A central limit theorem for Lebesgue integrals of random fields

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by  Jürgen Kampf

A central limit theorem for Lebesgue integrals of random fields
Ju¨rgen Kampf
January 5, 2016
6
1
0 Abstract
2
InthispaperweshowacentrallimittheoremforLebesgueintegralsofstationaryBL(θ)-dependent
n
randomfieldsastheintegrationdomaingrowsinVanHove-sense. Ourmethodistousethe(known)
a
analogue result for discrete sums. As applications we obtain various multivariate versions of this
J
central limit theorem.
4

] 1 Introduction
R
P
Random fields are collections of random variables indexed by the Euclidean space Rd. They have appli-
.
h
cationsinvariousbranchesofscience,e.g.inmedicine[1,13],ingeostatistics[5,15]orinmaterialscience
t
a [10, 14].
m
The aim of the present paper is to establish a central limit theorem for integrals X(t)dt, where
Wn
[ (Wn)n N isasequenceofcompactsubsetsofRd and(X(t))t Rd isarandomfield. TheRsequence(Wn)n N
1 of inte∈gration domains is assumed to grow in Van Hove-sen∈se (VH-sense), i.e. ∈
v
3 lim λd((bdWn)+Bd)/λd(Wn)=0,
n
1 →∞
5 whereλ denotestheLebesguemeasure,bdW istheboundaryofW Rd,A+B := a+b a A, b B
d
0 for two subsets A,B Rd and Bd := x Rd x 1 is the close⊆d Euclidean un{it bal|l. ∈ ∈ }
0 ⊆ { ∈ |k k≤ }
Themainresultofthe presentpaperisthe following(the notionofBL(θ)-dependence willbe defined
.
1 in Subsection 2.1).
0
6 Theorem 1. Let θ =(θr)r N be a monotonically decreasing zero sequence. Let (X(t))t Rd be a measur-
1 able, stationary, BL(θ)-dep∈endent R-valued random field such that ∈
:
v
i Cov X(0),X(t) dt< .
X
ZRd| | ∞
r (cid:0) (cid:1)
a Let (Wn)n N be a VH-growing sequence of subsets of Rd. Then
∈
X(t)dt EX(0)λ (W )
Wn − d n (0,σ2), n ,
R λd(Wn) →N →∞
in distribution, where p
σ2 := Cov X(0),X(t) dt.
ZRd
(cid:0) (cid:1)
There is a wide literature on similar results, where mixing conditions are assumed instead of BL(θ)-
dependence, see e.g. [4, 7, 8, 9]. For BL(θ)-dependent random fields there are no central limit theorems
for Lebesgue integrals up to now. However, there are such results for discrete sums [2] and for Lebesgue
measures of excursion sets [3]. In the latter paper the random field is in fact assumed to be quasi-
associated, which is a slightly stronger assumption than BL(θ)- dependence.
This paper is organized as follows: In Section 2 we collect preliminaries about associated random
variables,randomfieldsandfunctionsofboundedvariation. Section3isdevotedtotheproofofthemain
theorem. InSection4wepresentseveralexampleshowthemainresultcanbeextendedtoamultivariate
centrallimittheorem. Thecasethatthe randomfieldisofthe form(f(X(t)))t Rd forsomedeterministic
function f :R Rs and some random R-valued field (X(t))t Rd will be of par∈ticular interest.
→ ∈
1
2 PRELIMINARIES 2
2 Preliminaries
2.1 Association concepts
In this subsection we introduce different association concepts and discuss their relations.
We start with the broadest appearing in this paper, namely BL(θ)-dependence.
For finite subsets I,J Rd we put dist(I,J) := min x y : x I, y J , where is the
1 1
ℓ1-norm. For two Lipschitz⊆functions f :Rn1 R and g :{Rkn2− kR we p∈ut ∈ } k·k
→ →
Ψ(n ,n ,f,g)=min n ,n Lip(f)Lip(g),
1 2 1 2
{ }
where
f(x) f(y)
Lip(f):=sup | − | x,y Rn, x=y
x y | ∈ 6
1
n k − k o
denotes the (optimal) Lipschitz constant of a Lipschitz function f :Rn R.
Fora randomfield (X(t))t Rd, a finite subsetI = t1,...,tn Rd w→ith n elements anda function f
on Rn we abbreviate f(X ):=∈f(X(t ),...,X(t )). I{f such an a}b⊆breviation X appears more than once
I 1 n I
within one formula, then always the same enumeration of the elements of I has to be used.
For a set M let #M denote the number of elements of M.
Furthermore, for ∆>0 we put
T(∆):= (j /∆,...,j /∆) (j ,...,j ) Zd .
1 d 1 d
{ | ∈ }
Def. 2. Let θ =(θr)r N be a monotonically decreasing sequence with limr θr =0.
∈ →∞
(i) An Rs-valuedrandomfield (X(t))t Rd is called BL(θ)-dependent if for any ∆>1 andany disjoint,
finite sets I,J T(∆) with dist(I,∈J) r and all bounded Lipschitz functions f : Rs#I R and
·
g :Rs#J R ⊆we have ≥ →
·
→ Cov(f(X ),g(X )) Ψ(#I,#J,f,g)∆dθ .
I J r
≤
(ii) An Rs-valued random field (X(t))t Zd is called BL(θ)-dependent if for any disjoint, finite sets
I,J Zd with dist(I,J) r andall∈boundedLipschitz functions f :Rs#I R andg :Rs#J R
· ·
⊆ ≥ → →
we have
Cov(f(X ),g(X )) Ψ(#I,#J,f,g)θ .
I J r
≤
Lemma 3. Let θ = (θr)r N be a monotonically decreasing sequence with limr θr = 0. For T = Zd
or T =Rd, let (X(n)(t)) ∈,n N, be a sequence of BL(θ)-dependent random fi→e∞lds such that the finite-
t T
∈ ∈
dimensionaldistributionsconvergetothoseofafield(X(t)) . Then(X(t)) isalsoBL(θ)-dependent.
t T t T
∈ ∈
Proof: Bythedefinitionofconvergenceinprobabilitywegetlim Ef(X(n))g(X(n))=Ef(X )g(X ),
n I J I J
lim Ef(X(n))=Ef(X ) andlim Eg(X(n))=Eg(X )f→or∞any finite sets I,J T andbounded
Lipnsc→h∞itz contiInuous functioIns f :R#In→∞R andJg :R#J RJ, which yields the asserti⊆on.
→ →
Lemma 4. Let θ = (θr)r N be a monotonically decreasing sequence with limr θr = 0. Let (X(t))t T
be a BL(θ)-dependent ran∈dom field and let f : Rs Rs′ be a Lipschitz fu→n∞ction. Then there is∈a
→
monotonically decreasing sequence θ′ = (θr′)r N with limr θr′ = 0 such that (f(X(t)))t T is BL(θ′)-
∈ →∞ ∈
dependent.
Proof: For a function f : V W and a finite set I let f denote the function V#I W#I,
I
→ →
(x ,...,x ) (f(x ),...,f(x )). We put θ := Lip(f)2 θ . Let I,J be two disjoint finite sets
wi1th dist(#I,IJ)7→ r and1 let f˜:Rs#′#II R and g :r′Rs′#J R b·ertwo Lipschitz functions. Then
· ·
≥ → →
Cov(f˜(f (X )),g(f (X ))) min #I,#J Lip(f˜ f ) Lip(g f ) ∆d θ
I I J J I J r
≤ { }· ◦ · ◦ · ·
min #I,#J Lip(f˜) Lip(g) ∆d θ .
≤ { }· · · · r′
An Rs-valued random field (X(t)) is called positively associated (PA) if
t T
∈
Cov(f(X ),g(X )) 0
I J
≥
2 PRELIMINARIES 3
for any finite sets I,J T and functions f : Rs#I R and g : Rs#J R which are bounded and
· ·
⊆ → →
monotonically increasing in every coordinate.
For a Lipschitz function f :Rn R we define coordinate-wise Lipschitz constants by
→
f(x ,...,x ,y ,x ,...,x ) f(x ,...,x ,z ,x ,...,x )
1 k 1 k k+1 n 1 k 1 k k+1 n
Lipk(f)=sup | − y −z − | |
k k
n | − |
x ,...,x ,x ,...,x ,y ,z R, y =z , k 1,...,n .
1 k 1 k+1 n k k k k
− ∈ 6 ∈{ }
o
An Rs-valued random field (X(t)) with E[X (t)2] < ,t T,k = 1,...,s, is called quasi-associated
t T k
∈ ∞ ∈
(QA) if
s s
Cov(f(X ),g(X )) Lip (f) Lip (g) Cov(X (t),X (u))
| I J |≤ t,k · u,l | k l |
t I k=1u J l=1
X∈ XX∈ X
for any finite sets I,J T and Lipschitz continuous functions f :Rs#I R and g :Rs#J R.
· ·
⊆ → →
ItiswellknownthateveryPArandomfieldisalsoQA,seee.g.Theorem5.3in[2,p.89](thistheorem
is only formulated in the special case s=1 and T =Zd, but the proof holds in the present setting).
Lemma 5. Let (X(t))t Rd be an Rs-valued QA random field. Assume that there are c > 0 and ǫ > 0
with ∈
Cov(Xi(t1),Xj(t2)) c t1 t2 −d−ǫ
≤ ·k − k∞
fort1,t2 Rd andi,j =1,...,s. Then(X(t))t Rd isBL(θ)-dependentforsomemonotonicallydecreasing
zero sequ∈ence θ. ∈
Proof: Let r>0, ∆>1 and let I,J T(∆) be finite with dist(I,J) r, w.l.o.g. #I #J. Moreover,
let f :Rs#I R and g :Rs#J R b⊆e bounded and Lipschitz contin≥uous. Then ≤
· ·
→ →
s s
Cov(f(X ),g(X )) Lip (f) Lip (g)Cov(X (t),X (u))
I J ≤ t,k · u,l k l
t I k=1u J l=1
X∈ XX∈ X
s2 #I Lip(f) Lip(g) max Cov(X (t),X (u))
k l
≤ · · · · t,k,l
u J
X∈
s2 #I Lip(f) Lip(g) max c t u d ǫ.
− −
≤ · · · · t I ·k − k∞
∈ u J
X∈
We have for fixed t I, if r >1,
∈
t u −d−ǫ ∞ s −d−ǫ # v T(∆) v = s
k − k∞ ≤ ∆ · { ∈ |k k∞ ∆}
uX∈J s=X⌈r∆⌉(cid:0) (cid:1)
= ∞ s −d−ǫ (2s+1)d (2s 1)d
∆ · − −
s=X⌈r∆⌉(cid:0) (cid:1) (cid:0) (cid:1)
d 1
∞ − d
=∆d+ǫ s d ǫ (1+( 1)d ι 1)(2s)ι
− − − −
ι −
s=X⌈r∆⌉Xι=0 (cid:18) (cid:19)
d 1
∆d+ǫ ∞ − d (1+( 1)d ι 1)2ιs d ǫ+ιds
− − − −
≤ ι −
Z⌈r∆⌉−1Xι=0(cid:18) (cid:19)
∆d+ǫd−1 d (1+( 1)d−ι−1)2ι(r∆−∆)−d−ǫ+ι+1
≤ ι − d+ǫ ι 1
ι=0(cid:18) (cid:19) − −
X
∆dd−1 d (1+( 1)d−ι−1)2ι(r−1)−d−ǫ+ι+1.
≤ ι − d+ǫ ι 1
ι=0(cid:18) (cid:19) − −
X
2 PRELIMINARIES 4
Putting
θr :=(c∆3d·ds·2mPadιx=−i=011(cid:0),d.ι.(cid:1).,(s1V+ar((−X1i)(d0−))ι−+1)θ22ι(r−d1+)−ǫ−d−ι−ǫ+1ι+1 ffoorr rr >=11,,
we obtain
Cov(f(X ),g(X )) min #I,#J Lip(f) Lip(g) ∆dθ .
I J r
≤ { }· · ·
2.2 Random fields
After having introduced the association concepts in subsection 2.1, we will now collect various other
preliminaries concerning random fields.
The following theorem(see [6, Ch. III, 3]and [12, Prop.3.1])saysthat for stationaryrandomfields
§
stochastic continuity and measurability are essentially equivalent.
Theorem 6. (i) Let(X(t))t Rd beastochasticallycontinuousrandomfield. Thenthereisameasurable
modification of (X(t))t Rd∈.
∈
(ii) Let (X(t))t Rd be a stationary and measurable random field. Then (X(t))t Rd is stochastically
continuous.∈ ∈
Lemma 7. Let (Xt)t Rd be a stationary, stochastically continuous random field with EX(0)j < for
j >0. Then (Xt)t Rd∈is continuous in j-mean. ∞
∈
Proof: Let (tn)n N be a sequence of points in Rd converging to a point t Rd. Then
∈ ∈
lim E X(t ) X(t)j = lim ∞P(X(t ) X(t)j x)dx= ∞ lim P(X(t ) X(t) √j x)dx=0.
n n n
n | − | n | − | ≥ n | − |≥
→∞ →∞Z0 Z0 →∞
We have been allowed to interchange limit and integral, since
P(X(t ) X(t) √j x) P(X(t ) √jx)+P(X(t) √jx)=2P(X(t) √jx)
| n − |≥ ≤ | n |≥ 2 | |≥ 2 | |≥ 2
due to the stationarity and
∞2P(X(t) √jx)dx= ∞2P(2 X(t)j x)dx=2j+1E X(0)j < .
| |≥ 2 | · | ≥ | | ∞
Z0 Z0
Lemma 8. Let (X(t))t Rd and (Y(t))t Rd be two stochastically continuous and measurable random fields
having the same distrib∈ution. Let A ,∈...,A Rd be bounded Borel sets. Assume that X(t)dt is
1 m ⊆ Ai
defined a.s. for i = 1,...,m, i.e. not both the positive part and the negative part of these integrals are
R
infinite. Then Y(t)dt,..., Y(t)dt are defined a.s. as well and
A1 Am
R R
d
X(t)dt,..., X(t)dt = Y(t)dt,..., Y(t)dt .
(cid:18)ZA1 ZAm (cid:19) (cid:18)ZA1 ZAm (cid:19)
Proof: By the Monotone ConvergenceTheorem, we may assume w.l.o.g. that there is some N N such
that X(t) [ N,N] and Y(t) [ N,N] for all t Rd. ∈
∈ − ∈ − ∈
We define processes (Xn(t))t Rd and (Yn(t))t Rd by putting
∈ ∈
z z z z +1 z z +1
Xn(t ,...,t ):=X 1,..., d , for all t 1, 1 ,...,t d, d , z ,...,z Z.
1 d 1 d 1 d
n n ∈ n n ∈ n n ∈
(cid:16) (cid:17) h (cid:17) h (cid:17)
We get
Xn(t)dt i=1,...,m (1)
nZAi (cid:12) o
(cid:12) z1 z1+1 zd zd+1 z1 zd
(cid:12)= λ A , , X ,..., i=1,...,m
d i
∩ n n ×···× n n n n
nz1,.X..,zd∈Z (cid:16) (cid:2) (cid:1) (cid:2) (cid:1)(cid:17) (cid:0) (cid:1)(cid:12)(cid:12) o
d z1 z1+1 zd zd+1 z1 zd (cid:12)
= λ A , , Y ,..., i=1,...,m
d i
∩ n n ×···× n n n n
nz1,.X..,zd∈Z (cid:16) (cid:2) (cid:1) (cid:2) (cid:1)(cid:17) (cid:0) (cid:1)(cid:12)(cid:12) o
= Yn(t)dt i=1,...,m . (cid:12) (2)
nZAi (cid:12) o
(cid:12)
(cid:12)
2 PRELIMINARIES 5
For ǫ,δ >0 we get
m m
P Xn(t)dt X(t)dt >ǫ P Xn(t) X(t) dt>ǫ
− ≤ | − |
(cid:16)Xi=1(cid:12)ZAi ZAi (cid:12) (cid:17) (cid:16)Xi=1ZAi (cid:17)
(cid:12) (cid:12) E m Xn(t) X(t) dt
i=1 Ai| − |
≤ ǫ
P R
m E Xn(t) X(t) dt
= i=1 Ai | − |
ǫ
P R
m δ+P(Xn(t) X(t) >δ) 2N dt
i=1 Ai | − | ·
≤ ǫ
P R (cid:0) (cid:1)
m δdt
n→∞ i=1 Ai
−→ ǫ
P R
m
δ λ (A )
= · i=1 d i .
ǫ
P
The limit relation holds by the MajorizedConvergenceTheorem, since the assumption that (X(t))t R is
stochastically continuous implies that Xn(t) converges to X(t). Since δ >0 was arbitrary,we get ∈
m
P Xn(t)dt X(t)dt >ǫ n→∞0
− −→
(cid:16)Xi=1(cid:12)ZAi ZAi (cid:12) (cid:17)
and the same way (cid:12) (cid:12)
m
P Yn(t)dt Y(t)dt >ǫ n→∞0.
− −→
(cid:16)Xi=1(cid:12)ZAi ZAi (cid:12) (cid:17)
Now (2) yields the assertion. (cid:12) (cid:12)
2.3 Functions of bounded variation
A function f : R R is said to be of locally bounded variation if there is a monotonically increasing
function α:R R→and a monotonically decreasing function β :R R such that f =α+β. We denote
→ →
the set of such functions α and β by A resp. B. We put
inf α(x) α A, α(0)=f(0) if x>0
{ | ∈ }
f+(x):= f(0) if x=0

sup α(x) α A, α(0)=f(0) if x<0.
{ | ∈ }
It is easy to see that f+ A and f− :=f f+ B. We put hf :=f+ f−.
∈ − ∈ −
Lemma 9. Let f : R R be a function of locally bounded variation. Then f = g h for a Lipschitz
f
continuous function g :→R R of Lipschitz constant 1. ◦
→
Proof: Foreachx R,forwhichthereist Rwithh (t)=x,defineg(x):=f(t). Nowgiswell-defined,
f
sincefort ,t Rw∈ithh (t )=h (t ),f is∈constanton[t ,t ]. Clearly,f =g h . Moreover,g -defined
1 2 f 1 f 2 1 2 f
onasubsetof∈Rsofar-isLipschitzcontinuouswithLipschitzconstant1. Indeed◦,letx ,x R, x <x ,
1 2 1 2
be two points for which there are t ,t R with h (t )=x and h (t )=x . Then ∈
1 2 f 1 1 f 2 2
∈
hf(t2) hf(t1)=f+(t2) f+(t1) (f−(t2) f−(t1)) f+(t2) f+(t1)+(f−(t2) f−(t1)) = f(t2) f(t1).
− − − − ≥| − − | | − |
Hence x x g(x ) g(x ).
2 1 2 1
It rem−ains t≥o |show t−hat g h|as a Lipschitz continuous extension to the whole of R. The domain of g
is R minus the union of countable many, disjoint intervals. For a point x lying on the boundary of the
domain of g but not in the domain of g, choose a sequence (xn)n N such that g(xn) is defined for all
n N. Then (g(xn))n N is a Cauchy sequence, since g is Lipschit∈z continuous, and hence convergent.
∈ ∈
Since (g(xn))n N is convergentfor every such sequence (xn)n N, the limit is independent of the choice of
∈ ∈
thesequence. Sowecanputg(x):=lim g(x ). ItiseasytoseethatthisextensionstillhasLipschitz
n n
→∞
constant 1. Now all gaps in the domain of g are open intervals. So they can be filled by affine functions.
Clearly, the Lipschitz constant is preserved again.
3 THE UNIVARIATE CLT 6
3 The univariate CLT
In this section we will prove Theorem 1.
Proof: For j = (j ,...,j ) Zd we put Q = d [j ,j +1) and Z(j) := X(t)dt EX(0). We
1 d ∈ j ×i=1 i i Qj −
will show that this random field (Z(j))j Zd fulfills the assumptions of TheoreRm 1.12 of [2, p. 178]. The
collection ∈
n
1
Zn(j):= nd X(j1+ kn1,...,jd+ knd)−EX(0), j ∈Zd,
k1,.X..,kd=1
is BL(θ )-dependent for any n N, where θ := θ . Indeed, let I,J Zd and let f : R#I R and
g :R#J′ R beboundedLipsch∈itzfunctions.r′ PutI˜r−=dI+ 1/n,2/n,...,⊆1 d,J˜=J+ 1/n,2/n→,...,1 d,
→ { } { }
nd nd
1 1
f˜:R#Ind R, (x ,...,x ) f x EX(0),..., x EX(0)
· → 1,1 #I,nd → nd 1,ℓ− nd #I,ℓ−
(cid:16) Xℓ=1 Xℓ=1 (cid:17)
and
nd nd
1 1
g˜:R#J·nd →R, (x1,1,...,x#J,nd)→g nd x1,ℓ−EX(0),...,nd x#J,ℓ−EX(0) .
(cid:16) Xℓ=1 Xℓ=1 (cid:17)
Then we have f(Z ) = f˜(X ), g(Z ) = g˜(X ), Lip(f˜) = Lip(f)/nd, Lip(g˜) = Lip(g)/nd and
n,I I˜ n,J J˜
dist(I˜,J˜) dist(I,J) d. So
≥ −
Cov(f(Z ),g(Z ))=Cov(f˜(X ),g˜(X ))
n,I n,J I˜ J˜
min #I nd,#J nd Lip(f˜)Lip(g˜)ndθ
r d
≤ { · · } −
=min #I,#J Lip(f)Lip(g)θ .
{ } r′
By Lemma 3, the field (Z(j))j Zd is BL(θ′)-dependent if we can show that the finite-dimensional
distributions of (Zn(j))j Zd converg∈e to those of (Z(j))j Zd. First we will show
∈ ∈
lim E Z (j) Z(j) =0, j Zd. (3)
n
n | − | ∈
→∞
Let ǫ>0. Due to Lemma 7, the field (X(t))t Rd is continuous in 1-mean and hence there is n such that
∈
E X(0) X(t) <ǫ for all t [0, 1]d.
| − | ∈ n
Since (Xt)t Rd is stationary, this implies
∈
E|X(j1+ kn1,...,jd+ knd)−X(t)|<ǫ for all t∈[j1+ k1n−1,j1+ kn1]×···×[jd+ kdn−1,jd+ knd].
Hence E Z (j) Z(j) <ǫ, which finishes the proof of (3).
n
Now l|et j(1),−...,j(r|) Zd and let δ >0. From Markov’s inequality we get
∈
r r E Z (j(l)) Z(j(l))
P Z (j(l)) Z(j(l)) >δ l=1 | n − | 0, n .
n
| − | ≤ δ → →∞
(cid:16)Xl=1 (cid:17) P
Sothefinite-dimensionaldistributionsof(Zn(j))j Zd convergetothoseof(Z(j))j Zd andhence(Z(j))j Zd
is BL(θ)-dependent. ∈ ∈ ∈
By Lemma 8, the assumption that (X(t))t Rd is stationary implies that (Z(j))j Zd is stationary.
Moreover,(Z(j))j Zd is centered, since ∈ ∈
∈
EZ(0)=E X(t)dt EX(0)= EX(t)dt EX(0)=0.
− −
Z[0,1)d Z[0,1)d
4 THE MULTIVARIATE CLT 7
Further,
Cov Z(0),Z(j) = Cov X(s),X(t) dtds
jX∈Zd (cid:0) (cid:1) jX∈ZdZ[0,1)dZj+[0,1)d (cid:0) (cid:1)
= Cov X(0),X(t s) dtds
Z[0,1)dZRd −
(cid:0) (cid:1)
= Cov X(0),X(t) dtds
Z[0,1)dZRd
(cid:0) (cid:1)
= Cov X(0),X(t) dt.
ZRd
(cid:0) (cid:1)
We put Q := j Zd j +[0,1)d W and W := j +[0,1)d . As explained in the
proof of [3, Thneorem{ 1∈.2], th|e assumption⊆thatn}(Wn)n Nn−is VH-gjr∈oQwning implies that (Qn)n N is regular
∈ S (cid:0) (cid:1) ∈
growing. Now Theorem 1.12 of [2, p. 178] implies that
Wn−X(t)dt−λd(Wn−)EX(0) = j∈QnZ(j) (0,σ2), n .
√#Q →N →∞
R λd(Wn−) P n
q
If we can show that
Wn\Wn−X(t)dt−λd(Wn\Wn−)EX(0) P 0, n , (4)
R λd(Wn) → →∞
p
then Slutzki’s theorem will imply the assertion, since, clearly, λd(Wn−)/ λd(Wn) 1. We get
→
q
p
Var X(t)dt = Cov(X(s),X(t))dtds
(cid:16)ZWn\Wn− (cid:17) ZWn\Wn−ZWn\Wn−
Cov(X(s),X(t)) dtds
≤ZWn\Wn−ZRd| |
=λ (W W ) Cov(X(0),X(t)) dt.
d n\ n− ZRd| |
Since (Wn)n N is VH-growing,we get
∈
Var Wn\Wn−X(t)dt = Var Wn\Wn−X(t)dt 0, n .
(cid:18)R λd(Wn) (cid:19) (cid:0)R λd(Wn) (cid:1) → →∞
By the Chebyshev inequalitypthis implies (4).
4 The multivariate CLT
In this section we extend Theorem 1 in various ways to multivariate central limit theorems.
Theorem 10. Let θ = (θr)r N be a monotonically decreasing zero sequence. Let (X(t))t Rd be an Rs-
valued random field. Assume∈that (X(t))t Rd is stationary, measurable, BL(θ)-dependent a∈nd fulfills
∈
Cov(X (0),X (t)) dt< , i,j =1,...,s.
i j
ZRd| | ∞
Let (Wn)n N be a VH-growing sequence of subsets of Rd. Then
∈
X (t)dt EX (0)λ (W ) X (t)dt EX (0)λ (W )
Wn 1 − 1 d n ,..., Wn s − s d n (0,Σ), n ,
R λd(Wn) R λd(Wn) →N →∞
(cid:16) (cid:17)
in distribution, whpere Σ is the matrix with entries p
Cov(X (0),X (t))dt, i,j =1,...,s.
i j
ZRd
4 THE MULTIVARIATE CLT 8
Proof: Let u = (u1,...,us) Rs. Then ( X(t),u )t Rd is BL(θ′)-dependent for a monotonically
sdteactrieoansainrygasneqdumenecaesuθr′ab=le(.θWr′)re∈∈hNavweith limr→∞hθr′ = 0idu∈e to Lemma 4. Obviously, (hX(t),ui)t∈Rd is
s s
Cov( X(0),u , X(t),u )dt= u u Cov(X (0),X (t))dt=uTΣu.
i j i j
ZRd h i h i i=1j=1 ZRd
XX
In particular, the integral is defined. So Theorem 1 implies
X (t)dt EX (0)λ (W ) X (t)dt EX (0)λ (W )
Wn 1 − 1 d n ,..., Wn s − s d n ,u
R λd(Wn) R λd(Wn)
D(cid:16) (cid:17) E
X(t),u dt E X(0),u λ (W )
= Wnh p i − h i d n (p0,uTΣu), n .
R λd(Wn) →N →∞
Since Y,u (0,uTΣu) for a rapndom vector Y (0,Σ), the Theorem of Cram´er and Wold implies
h i∼N ∼N
the assertion.
Corollary 11. Let (X(t))t Rd be a stationary, measurable R-valued random field and let f1,...,fs :
R R be functions. Let (W∈n)n N be a VH-growing sequence of subsets of Rd. Assume that one of the
→ ∈
following conditions holds:
(i) The field (X(t))t Rd is BL(θ)-dependent for a monotonically decreasing zero sequence θ =(θr)r N,
the maps f ,...,∈f are Lipschitz continuous and ∈
1 s
Cov f (X(0)),f (X(t)) dt< , i,j =1,...,s.
i j
ZRd ∞
(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
(ii) The field (X(t))t Rd is QA and there are c>0 and ǫ>0 with
∈
Cov(X(0),X(t)) c t d ǫ, t Rd. (5)
− −
≤ ·k k∞ ∈
The maps f ,...,f are Lipschitz continuous.
1 s
(iii) The field (X(t))t Rd is PA with EX(0)2 < . The maps f1,...,fs are of locally bounded variation
with E[h (X(0))∈2]< , i=1,...,s, and t∞here are c>0 and ǫ>0 with
fi ∞
Cov h (X(0)),h (X(t)) c t d ǫ, t Rd, i,j =1,...,s. (6)
fi fj ≤ ·k k∞− − ∈
(cid:0) (cid:1)
Then
f (X(t))dt Ef (X(0))λ (W ) f (X(t))dt Ef (X(0))λ (W )
Wn 1 − 1 d n ,..., Wn s − s d n (0,Σ),
R λd(Wn) R λd(Wn) →N
(cid:16) (cid:17)
as n in distribuption, where Σ is the matrix with entries p
→∞
Cov f (X(0)),f (X(t)) dt, i,j =1,...,s.
i j
ZRd
(cid:0) (cid:1)
Part (i) of this corollary is an immediate consequence of Lemma 4 and Theorem 10.
Proof of Corollary 11(ii): The field (X(t))t Rd is BL(θ)-dependent by Lemma 5 and thus Lemma 4
implies that the field (f1(X(t)),...,fs(X(t)))t R∈d is also BL(θ)-dependent.
In order to check the integrability assumpt∈ions from part (i), we put
N if f (x)< N
j
− −
f(N) :x f (x) if f (x) [ N,N]
j 7→ j j ∈ −
N if fj(x)>N.

REFERENCES 9
Since (X(t))t Rd is QA, we get
∈
Cov f(N)(X(0)),f(N)(X(t)) Lip(f(N)) Lip(f(N)) Cov(X(0),X(t))
| i j |≤ i · j ·| |
(cid:0) (cid:1) Lip(fi) Lip(fj) Cov(X(0),X(t)).
≤ · ·| |
By the Monotone Convergence Theorem, applied to both summands of E[f(N)(X(0))f(N)(X(t))]
i j −
E[f(N)(X(0))] E[f(N)(X(t))], this yields
i · j
Cov f (X(0)),f (X(t)) Lip(f ) Lip(f ) Cov(X(0),X(t)).
i j i j
| |≤ · ·| |
Moreover,(5) implies (cid:0) (cid:1)
Cov X(0),X(t) dt<
ZRd ∞
(cid:12) (cid:0) (cid:1)(cid:12)
and hence (cid:12) (cid:12)
Cov f (X(0)),f (X(t)) dt< , i,j =1,...,s.
i j
ZRd ∞
(cid:12) (cid:0) (cid:1)(cid:12)
So part (i) yields the assertio(cid:12)n. (cid:12)
Proof of Corollary 11(iii): Since (X(t))t Rd is PA, the random field (hf1(X(t)),...,hfs(X(t)))t Rd
is also PA, see Theorem 1.8(d) of [2, p. 7], a∈nd therefore QA. By Lemma 5 it is BL(θ)-dependent∈for
somemonotonicallydecreasingzerosequenceθ. Hence (f1(X(t)),...,fs(X(t)))t Rd isBL(θ′)-dependent
for some monotonically decreasing zero sequence θ by Lemma 9 and Lemma 4.∈
′
Clearly, the field (f1(X(t)),...,fs(X(t)))t Rd is also stationary and measurable.
Moreover,(6) implies ∈
Cov h (X(0)),h (X(t)) dt< , i,j =1,...,s.
ZRd fi fj ∞
(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
Now Lemma 9 and the QA property of (hf1(X(t)),...,hfs(X(t)))t Rd give
∈
Cov f (X(0)),f (X(t)) dt Cov h (X(0)),h (X(t)) dt< , i,j =1,...,s.
ZRd i j ≤ZRd fi fj ∞
(cid:12) (cid:0) (cid:1)(cid:12) (cid:12) (cid:0) (cid:1)(cid:12)
So Th(cid:12)eorem 10 yields the assert(cid:12)ion. (cid:12) (cid:12)
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