j ∈ U, (13)

then Theorem 2 gives an estimate of ∆(h, X_U) in terms of the Lyapunov fraction instead of Lε.

Using the smoothing techniques we establish (Theorem 3) the upper bound for

∆(B, X_U) = |P(W ∈ B) − P(Z ∈ B)| (14) where B is an arbitrary convex set in R^s.

It is also shown that the Bernstein block techniques is useful in combination with the above mentioned theorems (see Theorem 4).

An application to kernel estimates of unknown density of a vector-valued stationary random field is provided as well. Theorem 5 extends some results obtained by Bosq et al. (1999), Roussas (2000, 2001), Veretennikov (2000), Bulinski and Millionshchikov (2002).

2. Results and proofs

Here we keep the notation used in the Introduction.

Theorem 1. Let X = {X_j, j ∈ Z^d} be a random field with values in R^s satisfying condition (8) where U is a finite subset of Z^d. Assume that for a function h condition (10) holds. Then for every ε > 0

∆(h, X_U) ≤ s(D₁ + ε(s + 1)D₂)R + 2εc(s)D₂

+(2ε⁻¹D₀ + (6s + 1)D₁ + (ε/2)s(s + 1)D₂)L_ε (15) where c(s) = P_s

k=1k^3/2 ≤ s^5/2, D₀ = √

2πM₀, D₁ = max{4M₀,√

2πM₁}, D₂ = √

2 max{√

2πM₀ + 2M₁, 4√

sM₁,√

2πM₂} (16)

and the constants M₀, M₁, M₂ appear in (10).

Proof. For i = 1, . . . , s (s ≥ 1) and xi, . . . , xs ∈ R introduce functions H_i(x_i, . . . , x_s) = E(h(Z₁, . . . , Z_i−1, x_i, . . . , x_s) − h(Z₁, . . . , Z_i, x_i+1, . . . x_s)),

(17) here Z = (Z₁, . . . , Z_s) is a standard normal vector in R^s (as usual if s = 1 one has H₁(x₁) = E(h(x₁)−h(Z₁)), if s ≥ 2 then H₁(x₁, . . . , x_s) = E(h(x₁, . . . , x_s)−

h(Z1, x2, . . . , xs)) and Hs(xs) = E(h(Z1, . . . , Zs−1, xs) − h(Z1, . . . , Zs)) ).

For i = 1, . . . , s consider a differential equation

∂fi

∂x_i − x_if_i = H_i (18)

where functions H_i are defined in (17).

Below we employ the solution of this equation given by the formula f_i = f_i(x_i, . . . , x_s) = e^x2i/2

xi

Z

−∞

H_i(u, x_i+1, . . . , x_s)e^−u2/2du

(for i = s one has fs = fs(xs) = e^x2s/2R_xs

−∞Hs(u)e^−u2/2du).

Lemma 1. For all x = (x_i, . . . , x_s), x⁰ = (x⁰_i, . . . , x⁰_s) ∈ R^s−i+1 and any i = 1, . . . , s, k = i, . . . , s the following inequalities are valid

|f_i(x)| ≤ D₀, |∂f_i(x)/∂x_k| ≤ D₁, (19)

|∂f_i(x)/∂x_k − ∂f_i(x⁰)/∂x_k| ≤ D₂kx − x⁰k (20) where D₀, D₁, D₂ are indicated in (16).

Proof. Note that Eh(Z1, . . . , Zi, xi+1, . . . , xs) for a Borel function h : R^s → R is given by the expression

(2π)^−i/2 Z

Rⁱ

e^−u21+...+u

2i

2 h(u₁, . . . , u_i, x_i+1, . . . , x_s)du₁. . . du_i and, for h having bounded partial derivatives in xi+1, ..., xs,

∂

∂x_kEh(Z1, . . . , Zi, xi+1, . . . , xs) = E ∂

∂x_kh(Z1, . . . , Zi, xi+1, . . . , xs)

for every k = i + 1, . . . , s. A simple calculation shows that, if K : R → R is a bounded Borel function, |K(x)| ≤ K0, x ∈ R, and R

RK(u)e^−u2/2du = 0, then for all x ∈ R

e^x2/2

¯¯

¯¯

¯¯ Zx

−∞

K(u)e^−u2/2du

¯¯

¯¯

¯¯ ≤ K0

pπ/2, |x|e^x2/2

¯¯

¯¯

¯¯ Zx

−∞

K(u)e^−u2/2du

¯¯

¯¯

¯¯ ≤ K0. (21) Thus to establish (19) we use the upper bounds for the absolute values of func- tions appearing under the signs of integrals in representations for f_i(x_i, . . . , x_s) and ∂f_i(x_i, . . . , x_s)/∂x_k.

To obtain (20) consider separately the cases k > i and k = i. Moreover, each time we have to consider whether x_i = x⁰_i or x_k = x⁰_k, k = i + 1, . . . , s.

As for the case of x_k = x⁰_k, k = i + 1, . . . , s, let us note the existence of the second partial derivatives ∂²f_i/∂x_i∂x_k, k = i, ..., s. Clearly from the equation (18) we have

∂²f_i/∂x_i∂x_k = ∂H_i/∂x_k + x_i∂f_i/∂x_k, k = i + 1, . . . , s;

∂²f_i/∂x²_i = ∂H_i/∂x_i + (1 + x²_i)f_i + x_iH_i.

The estimate for the second partial derivative in x_i and x_k, k > i, follows now from (21). To estimate ∂²f_i/∂x²_i it suffices to integrate by parts in the integral representation for x²_if_i. In the case x_i = x⁰_i we use again the representations for f_i and ∂f_i/∂x_k, k = i, . . . , s. The Lemma is proved.

Continuing the proof of Theorem 1, observe that (17) and (18) imply Xs

i=1

E µ ∂

∂x_i − W_i

¶

f_i(W_i, . . . , W_s) = Eh(W ) − Eh(Z) (22)

where the vector W is defined in (9).

For each fixed i ∈ {1, . . . , s} we estimate the summand E

µ ∂

∂x_i − W_i

¶

f_i(W_i, . . . , W_s)

in the left-hand side of (22). Analogously to Bulinski and Suquet (2001) introduce for a given ε > 0 auxiliary random vectors

T_j = (T_j1, . . . , T_js) = (b(Y_j1), . . . , b(Y_js)), V_j = (V_j1, . . . , V_js) = Y_j − T_j where b(y) = sign(y) min{|y|, ε}, y ∈ R. For the sake of brevity we write W = (W_i, . . . , W_s) and T_j = (T_ji, . . . , T_js). Set

W^(j) = W − (Y_ji, . . . , Y_js), j ∈ U.

It can be seen that

EW_if_i(W) = X4

q=1

R_iq where

R_i1 = X

j∈U

EY_jif_i(W^(j)), Ri2 = X

j∈U

EVji(fi(W) − fi(W^(j))), R_i3 = X

j∈U

ET_ji(f_i(W) − f_i(W^(j)+ T_j)), R_i4 = X

j∈U

ET_ji(f_i(W^(j) + T_j) − f_i(W^(j))).

Note that for a Lipschitz function G : R^m → R and a linear map A : Rⁿ → R^m (m, n ∈ N) the composition G(A(·)) is a Lipschitz function with Lip(GA) ≤ Lip(G)kAk1. Using this fact, definitions (7), (12) and inequalities (19) we derive the following estimates

|Ri1| ≤ X

j∈U

|cov(Yji, fi(W^(j)))| ≤ X

j∈U

|cov(Tji, fi(W^(j)))| + 2D0

X

j∈U

E|Vji|

≤ D₁kV⁻¹k²₁|U|θ₁ + 2D₀ε⁻¹X

j∈U

EY_ji²1{kY_jk > ε}

= D1R + 2D0ε⁻¹X

j∈U

EY_ji²1{kYjk > ε},

|R_i2| ≤ X

j∈U

|EV_ji(f_i(W) − f_i(W^(j)+ T_j))| + |EV_ji(f_i(W^(j) + T_j) − f_i(W^(j)))|

≤ D₁X

j∈U

Xs k=i

E(|V_jiV_jk| + |V_jiT_jk|)

≤ D₁X

j∈U

Xs k=i

µ1

2E(Y_ji² + Y_jk²)1{kY_jk > ε} + εE|Y_ji|1{|Y_ji| > ε}

¶

≤ 3

2D₁(s − i + 1)X

j∈U

EY_ji²1{kY_jk > ε} + 1

2D₁L_ε. In a similar way

|R_i3| ≤ D₁X

j∈U

Xs k=i

E|T_jiV_jk| ≤ D₁X

j∈U

Xs k=i

εE|Y_jk|1{kY_jk > ε} ≤ D₁L_ε. Due to differentiability of f_i one has

¯¯

¯fi(W^(j)+ T_j) − f_i(W^(j)) − Xs

k=i

∂f_i(W^(j))

∂x_k T_jk

¯¯

¯

≤ Xs

k=i

¯¯

¯¯∂f_i(W^(j)+ τ₀T_j)

∂xk − ∂f_i(W^(j))

∂xk

¯¯

¯¯ |T^jk| ≤ D₂(s−i+1)^1/2 Xs

k=i

i≤p≤smax |T_jp||T_jk| where τ₀ ∈ [0, 1]. Taking into account the relation P

j∈U V ar(Y_j) = I where I is a unit matrix of order s we conclude that

R_i4 = X

j∈U

Xs k=i

ET_jiT_jk∂f_i(W^(j))

∂x_k + 4_i1. One has

|4_i1| ≤ εD₂(s − i + 1)^1/2X

j∈U

Xs k=i

E|T_jiT_jk|

≤ εD₂(s + i − 1)^1/2 Xs

k=i

X

j∈U

E(Y_ji² + Y_jk²)/2 ≤ εD₂(s − i + 1)^3/2

and X

j∈U

Xs k=i

ET_jiT_jk∂f_i(W^(j))

∂xk = X

j∈U

Xs k=i

cov µ

T_jiT_jk, ∂f_i(W^(j))

∂xk

¶

+X

j∈U

Xs k=i

ET_jiT_jkE

µ∂f_i(W^(j))

∂x_k − ∂f_i(W)

∂x_k

¶

+X

j∈U

Xs k=i

ET_jiT_jkE∂f_i(W)

∂xk = X3

q=1

C_iq.

For any i, k = 1, . . . , s the function b_ik(x) = b(x_i)b(x_k), x = (x₁, . . . , x_s) ∈ R^s, is a Lipschitz one with Lip(b_ik) ≤ 2ε. Consequently

|C_i1| ≤ X

j∈U

Xs k=i

¯¯

¯cov

³

T_jiT_jk,∂fi(W^(j))

∂x_k

´¯¯

¯ ≤ 2D2εkV⁻¹k²₁(s − i + 1)|U|θ₁

≤ 2D₂(s − i + 1)ε R.

|Ci2| ≤

¯¯

¯¯

¯¯ X

j∈U

Xs k=i

ETjiTjkE

µ∂f_i(W^(j))

∂x_k − ∂f_i(W^(j) + T_j)

∂x_k

¶¯

¯¯

¯¯

¯

+

¯¯

¯¯

¯¯ X

j∈U

Xs k=i

ET_jiT_jkE

µ∂fi(W^(j)+ Tj)

∂x_k − ∂fi(W)

∂x_k

¶¯

¯¯

¯¯

¯

≤ D₂(s − i + 1)^1/2X

j∈U

Xs k=i

E|T_jiT_jk|E max

p=i,...,s|T_jp| + D₂X

j∈U

Xs k=i

Xs p=i

E|T_jiT_jk|E|V_jp|

≤ εD₂(s − i + 1)^3/2+ εD₂(s − i + 1)L_ε. To estimate C_i3 consider at first the case k = i. Then

X

j∈U

ET_ji²E∂f_i(W)

∂x_i = E∂f_i(W)

∂x_i

X

j∈U

E Y_ji² + 4_i2 = E∂f_i(W)

∂x_i + 4_i2,

|4_i2| =

¯¯

¯E∂fi(W)

∂x_i

X

j∈U

(E T_ji² − E Y_ji²)

¯¯

¯ ≤ D1

X

j∈U

E Y_ji²1{kY_jk > ε}.

The bounds for R_i2 and R_i3 imply that

|4_i3| =

¯¯

¯X

j∈U

Xs k=i+1

ET_jiT_jkE∂fi(W)

∂x_k

¯¯

¯

=

¯¯

¯ Xs k=i+1

E∂f_i(W)

∂xk

X

j∈U

E(Y_jiY_jk− T_jiV_jk − V_jiT_jk − V_jiV_jk)

¯¯

¯

≤ X

j∈U

Xs k=i+1

¯¯

¯¯E∂f_i(W)

∂x_k

¯¯

¯¯ (E|T^jiV_jk| + E|V_jiT_jk| + E|V_jiV_jk|)

≤ D₁((3/2)L_ε + (3s/2)X

j∈U

E Y_ji²1{kY_jk > ε}) in view of the relation P

j∈U V ar(Y_j) = I, that is P

j∈U

EY_jiY_jk = 0, k 6= i.

Finally we have

∆(h, X_U) ≤ Xs

i=1

¯¯

¯¯E µ ∂

∂x_i − W_i

¶

f_i(W_i, . . . , W_s)

¯¯

¯¯

≤ Xs

i=1

à ₃ X

q=1

(|R_iq| + |4_iq|) + X2

q=1

|C_iq|

! . Observing that P_s

i=1(s − i + 1)^3/2 = c(s) and P_s

i=1(s − i + 1) = s(s + 1)/2 we come to (15). The proof of Theorem 1 is complete.

Corollary 1. For a family of centered random fields X⁽ⁿ⁾ = {X_j⁽ⁿ⁾, j ∈ Z^d} (n ∈ N) with values in R^s and a family of finite subsets U_n of Z^d, the CLT holds, that is

W (X_U⁽ⁿ⁾_n ) −→ Z as n → ∞,^Law whenever, for every ε > 0,

L_ε(X_U⁽ⁿ⁾_n ) → 0 and R(X_U⁽ⁿ⁾_n ) → 0 as n → ∞.

If X_U⁽ⁿ⁾_n consists of independent random vectors then R(X_U⁽ⁿ⁾_n ) = 0. Thus The- orem 1 comprises the multidimensional Lindeberg theorem for independent summands. Analogously to Bulinski and Suquet (2001) one can also obtain from (15) the generalization of the classical Newman CLT for associated ran- dom fields.

Theorem 2. Assume that conditions of Theorem 1 are satisfied and, more- over, (13) holds. Then

∆(h, X_U) ≤ s(D₁ + D₂(s + 1))R

+(2D₀ + (6s + 1)D₁ + (2c(s) + s(s + 1)/2)D₂)L_2+δ (23) where the Lyapunov fraction

L_2+δ = L_2+δ(X_U) = X

j∈U

EkY_jk^2+δ ≤ kV⁻¹k^2+δ₁ X

j∈U

EkX_jk^2+δ₁ , random vectors Y_j are defined in (9) and c(s) appears in (15).

Proof. Observe that for ε = 1 and δ ∈ (0, 1] one has L_ε ≤ L_2+δ. To estimate ∆_i1 (and analogously C_i2) we use inequalities |abc| ≤ (|a|³ + |b|³ +

|c|³)/3 for all a, b, c ∈ R and |a|³ ≤ |a|^2+δ when |a| ≤ 1. Thus D₂(s − i + 1)^1/2X

j∈U

Xs k=i

E|T_jiT_jk max

p=i,...,sT_jp|

≤ 1

3D2(s + i − 1)^1/2X

j∈U

Xs k=i

(E|Tji|³ + E|Tjk|³ + E max

p=i,...,s|Tjp|³)

≤ D₂(s − i + 1)^3/2X

j∈U

EkY_jk^2+δ = D₂(s − i + 1)^3/2L_2+δ. Theorem is proved.

Now we need some new notation. Let B^(γ) be a γ-neighborhood of a set B ⊂ R^s with respect to Euclidean distance (that is B^(γ) = {x ∈ R^s : inf_y∈B kx − yk < γ}), ∂B being the boundary of B.

Remark 3. From Theorem 1.4 by Goldstein and Rinott (1996) the es- timate for ∆(h, X_U) (in our notation) can be established when h ∈ C_b³(R^s) and EX_j = 0, EkX_jk⁴ < ∞, j ∈ U. Using a function h ∈ C_b³ approximating the indicator function of a convex set B ⊂ R^s (more precisely, for a given γ ∈ (0, 1) let h(x) = 1 for x ∈ B, h(x) = 0 for x /∈ B^(γ) and 0 ≤ h(x) ≤ 1 for all x ∈ R^s) one can derive from the mentioned estimate that

∆(B, X_U) ≤ P(Z ∈ (∂ B)^(γ)) + G_γ(X_U) (24) where ∆(B, X_U) is defined in (15), G_γ(·) is a certain (nonrandom) functional in X_U. In Theorems 1 and 2 of our paper the estimates of ∆(h, X_U) are

obtained in other terms under lower moment assumptions and for a wider class of functions h satisfying the conditions (10). We have

∆(B, XU) ≤ P(Z ∈ (∂B)^(γ)) + Hγ(XU) (25) where H_γ(·) is a specified nonrandom functional in X_U, as the next theorem shows. For fixed U, ε and s one has G_γ(X_U) = O(γ⁻³) as γ → 0 whereas H_γ(X_U) = O(γ⁻²) as γ → 0.

Write

γ0(s) = min{1, 3p

π/(2s)}.

Theorem 3. Let conditions of Theorem 1 be satisfied and B be a convex set in R^s. Then for any γ ∈ (0, γ₀(s)] the estimate (25) holds with

H_γ(X_U) =√

2πsγ⁻¹(2 + 12√

2(s + 1)εγ⁻¹)R + 48√

πc(s)εγ⁻² +√

2π(ε⁻¹+ (12s + 2)γ⁻¹+ 6√

2s(s + 1)εγ⁻²)L_ε. (26) If, moreover, conditions of Theorem 2 are satisfied then one can take

H_γ(X_U) =√

2πsγ⁻¹(2 + 12√

2(s + 1)γ⁻¹)R +√

2π(1 + (12s + 2)γ⁻¹ + 12√

2(2c(s) + s(s + 1)/2)γ⁻²)L_2+δ. (27) Proof. Introduce a function ψ setting

ψ(x) =



















1, x ≤ 0,

1 − ^16x_3γ3³, x ∈ (0,^γ₄],

3

2 − ^2x_γ − _3γ¹⁶3(^γ₂ − x)³, x ∈ (^γ₄,^3γ₄ ],

16

3γ³(γ − x)³, x ∈ (^3γ₄ , γ],

0, x ≥ γ.

(28)

It is easy to verify that the following statement is true.

Lemma 2. The function ψ ∈ C²(R) and for all u ∈ R one has 0 ≤ ψ(u) ≤ 1, |ψ⁰(u)| ≤ 2γ⁻¹, |ψ⁰⁰(u)| ≤ 8γ⁻². For a convex set B ⊂ R^s define the function

h(x) = ψ(ρ(x, B)), x ∈ R^s, (29)

where ψ is given by (28) and ρ is the Euclidean distance in R^s. Obviously

0 ≤ h(x) ≤ 1, x ∈ R^s, h(x) = 1, x ∈ B, h(x) = 0, x /∈ B^(γ). (30) Lemma 3. For the nonnegative function h ∈ C¹(R^s) (see (29)) the condi- tion (10) is satisfied with M₀ = 1, M₁ = 2 γ⁻¹ and M₂ = 12 γ⁻².

To prove this result one can use the properties of ψ given in Lemma 2 and take into account that for all x /∈ [B] and any i = 1, . . . , s there exists

∂

∂x_iρ(x, B) = − cos(ei, n).

Here ei is the i-th unit vector of the natural orthonormal basis of R^s, n = y−x where y ∈ [B] and kx − yk = ρ(x, B); [B] is a closure of B in the Euclidean distance.

Lemma 4. If γ satisfies the conditions of Theorem 3 and the function h is given by (29), the statement of Lemma 1 is valid with

D₀ = p

π/2, D₁ = 2√

2πγ⁻¹, D₂ = 24√

πγ⁻². (31)

Proof. The indicated values for D₀, D₁, D₂ can be easily obtained from the formula (19), analogously to the proof of Lemma 1, using the fact that

|H_i(x)| ≤ 1 since 0 ≤ h(x) ≤ 1, x ∈ R^s.

Now we proceed with the proof of Theorem 3. Due to Theorem 1 and Lemma 4 we come to the estimate (15) with D₀, D₁, D₂ indicated in (31). In view of (30)

P(W ∈ B) − P(Z ∈ B^(γ)) ≤ Eh(W ) − Eh(Z) ≤ ∆(h, X_U). (32) In a similar way for the set B_(γ) = B \ (∂B)^(γ) one has

P(W ∈ B) − P(Z ∈ B_(γ)) ≥ −∆(h, X_U). (33) Now (32) and (33) imply (25) with H(X_U) given by (26). The second assertion of Theorem 3 follows in the same manner as Theorem 2 was obtained using Theorem 1. Theorem 3 is proved.

The next result gives the possibility to provide an estimate for ∆(B, X_U) which is uniform on the class C_s of convex sets of R^s. It is easy to derive im- mediately the following bound from Corollary 3.2 of Bhattacharia and Ranga Rao (1976).

Lemma 5. For any k ∈ N, all γ > 0 and every convex set B ⊂ R^s P(Z ∈ (∂B)^γ) ≤ a(s)γ

where a(1) = 2p

2/π and a(s) = 2^1/2(s − 1)Γ((s − 1)/2)/Γ(s/2) for s ≥ 2.

Thus

a(s) ≤ a₀s^1/2, a₀ = const, s ∈ N. (34) Corollary 2. Let conditions of Theorem 2 be satisfied. Then

sup

B∈Cs

∆(B, X_U) ≤ c(R(X_U) + L_2+δ(X_U))^1/3 where a factor c = c(s).

Remark 4. We are interested in asymptotical behaviour of random vec- tors W = W (XU) as U → ∞ in a sense. In this regard note that in general R = R(X_U) need not tend to zero for growing sets U (if for dependent sum- mands there are points of U which are ”rather close” to each other). So, it is natural to use the combination of the obtained results with the Bernstein block techniques. Our next two theorems provide examples of this approach.

Let X = {Xj, j ∈ Z^d} be a (BL, θ)-dependent random field with values in R^s such that (4) holds with a(I, J) = min{|I|, |J|} and some function u(r) & 0 as r → ∞. Define for a Lipschitz function ϕ : R^s → R^m (in Euclidean spaces we use the norm k · k₁) a random field eX = { eX_j, j ∈ Z^d} where eX_j = ϕ(X_j), j ∈ Z^d.

Lemma 6. A random field eX is (BL, eθ)-dependent where for any finite disjoints sets I, J ⊂ Z^d

θ(I, J) ≤ min{|I|, |J|}(Lip(ϕ))e ²u(ρ(I, J)). (35) Proof. Note that if f_k(f_k−1(. . . f₁)) is a composition of Lipschitz functions f₁, . . . , f_k then

Lip(f_k(f_k−1(. . . f₁))) ≤ Lip(f_k) . . . Lip(f₁).

Now (35) is obvious due to (BL, θ)-dependence of a field X.

Let U be a finite subset of Z^d such that U =

[N k=0

U^(k), N ∈ N, (36)

where U⁽⁰⁾, . . . , U^{(N )} are disjoint sets and for some positive b and q

|U^(k)| ≤ b and ρ(U^(k), U^(l)) ≥ q, k, l = 1, . . . , N, l 6= k. (37) Assume that

E eXj = 0 ∈ R^m, Ek eXjk² < ∞, j ∈ Z^d. (38) Set for k = 0, . . . , N

X_k = X

j∈U^(k)

Xe_j, V₁² = XN

k=1

V ar(X_k), V₀² = V ar(X₀). (39)

Using this notation we have eS = eS(U) =P

j∈U Xe_j = P_N

k=1X_k. Suppose that det V₁² > 0 and introduce for random vectors Yk = V₁⁻¹Xk (k = 1, . . . , N ) the Lindeberg function

L_ε = XN

k=1

EkY_kk²1{kY_kk > ε}, ε > 0. (40)

Theorem 4. Let eX be a random field satisfying all the above mentioned conditions and U be a finite set appearing in (36). Then for any nonrandom matrix A of order m and all ε > 0, γ ∈ (0, γ0(m)] one has

∆ := sup

B∈Cm

|P(A eS ∈ B) − P(Z ∈ B)|

≤ 2a(m)γ + γ⁻²{m√

2π(2 + 12√

2(m + 1)ε)Nb(Lip(ϕ))²u(q)kV₁⁻¹k²₁ +48c(m)√

πε +√

2π(ε⁻¹ + 12m + 2 + 6√

2m(m + 1)ε)L_ε} + γ⁻²∆ (41) where

∆ = 2m{kAV₁ − Ik²₁(m + kV₁⁻¹k²₁Nb(Lip(ϕ))²u(q)) + mkAk²₁kV₀²k₁}, (42) Cm is a class of convex sets in R^m, a(m) appears in (34) and Z is a Gaussian vector in R^m. If, moreover, for some δ ∈ (0, 1]

sup

j∈Z^d

Ek eX_jk^2+δ ≤ ec < ∞ (43) then

∆ ≤ 2a(m)γ + γ⁻²√

2π{a₁(m)kV₁⁻¹k²₁Nb(Lip(ϕ))²u(q)

+ ecb^2+δa₂(m)N kV₁⁻¹k^2+δ₁ } + γ⁻²∆ (44) where a₁(m) = m(2 + 12√

2(m + 1)) and a₂(m) = m^2+(3δ)/2(12m + 3 + 12√

2(2c(m) + m(m + 1)/2)).

Proof. We need the following two elementary results.

Lemma 7. Let ζ0, ζ1 be random vectors with values in R^m such that Ekζ₀k² < ∞. Then for any value γ > 0

sup

B∈Cm

|P(ζ₀ + ζ₁ ∈ B) − P(Z ∈ B)| ≤ sup

B∈Cm

|P(ζ₁ ∈ B) − P(Z ∈ B)|

+ γa(m) + γ⁻²Ekζ₀k².

Lemma 8. Let ζ be a centered random vector with values in R^m such that Ekζk² < ∞. Then for any nonrandom matrix A of order m one has

EkAζk² ≤ m²kAk²₁kV ar(ζ)k₁. To prove Theorem 4 note that A eS = ζ₀ + ζ₁ where

ζ1 = V₁⁻¹ XN

k=1

Xk, ζ0 = (AV1 − I)V₁⁻¹ XN

k=1

Xk + AX0, here I is a unit matrix of order m.

Using estimate (49) and Lemmas 5,7 we get

∆ ≤ sup

B∈Cm

|P(ζ₁ ∈ B) − P(Z ∈ B)| + γa(m) + γ⁻²Ekζ₀k². (45) Theorem 3 provides a bound

sup

B∈Cm

|P(ζ₁ ∈ B) − P(Z ∈ B)| ≤ a(m)γ + γ⁻²{m√

2π(2 + 12√

2(m + 1)ε)R + 48√

πc(m)ε +√

2π(ε⁻¹+ 12m + 2 + 6√

2m(m + 1)ε)L_ε} (46) where L_ε appears in (40) and R is defined for a collection of random vectors X₁, . . . , X_N in the same manner as R in (9). Namely

R = kV₁⁻¹k²₁Nθ₁

where θ₁ is given for X_U = {X_k, k = 1, . . . , N } accordingly to (7). Hence R ≤ Nb(Lip(ϕ))²u(q)kV₁⁻¹k²₁. (47)