Mathematical Notes, vol. 72, no. 4, 2002, pp. 569–575. Translated from Matematicheskie Zametki, vol. 72, no. 4, 2002, pp. 617–624. c Original Russian Text Copyright 2002 by A. P. Shashkin.
A Berry–Esseen Type Estimate for a Weakly Associated Vector Random Field A. P. Shashkin Received October 29, 2001; in final form, January 23, 2002
Abstract—A Berry–Esseen type estimate is established for a weakly associated vector random field when sums are taken over regularly growing sets. Key words: random field, weak association, Berry–Esseen estimate, family of sets growing in
the Fischer sense, covariance matrix.
1. INTRODUCTION Consider a random field {Xt , t ∈ Zd } formed by random vectors with values in Rs . This field possesses the weak association property (see [1]) if cov(f (Xt1 , . . . Xtm ), g(Xtm+1 , . . . Xtm+n )) ≥ 0
(1)
for all m, n ∈ N , for an arbitrary collection of different multi-indices t1 , . . . , tm+n from Zd , and any coordinate-wise nondecreasing functions f : Rms → R and g : Rns → R (provided that this covariance is defined). The association property (for details, see [2]) is a more rigid requirement when the functions f and g in (1) are taken from the same collection ( Xt1 , . . . , Xtm ). These and other contiguous notions are encountered in statistical physics, mathematical statistics, and reliability theory; see, for example, [2, 3]. Note that any family of independent random variables is automatically associated. There is a number of papers concerned with estimates of the rate of normal approximation for real random fields possessing the association property or its modifications (see, for example, [3–6] the references therein). Vector weakly associated processes were studied in [1]. The goal of this paper is to derive an analog of the Berry–Esseen estimate for a weakly associated vector field for the case in which sums are taken over regularly growing sets. 2. MAIN RESULT Suppose that {Xt , t ∈ Zd } is a weakly associated wide-sense stationary zero-mean field of random vectors taking values in Rs . Further, U denotes a finite subset of Zd with the number of elements |U | . Set Xt . SU = t∈ U
For a random vector X , its expectation and covariance matrix is denoted by EX and DX , respectively. The notation · indicates both the Euclidean norm of a vector in Rs and the spectral norm of the matrix corresponding to this norm. Let the following inequality be satisfied for some c0 > 0 and δ ∈ (0 ; 1]: (2) sup EXt 2+δ ≤ c0 < ∞. t∈Zd
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Suppose that for all k, l = 1, . . . , s the following series are convergent: EX0,k Xt,l . Akl =
(3)
t∈Zd
Let us introduce the matrices A = (Akl ) , ΣU = (σkl (U )) , where σkl (U ) =
ESU ,k SU ,l . |U |
(Here Xt,k and SU ,k are the kth components of the vectors Xt and SU , respectively.) By an integer cube with edge length p we mean the set B = V ∩ Zd , where V = [a1 ; a1 + p) × · · · × [ad ; ad + p),
p ∈ N,
aq ∈ Z,
q = 1, . . . , d.
Using [7], it is readily verified that for “regularly increasing” sets U (such as integer cubes) we have σkl (U ) → Akl for all k, l = 1, . . . , s . Thus the matrix A is symmetric and is nonnegative definite. Suppose that B1 , . . . , Bm are disjoint integer cubes contained in the set U and having edge length p . Let us also introduce the sets B0 =
m
Bj
and
R = U \ B0 .
j=1
In what follows, we shall drop the index j in the notation Bj unless this leads to confusion. Theorem 1. Suppose that {Xt , t ∈ Zd } is the random field described above. Then for all λ ∈ Rs we have the estimate −(Aλ,λ)/2 E exp i λ, SU − e |U | |R| |R| 1 λ √ 2 A − ΣB Tr A 2+ + λ s + ≤ 2 mpd mpd 2 1 (Aλ, λ) + λ2+δ m−δ/2 p1+δ/2 c0 , (4) + mφ − 2m where φ(x) = ex − 1 − x.
(5)
We need the following two well-known elementary assertions. Lemma 1. Suppose that zi , wi ∈ C and |zi | ≤ 1 , wi ≤ 1 , i = 1, . . . , k . Then |z1 · · · zk − w1 · · · wk | ≤
k
|zi − wi |.
i=1
Lemma 2. For all x ∈ R and δ ∈ (0 ; 1] , the following inequalities are valid: ix 1 2 ix 2+δ |e − 1| ≤ |x|, e − 1 − ix + 2 x ≤ |x| . Further, if |x| ≤ 1/2 , then |φ(x)| ≤ x2 , where φ(x) is defined by formula (5). MATHEMATICAL NOTES
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Proof of Theorem 1. Let us pass to the proof of the theorem, using the approach from [1]. Set Yj = (Xj , λ) , where ( · , · ) is the inner product on Rs . Then 1 1 −(Aλ,λ)/2 E exp i λ, SU − e − Y − 1 ≤ E exp i t d |U | |U | mp t∈ U
m Yt Yt Yt + Eexp i E exp i − 1 + E exp i − d d d mp mp mp j=1 t∈ R t∈ B0 t∈ Bj
4 m Yt −(Aλ,λ)/2 E exp i bq . (6) −e = + mpd q=1 j=1 t∈ Bj Let us estimate b1 , . . . , b4 separately. By Lemma 2, we have √ Y |U | 1 |R| t λ − 1 ≤ Tr A , b1 ≤ E d mp 2 mpd |U | U √ |R| , b2 ≤ λ Tr A mpd
(7)
(8)
where we have taken into account the fact that Tr ΣU ≤ Tr A in view of the weak association of the random field {Xt , t ∈ Rd } . To estimate b3 , let us use an analog of Newman’s inequality for the characteristic functions of a collection of weakly associated vectors (see [1]). Namely, suppose that Z1 , . . . , Zm are weakly associated random vectors in Rs , ϕ(m) is their joint characteristic function, and ϕk is the characteristic function of Zk , k = 1, . . . , m . Then for all r1 , . . . , rm ∈ Rs we have m
(m) ϕ (r1 , . . . , rm ) − ϕq (rq ) ≤ 2 q=1
s
|rq , k ||rj ,l | cov(Zq ,k , Zj ,l ).
1≤q<j≤m k ,l=1
Take r1 = · · · = rm = λ/ mpd and denote Zq = Bq Xj , q = 1, . . . , m . We have
b3 ≤ 2 ≤
s |λk λl | cov(Zq , k , Zj ,l ) mpd
1≤q<j≤m k ,l=1 s 2
λ mpd
≤ λ2
(cov(SB0 ,k , SB0 ,l ) − m cov(SB ,k , SB ,l ))
k ,l=1 s
(Akl − σkl (B)) ≤ λ2 sA − ΣB .
(9)
k ,l=1
Finally, to estimate b4 , set ηj = Lemma 2, we have
t∈Bj
Yt / mpd and ψj ,m,p = E(eiηj − 1 − iηj + ηj2 /2) . By
2+δ Yt ≤ λ2+δ p(2+δ)d (mpd )−(1+δ/2) c0 , |ψj ,m,p | ≤ (mpd )−(1+δ/2) E t∈Bj
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where c0 appears in (2). By Lemma 1, we have
m 1 2 −(Aλ,λ)/2 1 − Eηj + ψj ,m,p − e b4 = 2 j=1 m 1 −(Aλ,λ)/(2m) + ψj ,m,p ≤ 1 − 2m (ΣB λ, λ) − e j=1 1 1 ((ΣB − A)λ, λ) + φ − (Aλ, λ) + λ2+δ p−(1+δ/2)d m−δ/2 c0 . ≤m 2m 2m
(10)
The assertion of the theorem is a consequence of (6)–(10). 3. COROLLARIES
Estimates of the proximity of the characteristic function of the normalized (in the standard way) sum of random variables to the characteristic function of the normal law and the proper maximal inequalities allow us to obtain, for example, the law of the iterated logarithm (see [5]). Therefore, let us turn to some consequences of the result proved above. We consider the family of bounded Lebesgue measurable sets V ⊂ Rd , dropping the parameter index in our notation. For a set V of the family in question, |V | denotes its Lebesgue measure; ∂V is the boundary of V ; diam V is the diameter of V ; and U = V ∩ Zd . We take a function γ : [0, +∞) → [0, +∞) such that limα→0+ γ(α) = 0 . Let us recall (see, for example, [8, pp. 30– 31 of the Russian translation]) that a family of sets V tends to infinity in the Fischer sense if |V | → ∞ and for all V for all sufficiently small α |(∂V )(α diam V ) | ≤ γ(α) ; |V |
(11)
here ∂V (τ ) is the τ -neighborhood of the boundary of the set V in the Euclidean metric. Note that in limit theorems other notions of growth for a family of sets are also used (see, for example, [9]). Corollary 1. Suppose that V → ∞ in the Fischer sense and for some ρ > 0 the function γ( · ) in (11) satisfies the condition α→0+. (12) γ(α) = O(αρ ), Suppose that for a vector random field {Xt , t ∈ Zd } satisfying the assumptions of Theorem 1, for some β > 0 , and for an integer cube B the following condition is satisfied: A − ΣB = O(|B|−β ),
|B| → ∞.
(13)
Then there exist positive constants C1 , C2 , ν , and µ such that for all λ ∈ Rs with λ ≤ C1 |U |ν (where U = V ∩ Zd ) we have the estimate −(Aλ,λ)/2 E exp i λ, SU − e ≤ C2 |U |−µ . |U | Proof. In the space Rd , let us construct hyperplanes perpendicular to the coordinate axes so that the lattice Zd is divided into integer cubes with edge p = [|U |θ ] which contain pd integer points each. The function [ · ] denotes the integer part of the number; the parameter θ ∈ (0 ; 1/d) will be chosen later. Just as above, suppose that m denotes the number of cubes B1 , . . . , Bm entirely MATHEMATICAL NOTES
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contained in U and m is the number of cubes B1 , . . . , Bm containing points from both U and d Z \ U . We take C1 , ν > 0 . Let us apply Theorem 1, and for λ ≤ C1 |U |ν let us estimate all summands appearing on the right-hand side of inequality (2); next, let us choose parameters so that each summand is of the form O(|U |−µ ), |U | → ∞ for some µ > 0 . Further, let C denote, in general, different (for each inequality) positive factors independent of U , but, possibly, dependent on C1 , ν , ρ , d , and the distributions {Xt , t ∈ Zd } . In the notation of Theorem 1, we have √
|R| ≤ m pd ≤ |(∂V )(p
d)
|,
√
mpd + |(∂V )(p
d)
| ≥ |V |.
(14)
Therefore, for all sufficiently large |U | , we have √
√
|(∂V )(p d) | |R| |(∂V )(p d) | √ ≤ C ≤ mpd |V | |V | − |(∂V )(p d) | ρ √ −1 p p d ≤C ≤ C|U |ρ(θ−d ) ≤ Cγ diam V diam V and, therefore,
λ
(15)
−1 |R| ≤ Cλ |U |ρ(θ−d )/2 . mpd
By assumption, we have λ2 A − ΣB ≤ Cλ2 p−dβ ≤ Cλ2 |U |−dβθ . If λ ≤ C1 |U |ν , 2ν ≤ 1 − θd , then |(Aλ, λ)/2m| ≤ 1/2 for all sufficiently large |U | ; by Lemma 2, we can write λ4 1 (Aλ, λ) ≤ C . mφ − 2m m Finally,
λ2+δ pd(1+δ/2) m−δ/2 ≤ Cλ2+δ |U |dθ(1+δ)−δ/2 .
It remains to choose θ < δ/(2d(1 + δ)) and then ν > 0 so that dθβ δ − 2dθ(1 + δ) ρ 1 −θ , , . ν < min 2 d 2 2(2 + δ)
(16
)
Remark. If VN = [−N ; N ]d , N ≥ 1 , is a family of cubes, then condition (12) is valid with ρ = 1 . If for some ζ > d and all k, l = 1, . . . , s the following estimate is valid: cov(X0,k , Xt, l ) = O(t−ζ ),
t → ∞,
then condition (13) is satisfied for β = (ζ/d) − 1 . Corollary 2. Suppose that det A = 0 under the assumptions of Theorems 1. Then for V and for all sufficiently large |U | appearing in the assumptions of Corollary 2, we have det DSU = 0 , and there exist constants C3 , C4 , ν1 , µ1 such that for λ ≤ C3 |U |ν1 the following inequality holds: 2
|E exp{i(λ, (DSU )−1/2 SU )}| − e−λ
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| ≤ C4 |U |−µ1 .
(17)
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A. P. SHASHKIN
Proof. Since DSU /|U | = ΣU → A as |U | → ∞ (this follows from condition (13)), we have det DSU = 0 for all sufficiently large |U | . In this case, substituting A−1/2 λ for λ into the previous corollary, for λ ≤ C1 |U |ν we have −λ2 /2 −µ E exp i λ, A−1/2 SU −e µ > 0. ≤ |U | , |U | In view of Lemma 2, we obtain −1/2 E exp i λ, A−1/2 SU − E exp{i(λ, (DSU ) SU )} |U | SU −1/2 −1/2 ≤ E exp i λ, (A − 1 − (DSU ) |U |) |U | 2 √ SU −1/2 − (DSU )−1/2 |U | ≤ λ Tr AC|U |−β ; ≤ λ E |U | A therefore, we can take ν1 = ν , µ1 = min{µ, β − ν} .
(18)
Corollary 3. Let the assumptions of Corollary 2 be satisfied and G ⊂ Rs is a bounded convex set. Then |P((DSU )−1/2 SU ∈ G) − P(ξ ∈ G)| ≤ M0 (1 + |G(1) |)|U |−ϑ , where the constants ϑ > 0 and M0 > 0 are independent of U and G , ξ ∼ N (0, Is ) , and Is is the unit matrix of order s . Proof. This assertion follows from (17) and the well-known von Bar inequality [10].
4. FURTHER GENERALIZATIONS The proofs given above can be carried over without essential changes to the case in which instead of sums over finite subsets of the lattice Zd we consider additive random functions given on the algebra Bb (Rd ) of bounded Borel sets of the space Rd . Namely, suppose that A(V ) , V ∈ Bb (Rd ) , is a random function with values in Rs such that A(V1 ∪ V2 ) = A(V1 ) + A(V2 ) almost everywhere if V1 , V2 ∈ Bb (Rd ) and V1 ∩ V2 = ∅ . Let us take the partition of Rd obtained by the translations of the cube Π(a) = {x ∈ Rd : 0 < xk ≤ a, k = 1, . . . , d} , a ∈ R+ , i.e., consider the cubes Πk (a) = Π(a) + ak , where k ∈ Zd , ak = (ak1 , . . . , akd ) . Suppose that for all sufficiently large a the variables Xk (a) = A(Πk (a)) , k ∈ Zd , form a stationary weakly associated vector field. Then, under the corresponding conditions on the field {Xk (a), k ∈ Zd } , we obtain asymptotic results for A(V ) as V → ∞ in the Fischer sense. In particular, suppose that {Zv , v ∈ Rd } is a weakly associated random field with values in Rs such that for some δ ∈ (0 ; 1] and for all v ∈ Rd we have EZv 2+δ < ∞. (19) EZv = 0 ∈ Rs ,
Let
Zv dv,
A(V ) =
V ∈ Bb (Rd ).
(20)
V
The moment conditions on the field {Zv , v ∈ Rd } ensure that the last integral can be regarded both in the sense of L2 -convergence and trajectory-wise. If condition (19) is satisfied, then all the estimates of asymptotic behavior proved in Secs. 2 and 3 (with the substitution of A(V ) for SU , where the sets V → ∞ in the Fischer sense and satisfy the requirement (12)) are valid for the integration scheme (20) if instead of (3) we use the condition
Fk ,l (u) du = Akl ∈ [0 ; +∞), Rd
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where Fk ,l (u) = cov(Z0,k , Zu,l ) , k, l = 1, . . . , s . Obviously, the field {Zv , v ∈ Rd } can be integrated with a positive bounded weight function. The summation scheme for the field {Xt , t ∈ Zd } can be reduced to the integration scheme for the field {Zv , v ∈ Rd } in the standard way. It suffices to set Zv = Xt for v ∈ [t ; t + 1) , t ∈ Zd . Note that the field {Zv , v ∈ Rd } thus introduced is not weakly associated (if the points v1 , . . . , vd belong to one cube [t ; t + 1)). However, as was noted above, we can only assume that the variables obtained by integration over sufficiently large disjoint cubes are weakly associated. Moreover, generalizations are possible also under the assumptions of quasiassociation type proposed in [11] (see also [12]), since we use analogs of Newman’s inequality for joint characteristic functions. ACKNOWLEDGMENTS The author wishes to thank Professor A. V. Bulinskii for permanent attention to the work on this paper. REFERENCES 1. A. R. Dabrowski and H. Dehling, “A Berry–Esseen theorem and a functional law of the iterated logarithm for weakly associated random vectors,” Stochastic Processes Appl., 30 (1988), no. 2, 277– 289. 2. J. Esary, F. Proschan, and D. Walkup, “Association of random variables with applications,” Ann. Math. Statistics, 38 (1967), no. 5, 1466–1474. 3. C. M. Newman, “Asymptotic independence and limit theorems for positively and negatively dependent random variables,” in: Inequalities in Statistics and Probability (Y. L. Tong, editor), Hayward, 1984, pp. 127–140. 4. A. V. Bulinskii, “Rate of convergence in the central limit theorem for fields of associated variables,” Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.], 38 (1995), no. 2, 417–425. 5. A. V. Bulinskii, “A functional law of the iterated logarithm for associated random fields,” Fund. Prikl. Mat., 1 (1995), no. 3, 623–639. 6. A. V. Bulinskii, “On the convergence rates in the central limit theorem for positively and negatively dependent random fields,” in: Probability Theory and Math. Statistics (I. A. Ibragimov and A. Yu. Zaitsev, editors), Gordon and Breach, 1996, pp. 3–14. 7. E. Bolthausen, “On the central limit theorem for stationary mixing random fields,” Ann. Probability, 10 (1982), no. 4, 1047–1050. 8. D. Ruelle, Statistical Mechanics. Rigorous Results, Benjamin–Cummings, New York, 1969. 9. A. V. Bulinskii and M. A. Vronskii, “A statistical version of the central limit theorem,” Fund. Prikl. Mat., 2 (1996), no. 4, 999–1018. 10. B. von Bar, “Multi-dimensional integral limit theorem,” Ark. Math., 7 (1967), no. 1, 71–88. 11. A. V. Bulinski and Ch. Suquet, “Normal approximation for quasiassociated random fields,” Statistics and Probability Letters, 54 (2001), no. 2, 215–226. 12. A. V. Bulinskii, “Asymptotic Gaussian property of quasiassociated vector random fields,” Obozrenie Prikl. Promyshl. Mat., 7 (2000), no. 2, 482–483. M. V. Lomonosov Moscow State University E-mail:
[email protected]
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