Normal approximation for quasi associated random fields

Alexander Bulinski^{a,1,3} Charles Suquet^{b,2}

aDepartment of Mathematics and Mechanics, Moscow State University, Moscow

119 899, Russia

bStatistique et Probabilités, FRE CNRS 2222, Bât M2, U.F.R. de Mathématiques, University Lille 1, F-59655 Villeneuve d’Ascq Cedex, France

AMS classification: Primary: 60F05

Abstract

For quasi associated random fields (comprising negatively and positively dependent fields) on ℤ^{d} we use Stein’s method to establish the rate of normal approximation for partial sums taken over arbitrary finite subsets U of ℤ^{d}.

Key words: Random fields; Dependence conditions; Positive and negative association; Lindeberg function; CLT; Convergence rates; Maximum of partial sums.

1 Introduction

There are a number of interesting stochastic models described by means of families of random variables possessing properties of positive or negative dependence or their modifications. One can refer to the pioneering papers by Harris (1960), Lehman (1966), Esary et al. (1967), Fortuin et al. (1971), Joag-Dev and Proschan (1983). Definition 1 (Esary et al. (1967)) A finite collection Y = (Y_{1}, …, Y_{n}) of real valued random variables Y_{k}, k = 1, …, n, is called associated or positively dependent if Cov(f(Y), g(Y)) ≥ 0 for any coordinate-wise nondecreasing functions f, g : ℝ^{n} → ℝ, whenever the covariance exists. An infinite family of random variables is associated if this is valid for every finite sub-family.

The association and related concepts (e.g. positive quadrant dependence, etc.) were initially connected with reliability theory and mathematical statistics only. Percolation

1 Corresponding author, e-mail: bulinski@mech.math.msu.su (A. V. Bulinski)

2 e-mail: Charles.Suquet@univ-lille1.fr

3 Partially supported by INTAS grant 99-01317 and RFBR grant 99-01-00112

Preprint submitted to Elsevier Preprint 25 July 2000

theory and statistical mechanics, where one considers random variables “satisfying the FKG inequalities” (Fortuin et al. (1971)) implying the association, provide a different domain of applications of this notion. Definition 2 (Joag-Dev and Proschan (1983)) Real valued random variables Y_{k}, k = 1, …, n, and a family thereof, are called negatively dependent if, each time the covariance exists,

Cov(f(Y_{i}, i ∈ I), g(Y_{j}, j ∈ J)) ≤ 0,  (1)

for every pair of disjoint subsets I, J of {1, …, n} and for any coordinate-wise nondecreasing functions f : ℝ^{I} → ℝ, g : ℝ^{J} → ℝ. Sampling without replacement provides an example of negatively dependent random variables (see Joag-Dev and Proschan (1983) for this and other examples).

Newman (1984) calls a family of random variables weakly associated if it satisfies the requirement of Definition 2, but with reversed sign in the inequality (1).

Evidently, associated families of random variables are weakly associated. Note also that any family of independent random variables is automatically associated and negatively dependent. Instead of the terms negative dependence and weak association one uses also negative association (NA) and positive association (PA). It is worth mentioning that concepts of mixing or positive (negative) dependence offer complementary approaches to analysis of dependent random variables. The main advantage of dealing with positively or negatively dependent random fields is due to the fact that most of their properties are determined by the covariance structure whereas the calculation of mixing coefficients is in general a nontrivial problem (we refer to the book on mixing by Doukhan (1994)).

Starting from the seminal paper by Newman (1980), during the last two decades, various classical limit theorems of probability theory (CLT, SLLN, weak and strong invariance principles, LIL and FLIL, Glivenko-Cantelli type theorems, etc.) were established for stochastic processes and random fields under the positive or negative dependence conditions mentioned above.

In this paper, we prove more general variants of CLT for random fields involving wider classes of random variables (Theorems 4-13 of Section 2). Section 3 is devoted to an important new result by Shao (2000) showing that the expectations of convex increasing functions of maxima of partial sums of negatively dependent random variables can be estimated by means of the independent copies of summands. We demonstrate that it is impossible to get an exact analogue of this result for negatively dependent random fields {X_{j}, j ∈ ℤ^{d}} with d > 1. Definition 3 Call a collection of real valued random variables Y = {Y_{t}, t ∈ T } with IE Y_{t}^{2} < ∞ (t ∈ T) quasi associated if, for all finite disjoint subsets I, J of T and any Lipschitz functions f : ℝ^{I} → ℝ, g : ℝ^{J} → ℝ, one has

| Cov(f(Y_{i}, i ∈ I), g(Y_{j}, j ∈ J))| ≤ ∑_{i∈I} ∑_{j∈J} L_{i}(f)L_{j}(g)| Cov(X_{i}, X_{j})|,  (2)

where the coordinate-wise Lipschitz constants L_{i}(f) are such that, for all x = (x_{i}, i ∈ I), y = (y_{i}, i ∈ I) in ℝ^{I},

|f(x) − f(y)| ≤ ∑_{i∈I} L_{i}(f)|x_{i} − y_{i}|.

Obviously, if a random field {Y_{t}, t ∈ T } is quasi associated, the same is true for the centered field {Y_{t} −IE Y_{t}, t ∈ T }. Inequality (2) is satisfied for PA or NA random fields, see Bulinski and Shabanovich (1998). An analogue of (2) for smooth functions f and g was firstly proved in Birkel (1988) for associated random variables (related results appeared in Newman (1984), Roussas (1994), Peligrad and Shao (1995) and Bulinski (1996)). Note that covariance inequalities are powerful tools in establishing moment inequalities and limit theorems for sums of dependent r.v.’s. We refer, e.g., to Ibragimov and Linnik (1971), Withers (1981), Bradley and Bryc (1985), Doukhan (1994), Bakhtin and Bulinski (1997), Louhichi (1998), Rio (2000). Interesting examples of application of covariance inequalities in statistical problems are provided by Doukhan and Louhichi (1999).

2 Normal Approximation

Let X = {X_{j}, j ∈ ℤ^{d}}, d ≥ 1, be a centered random field such that IE |X_{j}|^{2} < ∞ for all j ∈ ℤ^{d}. For a finite subset U of ℤ^{d} denote

W = B^{−1} ∑_{j∈U} X_{j},  B^{2} = ∑_{j∈U} IE X_{j}^{2},  (3)

where the trivial case B^{2} = 0 is excluded. Introduce further

R = B^{−2} ∑_{\begin{matrix}j,q∈U \\ j≠q \end{matrix}} | Cov(X_{j}, X_{q})|

and, for ε > 0, the Lindeberg function

L_{ε} = B^{−2} ∑_{j∈U} IE X_{j}^{2}1{|X_{j}| > εB}.

Evidently, B^{2}, W, R and L_{ε} are functions of X_{j}, j ∈ U, and we use also notations B^{2}(X, U), W (X, U), R(X, U) and L_{ε}(X, U). Theorem 4 If X = {X_{j}, j ∈ ℤ^{d}}, d ≥ 1, is a quasi associated centered random field, then, for any finite subset U of ℤ^{d}, every x ∈ ℝ and arbitrary positive ε, γ,

|P (W ≥ x)−P (Z ≥ x)| ≤ P (x−γ ≤ Z ≤ x+γ)+C{(3/2)ε+(4+ε)L_{ε}+(1+2ε)R}, (4)

where Z is a standard normal random variable, and one can take

C = C(γ) = 2 + 2/γ.  (5)

PROOF. It is well known (see Stein (1986)) that, for any bounded continuous function g : ℝ → ℝ, the unique bounded solution of the equation

f′(w) − wf(w) = g(w) − IE g(Z)  (6)

is determined by the formula

f(w) = exp(w^{2}/2) ∫_{−∞}^{w}(g(t) − IE g(Z)) exp(−t^{2}/2) dt.  (7)

For fixed x ∈ ℝ and γ > 0 define a smooth nondecreasing function g(w) = g_{x,γ}(w) in such a way that g(w) = 0 if w < x, g(w) = 1 if w > x + γ and g′(w) ≤ 2/γ for all x ∈ ℝ. Hence, for any x, t ∈ ℝ, γ > 0,

1_{[x,∞)}(t) ≥ g(t) ≥ 1_{[x+γ,∞)}(t).  (8)

Then one can prove that for the function f(w) (depending on x and γ) given by (7) there exists f′′(w) for all w ∈ ℝ and, for every x ∈ ℝ, γ > 0,

∥ f ∥_{∞} ≤ \sqrt{π/2}, ∥ f′ ∥_{∞} ≤ 2, ∥ f′′ ∥_{∞} ≤ \sqrt{π/2} + ∥ g′ ∥_{∞}.

Consequently f, f′ are Lipschitz functions and, for the corresponding Lipschitz constants,

max{Lip(f), Lip(f′)} ≤ C  (9)

the bound C being given in (5).

The ingenious Stein’s idea is that substituting in (6) instead of w a random variable W and taking the expectation one gets

IE f′(W) − IE W f(W) = IE g(W) − IE g(Z).  (10)

So the left hand side of (10) gives an accuracy of normal approximation for IE g(W). Note that we can not use here the discontinuous indicator function g(w) = 1_{(−∞,x]}(w) to measure the Kolmogorov distance between distribution functions of W and Z.

Our further steps consist of evaluating IE f′(W) − IE W f(W). For a given ε > 0, define the Lipschitz function

h(t) = h_{ε}(t) = \{ \begin{matrix}−ε & if t < −ε, \\ t & if − ε ≤ t ≤ ε, \\ ε & if t > ε. \end{matrix}  (11)

Set for j ∈ U

ξ_{j} = X_{j}/B, ξ_{j,1} = h(ξ_{j}), ξ_{j,2} = ξ_{j} − ξ_{j,1}, W^{(j)} = W − ξ_{j}.  (12)

Clearly ξ_{j,1} and ξ_{j,2} depend on ε. Using the obvious relations

ξ_{j} = ξ_{j,1} + ξ_{j,2}, W = W^{(j)} + ξ_{j} = W^{(j)} + ξ_{j,1} + ξ_{j,2},

one can write

IE W f(W) = ∑_{j∈U} IE ξ_{j}f(W) = R_{1} + R_{2} + R_{3} + R_{4},  (13)

where

R_{1} = ∑_{j∈U} IE ξ_{j}f(W^{(j)}),
R_{2} = ∑_{j∈U} IE{ξ_{j,2}[f(W) − f(W^{(j)})]},
R_{3} = ∑_{j∈U} IE{ξ_{j,1}[f(W^{(j)} + ξ_{j,1} + ξ_{j,2}) − f(W^{(j)} + ξ_{j,1})]},
R_{4} = ∑_{j∈U} IE{ξ_{j,1}[f(W^{(j)} + ξ_{j,1}) − f(W^{(j)})]}.

Here R_{2}, R_{3} and R_{4} depend also on ε.

Note that if F : ℝ → ℝ, G : ℝ^{I} → ℝ are Lipschitz functions, then F (G(.)) is Lipschitz as well and its coordinate-wise Lipschitz constants L_{i} can be chosen so that L_{i}(F (G(.))) ≤ Lip F · L_{i}(G), i ∈ I. Therefore, (2) and (9) imply

|R_{1}| ≤ ∑_{j∈U} | Cov(ξ_{j}, f(W^{(j)}))| ≤ \frac{C}{B^{2}} ∑_{j∈U} ∑_{\begin{matrix}j,q∈U \\ j≠q \end{matrix}} | Cov(X_{j}, X_{q})| = CR.  (14)

Taking into account (9), we observe that

|R_{2}| ≤ ∑_{j∈U}|IE{ξ_{j,2}[f(W^{(j)} + ξ_{j,1} + ξ_{j,2}) − f(W^{(j)} + ξ_{j,1})]}|

+ ∑_{j∈U}|IE{ξ_{j,2}[f(W^{(j)} + ξ_{j,1}) − f(W^{(j)})]}|

≤ C ∑_{j∈U}(IE |ξ_{j,2}|^{2} + IE |ξ_{j,2}||ξ_{j,1}|)

≤ C(∑_{j∈U} IE |ξ_{j}|^{2}1_{{|ξ_{j}|>ε}} + ε ∑_{j∈U} IE |ξ_{j}|1_{{|ξ_{j}|>ε}})
≤ 2CL_{ε},  (15)

in view of the following estimates:

|ξ_{j,1}| ≤ ε, |ξ_{j,2}| ≤ |ξ_{j}|1_{{|ξ_{j}|>ε}} ≤ (ξ_{j}^{2}/ε)1_{{|ξ_{j}|>ε}}.  (16)

Analogously, we get

|R_{3}| ≤ C ∑_{j∈U} IE |ξ_{j,1}||ξ_{j,2}| ≤ CL_{ε}.  (17)

The Taylor formula yields

f(W^{(j)} + ξ_{j,1}) − f(W^{(j)}) = f′(W^{(j)})ξ_{j,1} + \frac{1}{2}f′′(η_{j})ξ_{j,1}^{2},  (18)

where η_{j} = η_{j}(ω) is a point between W^{(j)}(ω) and W^{(j)}(ω) + ξ_{j,1}(ω) (ω ∈ Ω, all random fields under consideration being defined on the same probability space (Ω, F, P)). Thus,

R_{4} = ∑_{j∈U} IE ξ_{j,1}^{2}f′(W^{(j)}) + Δ_{1},

where, because of relations |ξ_{j,1}| ≤ |ξ_{j}| for j ∈ U and ∑_{j∈U} IE ξ_{j}^{2} = 1, we have

|Δ_{1}| ≤ \frac{1}{2} ∑_{j∈U} IE |ξ_{j,1}|^{3}|f′′(η_{j})| ≤ \frac{1}{2}∥ f′′ ∥_{∞} ∑_{j∈U} IE |ξ_{j,1}|^{3} ≤ \frac{1}{2}Cε.  (19)

Further on,

∑_{j∈U} IE{ξ_{j,1}^{2}f′(W^{(j)})} = ∑_{j∈U} Cov(ξ_{j,1}^{2}, f′(W^{(j)})) + ∑_{j∈U} IE ξ_{j,1}^{2} IE f′(W^{(j)}).  (20)

Note that h^{2} is a Lipschitz function with Lip(h^{2}) = 2ε. By (2) and (9),

|∑_{j∈U} Cov(ξ_{j,1}^{2}, f′(W^{(j)}))| ≤ \frac{2Cε}{B^{2}} ∑_{\begin{matrix}j,q∈U \\ j≠q \end{matrix}} | Cov(X_{j}, X_{q})| = 2CεR.  (21)

Now,

∑_{j∈U} IE ξ_{j,1}^{2} IE f′(W^{(j)}) = ∑_{j∈U} IE ξ_{j,1}^{2} IE f′(W) + ∑_{j∈U} IE ξ_{j,1}^{2}(IE f′(W^{(j)}) − IE f′(W)) (22)

and

|∑_{j∈U} IE ξ_{j,1}^{2}(IE f′(W^{(j)}) − IE f′(W))| ≤ Δ_{2} + Δ_{3},  (23)

where

Δ_{2} = ∑_{j∈U} IE ξ_{j,1}^{2}| IE{f′(W^{(j)} + ξ_{j,1} + ξ_{j,2}) − f′(W^{(j)} + ξ_{j,1})}|,
Δ_{3} = ∑_{j∈U} IE ξ_{j,1}^{2}| IE{f′(W^{(j)} + ξ_{j,1}) − f′(W^{(j)})}|.

Relations (9) and (16) yield

Δ_{2} ≤ C ∑_{j∈U} IE ξ_{j,1}^{2} IE |ξ_{j,2}| ≤ Cε^{2} ∑_{j∈U} IE |ξ_{j,2}| ≤ CεL_{ε},  (24)
Δ_{3} ≤ C ∑_{j∈U} IE ξ_{j,1}^{2} IE |ξ_{j,1}| ≤ Cε.  (25)

Using again the relation ∑_{j∈U} IE ξ_{j}^{2} = 1, we have

IE f′(W) ∑_{j∈U} IE ξ_{j,1}^{2} = IE f′(W) + IE f′(W) ∑_{j∈U}(IE ξ_{j,1}^{2} − IE ξ_{j}^{2}).  (26)

On account of (9) and (12) we obtain

|IE f′(W) ∑_{j∈U}(IE ξ_{j,1}^{2}−IE ξ_{j}^{2})| ≤ ∥ f′ ∥_{∞} ∑_{j∈U} IE |ξ_{j}^{2}−ξ_{j,1}^{2}| ≤ C ∑_{j∈U} IE |ξ_{j}^{2}|1_{{|ξ_{j}|>ε}} = CL_{ε}. (27)

Hence (18)–(27) imply that

R_{4} = IE f′(W) + Δ_{4},

where

|Δ_{4}| ≤ (3/2)Cε + C(1 + ε)L_{ε} + 2CεR.  (28)

Finally, due to (8), for any x ∈ ℝ, γ > 0,

P (W ≥ x) − P (Z ≥ x + γ) ≥ IE g(W) − IE g(Z) = IE f′(W) − IE W f(W)
= IE f′(W) − R_{1} − R_{2} − R_{3} − IE f′(W) − Δ_{4}
= −R_{1} − R_{2} − R_{3} − Δ_{4}.  (29)

According to (14)–(17) and (28),

|∑_{i=1}^{3} R_{i} + Δ_{4}| ≤ (3/2)Cε + C(4 + ε)L_{ε} + C(1 + 2ε)R.  (30)

For the function g̃(t) = g_{x−γ,γ}(t), t ∈ ℝ, we get in a similar way

P (W ≥ x) − P (Z ≥ x − γ) ≤ IE g̃(W) − IE g̃(Z).  (31)

So we come to the same bounds as in (29) and (30). Thus, the estimate (4) is established. □

Remark 5 To prove Theorem 4 and other results concerning quasi associated random fields we actually need only the property (2) where the cardinality of the set I is equal to 1. Corollary 6 For a family of quasi associated centered random fields X^{(n)} = {X_{j}^{(n)}, j ∈ ℤ^{d}}, n ∈ ℕ, and a family of finite subsets U_{n} of ℤ^{d}, the CLT holds, i.e.

W (X^{(n)}; U_{n}) ⟶^{law} Z as n → ∞,

whenever, for every ε > 0,

L_{ε}(X^{(n)}; U_{n}) → 0 and R(X^{(n)}; U_{n}) → 0 as n → ∞.  (32)

Remark 7 If X^{(n)} are positively or negatively dependent random fields, the condition on R in (32) means that asymptotically the behaviour of sums is similar to the case of independent r.v.’s, since

∑_{j∈U_{n}} Var X_{j}^{(n)}/ Var( ∑_{j∈U_{n}} X_{j}^{(n)}) → 1, as n → ∞.

Remark 8 Our Theorem 4 comprises Theorem 1 by Bulinski and Vronski (1996) where a strictly stationary associated random field X = {X_{j}, j ∈ ℤ^{d}} was studied under the condition

σ^{2} = ∑_{j∈ℤ^{d}} Cov(X_{0}, X_{j}) < ∞

and summation was carried over finite sets U_{n} ⊂ ℤ^{d}, growing in the Van Hove sense (appropriate to the discrete case). The latter result generalized the classical Newman’s CLT where partial sums were taken over blocks. We use the renorm group approach, considering for m = (m_{1}, …, m_{d}) ∈ ℕ^{d} auxiliary random fields

Y_{q}^{(m)} = ∑_{j∈Π_{q}^{(m)}} X_{j}, q = (q_{1}, …, q_{d}) ∈ ℤ^{d},

where Π_{q}^{(m)} = {i = (i_{1}, …, i_{d}) ∈ ℤ^{d} : (q_{k} −1)m_{k} < i_{k} ≤ q_{k}m_{k}}. Note that here we consider more general dependence conditions and neither stationarity nor the existence of absolute moments of summands of order higher than two is required. Moreover, no conditions are imposed on the growth of the sets U_{n} used to form the partial sums and there are no hypotheses concerning the rates of decrease of the covariance functions of random fields under consideration (cf. Cox and Grimmett (1984), Roussas (1994), Bulinski and Keane (1996), Bulinski and Vronski (1996)).

Let now X = {X_{j}, j ∈ ℤ^{d}}, d ≥ 1, be a random field such that

IE X_{j} = 0, IE |X_{j}|^{s} < ∞ for some s ∈ (2, 3] and all j ∈ ℤ^{d}.  (33)

For a finite subset U of ℤ^{d}, denote by L_{s} the Lyapounov fraction

L_{s} = B^{−s} ∑_{j∈U} IE |X_{j}|^{s},

where B^{2} > 0 is defined in (3). Theorem 9 If X is a quasi associated centered random field satisfying (33), then, for any finite subset U of ℤ^{d}, every x ∈ ℝ and arbitrary positive γ, one has

|P (W ≥ x) − P (Z ≥ x)| ≤ P (x − γ ≤ Z ≤ x + γ) + 3CR + (13/2)CL_{s}, (34)

where W, Z and C are the same as in (3) and (5).

PROOF. The scheme of the proof follows that of Theorem 4. Take ε = 1 in the definition of the function h in (11) and note that L_{1} ≤ L_{s}. Instead of (25) one has now

Δ_{3} ≤ C ∑_{j∈U} IE ξ_{j,1}^{2} IE |ξ_{j,1}| ≤ C ∑_{j∈U} IE |ξ_{j,1}|^{3} ≤ CL_{s},

where the Lyapunov inequality and the bounds |ξ_{j,1}| ≤ 1, |ξ_{j,1}| ≤ |ξ_{j}| (j ∈ U) are used. To estimate |Δ_{1}| we modify (19) in the same way. □

Remark 10 The bound (34) is similar to Lemma 3 in Shao and Su (1999). Unfortunately in that paper, there is a gap in the proof (see p. 143) due to the application of the Hoeffding formula to discontinuous functions. Although in both papers Stein’s method is used with a similar splitting (R_{1}, R_{2}, R_{3}, R_{4}), our treatment of R_{4} is different.

Now we turn to the general dependence conditions for random fields proposed initially by Doukhan and Louhichi (1999) for stochastic processes. Note that for random fields there are no “future” and “past”, but it is natural to measure the dependence between a single random variable X_{j} and other X_{q}, q ∈ ℤ^{d}, when ∥q − j∥ is “large” (∥j∥ = max_{1≤k≤d} |j_{k}| for j = (j_{1}, …, j_{k}) ∈ ℤ^{d}), cf. Coulon-Prieur and Doukhan (2000), where for a process (d = 1) the dependence between pairs X_{i}, X_{j} and all X_{q} from the “past” was measured.

To simplify further exposition we shall from now on consider Lipschitz functions f : 𝕊 → 𝕋, where 𝕊 and 𝕋 are some metric spaces with distances ρ and τ respectively, which means that

Lip(f) = sup_{\begin{matrix}x,y∈𝕊 \\ x≠y \end{matrix}} \frac{τ (f(x), f(y))}{ρ(x, y)} < ∞.

Thus in the special case of Euclidean spaces we drop the distinction between the possibly different behavior of f in various directions.

For a finite subset U of ℕ^{d}, denote by BL(𝕊^{U}) the class of bounded Lipschitz functions f : 𝕊^{U} → ℝ. In a space 𝕊^{U} we apply the metric ∑_{j∈U} ρ(x_{j}, y_{j}) for x = (x_{j}, j ∈ U) ∈ 𝕊^{U}, y = (y_{j}, j ∈ U) ∈ 𝕊^{U} (if 𝕊 = ℝ , the Euclidean distance is used). Let θ = (θ_{r})_{r∈ℕ} be a sequence of nonnegative numbers tending to 0 as r → ∞. Definition 11 A random field X = {X_{j}, j ∈ ℤ^{d}}, d ≥ 1, with values in a metric space 𝕊 is called (BL, θ)-dependent if, for each j ∈ ℤ^{d} and any J ⊂ ℤ^{d} such that j ∉ J,

|Cov(f(X_{j}), g(X_{q}, q ∈ J))| ≤ θ_{r} Lip(f) · Lip(g)  (35)

for all f ∈ BL(𝕊), g ∈ BL(𝕊^{J}), where r = inf{∥j − q∥; q ∈ J }. Remark 12 Clearly a quasi associated random field X satisfies (35) with

θ_{r} = sup_{j∈ℤ^{d}} ∑_{\begin{matrix}q∈ℤ^{d} \\ ∥j−q∥≥r \end{matrix}} | Cov(X_{j}, X_{q})|,  (36)

whenever θ_{1} < ∞. The coefficient (36) was used in Bulinski (1995) to prove FLIL for associated random fields and in Bulinski (1996) to obtain the Berry-Esséen type estimates for PA and NA random fields. For positively dependent random fields (36) is the well known Cox-Grimmett coefficient (see Cox and Grimmett (1984)). A different coefficient was employed in Oliveira and Suquet (1995), Morel and Suquet (2000). It is worth mentioning that in Definition 11 no moment conditions are imposed on the field X. Furthermore, if 𝕊 is a normed vector space (with the corresponding distance), IE |X|_{j} < ∞ for all j ∈ ℤ^{d} and (35) is valid, the centered random field {X_{j} −IE X_{j}, j ∈ ℤ^{d}} is also (BL, θ)-dependent with the same sequence θ as for the field X. Theorem 13 Let X be a centered (BL, θ)-dependent random field such that IE X_{j}^{2} < ∞ for all j ∈ ℤ^{d}. Then, for any finite subset U of ℤ^{d}, the estimate of the type (4) holds with R replaced by |U|B_{n}^{−2}θ_{1} and (4 + ε)L_{ε} by (4 + ε + 2ε^{−1})L_{ε}. If, moreover, (33) is satisfied, then one can use in the right-hand side of (34) the same replacement for R and (17/2)CL_{s} instead of (13/2)CL_{s}.

PROOF. In view of (35) we have, instead of (14) and (21),

|R_{1}| ≤ ∑_{j∈U} | Cov(ξ_{j,1}, f(W^{(j)}))| + 2∥ f ∥_{∞} ∑_{j∈U} IE |ξ_{j,2}| ≤ C|U|B^{−2}θ_{1} + 2Cε^{−1}L_{ε},

|∑_{j∈U} Cov(ξ_{j,1}^{2}, f′(W^{(j)}))| ≤ 2εC|U|B^{−2}θ_{1}.

To prove the second statement of Theorem 13 take ε = 1 as in the proof of Theorem 9 and use again the estimate L_{1} ≤ L_{s}. □

If the function B^{2}(X, U) behaves ”regularly” (as in stationary case) then the term |U|B^{−2}θ_{1} is not vanishing for growing U. So (cf. Remark 8) it is reasonable to use the Bernstein blocks technique to invoke θ_{r} instead of θ_{1} where r is a distance between ”large” blocks. To this end, some modifications of the Definition 11 are appropriate. We provide here only the following one. Let Θ be a family of nonnegative functions θ(I, J) of finite sets I, J ⊂ ℤ^{d} such that θ(I, J) → 0 when I and J are shifted so that inf{∥i − j∥; i ∈ I, j ∈ J} → ∞. Definition 14 A random field X = {X_{j}, j ∈ ℤ^{d}}, d ≥ 1, with values in a metric space 𝕊 is called (BL, Θ)-dependent, if there exists a function θ ∈ Θ such that, for each pair of disjoint finite sets I, J ⊂ ℤ^{d},

| Cov(f(X_{i}, i ∈ I), g(X_{j}, j ∈ J))| ≤ θ(I, J) Lip(f) Lip(g)

for all f ∈ BL(𝕊^{I}), g ∈ BL(𝕊^{J}).

It is natural to restrict the class Θ and suppose that θ(I, J) ≤ u(|I|, |J|; r) for some function u : ℕ × ℕ × ℝ_{+} → ℝ_{+}, where |I| stands for cardinality of I and r = inf{∥i − j∥; i ∈ I, j ∈ J}. For quasi associated random fields θ(I, J) = ∑_{i∈I} ∑_{j∈J} | Cov(X_{i}, X_{j})| ≤ min{|I|, |J|}θ_{r} where θ_{r} is given by (36), whenever θ_{1} < ∞. Remark 15 Let ζ^{(n)} = {ζ_{j}^{(n)}, j ∈ ℤ^{d}} be (BL, θ^{(n)})-dependent random field with values in a metric space 𝕊, and θ^{(n)} = {θ_{r}^{(n)}, r ≥ 1}, n ∈ ℕ. Define X_{j}^{(n)} = F_{j}^{(n)}(ζ_{j}^{(n)}), j ∈ ℤ^{d}, n ∈ ℕ, taking Lipschitz functions F_{j}^{(n)} : S → ℝ (n ∈ ℕ, j ∈ ℤ^{d}). Define for a family of finite sets U_{n} ⊂ ℤ^{d}

λ_{n} = sup_{j∈U_{n}} Lip(F_{j}^{(n)}), n ∈ ℕ.

Then it is easily seen that X^{(n)} = {X_{j}^{(n)}, j ∈ U_{n}} is (BL, λ_{n}^{2}θ^{(n)})-dependent random field where λ_{n}^{2}θ^{(n)} = {λ_{n}^{2}θ_{r}^{(n)}, r ≥ 1}, n ∈ ℕ (to use the definitions for random fields on ℤ^{d} we set X_{j}^{(n)} = 0 for j ∈ ℤ^{d} \ U_{n}). Using Theorem 13 we come to analogues of the results by Coulon-Prieur and Doukhan (2000) but we do not assume that F_{j}^{(n)} are bounded, we need not impose conditions on variances of partial sums (see (6) in the cited paper) and the finite sets U_{n} of arbitrary configuration can be used as well. Remark 16 Recently Stein’s technique was used by Pruss and Szynal (2000) to establish the CLT for negatively correlated random variables with negatively correlated squares. Note that this interesting result does not comprise quasi associated random variables and is aimed at generalization of CLT for pairwise dependent families of r.v.’s.

3 On partial sums of negatively associated random fields

For a sequence of NA random variables X_{1}, X_{2}, …, let X_{1}^{∗}, X_{2}^{∗}, … be their independent copies, that is {X_{i}^{∗}, i ≥ 1} is a sequence of independent random variables and, for each i ≥ 1, X_{i}^{∗} has the same distribution as X_{i}. Set

S_{j} = ∑_{i=1}^{j} X_{i}, S_{j}^{∗} = ∑_{i=1}^{j} X_{i}^{∗}, M_{n} = max_{1≤i≤n} S_{i}, M_{n}^{∗} = max_{1≤i≤n} S_{i}^{∗}.

Recently Shao established the following useful result. Theorem 17 (Shao (2000)) Let X_{1}, X_{2}, … be a sequence of negatively dependent random variables. Then, for any convex function f : ℝ → ℝ and each n ∈ ℕ,

IE f(S_{n}) ≤ IE f(S_{n}^{∗}).  (37)

If, moreover, f is nondecreasing, then

IE f(M_{n}) ≤ IE f(M_{n}^{∗}).  (38)

It would be desirable to have an analogue of this theorem for negatively associated random fields. It is clear that (37) does not involve the partial order of the parameter set, so its counterpart is valid also for sums S(U) taken over arbitrary finite subsets U of ℤ^{d}. As for the maximum of partial sums, the situation is quite different in the case of random fields.

Let X = {X_{j}, j ∈ ℤ^{d}} be a NA random field. For j, q ∈ ℤ^{d}, the notation j ≤ q means that j_{k} ≤ q_{k} for all k = 1, …, d. Set

S_{q} = ∑_{1≤j≤q} X_{j}, M_{n} = max_{1≤q≤n} S_{q},

where q, n ∈ ℕ^{d}, 1 = (1, …, 1) ∈ ℕ^{d}. Denote by {X_{j}^{∗}, j ∈ ℤ^{d}} a random field consisting of independent random variables where each X_{j}^{∗} has the same distribution as the corresponding X_{j}. Set

S_{q}^{∗} = ∑_{1≤j≤q} X_{j}^{∗},  M̂_{n} = max_{1≤q≤n} S_{q}^{∗},  q, n ∈ ℕ^{d}.

Proposition 18 For every d > 1 and any n ∈ ℕ^{d}, one can construct a NA random field X = {X_{j}, j ∈ ℤ^{d}} such that, for all nondecreasing (not necessarily convex) functions f : ℝ → ℝ,

IE f(M_{n}) ≥ IE f(M̂_{n}).

PROOF. Clearly, it is sufficient to provide an example for d = 2 and n = (2, 2) (we can take other variables X_{j} = 0).

Let X_{1,1} = 0, X_{1,2} = −X_{2,1} where P (X_{1,2} = 1) = P (X_{1,2} = −1) = 1/2 and X_{2,2} = c < −2.

Recall (see, e.g., Joag-Dev and Proschan (1983)) that if {X_{i}, i ∈ I_{k}}, k = 1, …, N, are independent collections of NA random variables, then the whole collection {X_{i}, i ∈ I_{k}, k = 1, …, N} is NA. Since X_{1,2} and X_{2,1} are NA, the same is true for X_{1,1}, X_{1,2}, X_{2,1}, X_{2,2}.

For n = (2, 2) we see that

M_{n} = max{X_{1,1}; X_{1,1} + X_{1,2}; X_{1,1} + X_{2,1}; X_{1,1} + X_{1,2} + X_{2,1} + X_{2,2}}
= max{0; X_{1,2}; −X_{1,2}; c} = |X_{1,2}| = 1.

Because of X_{1,1}^{∗} = 0 and X_{1,2}^{∗} + X_{2,1}^{∗} + X_{2,2}^{∗} < 0, one has, for the same n,

M̂_{n} = max{X_{1,1}^{∗}; X_{1,1}^{∗} + X_{1,2}^{∗}; X_{1,1}^{∗} + X_{2,1}^{∗}; X_{1,1}^{∗} + X_{1,2}^{∗} + X_{2,1}^{∗} + X_{2,2}^{∗}}
= max{0; X_{1,2}^{∗}; X_{2,1}^{∗}}.

It follows clearly that

IE f(M̂_{n}) = IE f(max\{0; X_{1,2}^{∗}; X_{1,2}^{∗}\}) = \frac{1}{4}f(0) + \frac{3}{4}f(1).

Then from f(1) ≥ f(0) and IE f(M_{n}) = f(1) we obtain IE f(M_{n}) ≥ IE f(M̂_{n}). □

Corollary 19 For a < b (a, b ∈ ℝ) let F_{a,b} be the class of functions f : ℝ → ℝ such that f(a) < f(b). Then for every d > 1, each n ∈ ℕ^{d}, one can construct a NA random field Y = {Y_{j}, j ∈ ℤ^{d}} such that

IE f(M_{n}) > IE f(M̂_{n}) for any f ∈ F_{a,b}.

PROOF. Having observed, that the negative dependence of the random field used in the proof of Theorem 18 is preserved by the transformation g : x ↦ Ax + B (A > 0), which maps [0, 1] onto [a, b], it suffices to consider Y = {Y_{j} = g(X_{j}), j ∈ ℤ^{d}}. □

Acknowledgements

The authors are grateful to Professor Qi-Man Shao who kindly provided them in June 2000 the electronic version of his paper A comparison theorem on moment inequalities between negatively associated and independent random variables. The first author would also like to express his gratitude for hospitality to the “Laboratoire de Statistique et Probabilités” of University Lille 1 where in June 2000 this joint paper was written.

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