Multivariate central limit theorems for intrinsic volumes of Boolean mod- mod-els

Now (a) and (b) are immediate consequences of Theorem 1.1, while (c) follows from Theorem 1.2. One final technical remark is in order. To ensure thatγ5andγ6vanish in the event thatR

Xf_i(x)⁶λ(dx)is not finite for somei ∈ {1, ..., m}, we use the convention that0 · ∞ = 0. This convention is supported by the technical details of the proof of Theorem 1.2; in particular we haveV_jk⁽¹⁾ = V_jk⁽²⁾ = 0because the difference operator is a deterministic function.

The idea of the following proof of Corollary 1.3 is to show that it is only a special case of Corollary 4.1.

Proof of Corollary 1.3. LetX = R^m(equipped with its Borelσ-field) andλ(·) = sP(X¹∈

·), i.e.,λisstimes the probability measure ofX1. Fori ∈ {1, . . . , m}let us denote byπi

the projectionR^m3 (y1, . . . , ym) 7→ yi. Then we have that Zs= (I1(π1/√

s), . . . , I1(πm/√ s)).

Together with the observation that, fori ∈ {1, . . . , m},p ∈ (0, ∞), ands > 0, Z

X

|πi(x)/√

s|^pλ(dx) = E |X1⁽ⁱ⁾|^ps^1−p/2,

we see that conclusions (a) and (b) of Corollary 1.3 follow from conclusions (a) and (b) of Corollary 4.1, withp = 3, whereas conclusion (c) follows from its counterpart in Corollary 4.1 withp ∈ {3, 4}.

4.2 Multivariate central limit theorems for intrinsic volumes of Boolean mod-els

In the following, we derive quantitative multivariate central limit theorems for Boolean models, extending previous findings in [12] and [17, Chapter 22]. Our proofs rely on the general bounds from Subsection 1.2 as well as arguments from [12] and [17, Chapter 22].

We denote by K^d the set of compact convex sets in R^d, d ≥ 1. For a probability measureQ^onK^dsuch thatQ({∅}) = 0^andγ > 0letηbe a Poisson process onR^d× K^d with intensity measureγλd⊗ Q^{, where}λd is Lebesgue measure onR^d. Note thatηis a stationary Poisson process inR^dwith independent marks inK^d distributed according toQ. A random compact convex setZ₀distributed according toQis called the typical grain. Fromη we construct the random closed set

Z := [

(x,K)∈η

(x + K),

which is called the Boolean model. For more details on Boolean models and further references we refer to [29].

By the convex ringR^dwe mean the set of all finite unions of elements fromK^d. Let V0, V1, . . . , Vd: R^d→ Rbe the intrinsic volumes (see, for example, [29, Section 14.2] for a definition via the Steiner formula and additive extensions). In particular, forA ∈ R^d, Vd(A)is the volume ofA,Vd−1(A)is half the surface area ofA(ifAis the closure of its interior), andV₀(A)is the Euler characteristic ofA.

In the sequel we study the intersection of the Boolean model Z with a compact convex observation windowW ∈ K^d. Note thatZ ∩ W almost surely belongs toR^d if E Vⁱ(Z0) < ∞fori ∈ {1, . . . , d}. Questions of interest include finding the fraction ofW covered byZ and the surface area ofZ ∩ W. We address both problems simultaneously by considering

V(Z ∩ W ) := (V0(Z ∩ W ), V1(Z ∩ W ), . . . , Vd(Z ∩ W )).

Denote byr(K)the inradius ofK ∈ K^d. In [12, Theorem 3.1] it is shown that there exists a matrixΣ = (σ_i,j)i,j∈{0,...,d}∈ R(d+1)×(d+1)such that

Σ(W ) := 1

V_d(W )(Cov(Vi(Z ∩ W ), Vj(Z ∩ W )))i,j∈{0,...,d}→ Σ as r(W ) → ∞ ifE Vⁱ(Z0)² < ∞fori ∈ {1, . . . , d}. If, additionally,P(V^d(Z0) > 0) > 0, the asymptotic covariance matrix Σ is positive definite (see [12, Theorem 4.1]). We describe the asymptotic behavior ofV(Z ∩ W )asr(W ) → ∞with respect tod3, d2, anddconvex. Theorem 4.2. (a) If E Vⁱ(Z0)³ < ∞ fori ∈ {1, . . . , d}, there exists a constantC1 ∈

(0, ∞)depending ond,γ, andQ^{such that} d3 V(Z ∩ W ) − E V(Z ∩ W )

pVd(W ) , NΣ



≤ C1

1 r(W )^min{1,d/2}

for allW ∈ K^dwithr(W ) ≥ 1.

(b) IfE Vⁱ(Z0)³< ∞fori ∈ {1, . . . , d}andP(V^d(Z0) > 0) > 0, there exists a constant C2∈ (0, ∞)depending ond,γ, andQ^{such that}

d2 V(Z ∩ W ) − E V(Z ∩ W ) pV_d(W ) , NΣ



≤ C2

1 r(W )^min{1,d/2}

for allW ∈ K^dwithr(W ) ≥ 1.

(c) IfE Vi(Z₀)⁴< ∞fori ∈ {1, . . . , d}andP(Vd(Z₀) > 0) > 0, there exists a constant C3∈ (0, ∞)depending ond,γ, andQ^{such that}

dconvex V(Z ∩ W ) − E V(Z ∩ W ) pV_d(W ) , NΣ



≤ C3

1 r(W )^min{1,d/2}

for allW ∈ K^dwithr(W ) ≥ 1.

(d) IfN_Σis replaced byN_{Σ(W )}, the assertions (a)-(c) hold with the rate1/pV_d(W ). Theorem 4.2(a) improves upon the moment assumptions of [12, Theorem 9.1] by requiring existence of third moments (i.e., E Vi(Z₀)³ < ∞ fori ∈ {1, . . . , d}) and not fourth moments. Parts (b) and (c) extend [12, Theorem 9.1] to different distances, in particular, the non-smoothd_convex-distance. The findings of [12] as well as the univariate results in [17] consider so-called geometric functionals, which include intrinsic volumes.

Theorem 4.2 could be also generalized to these functionals, but for the sake of simplicity we consider only intrinsic volumes. Since our proof of Theorem 4.2 is based on second

order Poincaré inequalities, it does not require dealing with the whole chaos expansion as in [12]. For previous results on volume and surface area of Boolean models we refer the reader to [12]. Theorem 4.2 indicates that the slow convergence ofΣ(W ) toW weakens the rate of convergence ford ≥ 3 (see also [12, Remark 9.5]). The rate of convergence1/pVd(W )for the distance toN_{Σ(W )}is comparable to1/√

nin the classical central limit theorem for sums ofni.i.d. random vectors and, thus, presumably optimal.

We prepare the proof of Theorem 4.2 by two lemmas. In the sequel, we use the Wills functionalV (K) :=Pd

i=0κ_d−iV_i(K)forK ∈ K^d, whereκ_d−i is the volume of the (d − i)-dimensional unit ball. We write the difference operatorDwith respect to the pair (x, K), withx ∈ R^d, K ∈ K^d.

Lemma 4.3. There exists a constantC ∈ (0, ∞)only depending ond,γ, andQ^{such that,} forx, x1, x2∈ R^d,K, K1, K2∈ K^d,i, j ∈ {0, . . . , d}, andm, m1, m2∈ {1, . . . , 6},

E |D(x,K)Vi(Z ∩ W )|^m≤ C^mV ((x + K) ∩ W )^m,

E |D(x²1,K1),(x2,K2)Vi(Z ∩ W )|^m≤ C^mV ((x1+ K1) ∩ (x2+ K2) ∩ W )^m,

E |D(x₁,K₁)V_i(Z ∩ W )|^m1|D_(x2,K2)V_j(Z ∩ W )|^m2

≤ C^m1+m2V ((x1+ K1) ∩ W )^m1V ((x2+ K2) ∩ W )^m2, and

E |D(x²₁,K₁),(x,K)Vi(Z ∩ W )|^m1|D²_(x

2,K₂),(x,K)Vj(Z ∩ W )|^m2

≤ C^m1+m2V ((x1+ K1) ∩ (x + K) ∩ W )^m1V ((x2+ K2) ∩ (x + K) ∩ W )^m2.

Proof. Form ∈ {2, 3}ori = jandm1= m2= 2this is shown in [17] in Proposition 22.4 in connection with (22.30) and (22.31) (see also [12, Lemma 3.3]), but the proof can be extended toi 6= jand the other choices form, m₁, m₂.

Moreover, we will use the following translative integral formula from [17, Proposition 22.5] and [12, Lemma 3.4].

Lemma 4.4. For allK, L ∈ K^d, Z

R^d

V ((x + K) ∩ L)dx ≤ V (K)V (L).

Proof of Theorem 4.2. We deduce Theorem 4.2 from Theorem 1.1 and Theorem 1.2 by bounding γ1, . . . , γ6 from Subsection 1.2 as follows. We denote by γ˜1, . . . , ˜γ6 the corresponding terms without the normalization1/pVd(W )of the functionals. Without loss of generality we can assume thatγ = 1. In the sequel let(Zn)_n∈Nbe independent copies of the typical grainZ₀. It follows from Lemma 4.3, the monotonicity and the translation invariance of the Wills functional (i.e., V (K) ≤ V (L) forK, L ∈ K^d with K ⊆ LandV (K + x) = V (K)forK ∈ K^dandx ∈ R^d), and Lemma 4.4 that

˜

γ₁²≤ (d + 1)²C⁴E Z

(R^d)³

V ((x1+ Z1) ∩ (x3+ Z3) ∩ W )V ((x2+ Z1) ∩ (x3+ Z3) ∩ W ) V ((x₁+ Z₁) ∩ W )V ((x₂+ Z₂) ∩ W )d(x₁, x₂, x₃)

≤ (d + 1)²C⁴E Z

(R^d)³

V ((x1+ Z1) ∩ (x3+ Z3) ∩ W )V ((x2+ Z1) ∩ (x3+ Z3) ∩ W ) V (Z1)V (Z2)d(x1, x2, x3)

≤ (d + 1)²C⁴E Z

R^d

V (Z1)²V (Z2)²V ((x + Z3) ∩ W )²dx

≤ (d + 1)²C⁴E Z

R^d

V (Z1)²V (Z2)²V (Z3)V ((x + Z3) ∩ W )dx

≤ (d + 1)²C⁴E V (Z¹)²E V (Z²)²E V (Z³)²V (W )

≤ (d + 1)²C⁴(E V (Z0)²)³V (W ) and

˜

γ₂²≤ (d + 1)²C⁴E Z

(R^d)³

V ((x1+ Z1) ∩ (x3+ Z3) ∩ W )²V ((x2+ Z2) ∩ (x3+ Z3) ∩ W )² d(x1, x2, x3)

≤ (d + 1)²C⁴E Z

R^d

V (Z1)²V (Z2)²V ((x + Z3) ∩ W )²dx

≤ (d + 1)²C⁴E V (Z¹)²V (Z2)²V (Z3)²V (W )

= (d + 1)²C⁴(E V (Z⁰)²)³V (W ).

Hence, we see thatγ1andγ2are at most of the order q

V (W )/Vd(W ). From the same arguments as above we obtain that, fork ∈ N^,

E Z

R^d

V ((x + Z₀) ∩ W )^kdx ≤ E V (Z0)^k−1 Z

R^d

V ((x + Z₀) ∩ W )dx ≤ E V (Z0)^kV (W ), (4.2)

whenceγ₃is at most of orderV (W )/V_d(W )^3/2. We can also show that E

Z

(R^d)²

V ((x1+ Z1) ∩ (x2+ Z2) ∩ W )²

V ((x1+ Z1) ∩ (x2+ Z2) ∩ W )²+ V ((x1+ Z1) ∩ W )²

d(x1, x2)

≤ 2E Z

(R^d)²

V ((x1+ Z1) ∩ (x2+ Z2) ∩ W )V (Z2)V (Z1)²d(x1, x2)

≤ 2E V (Z1)³V (Z₂)²V (W )

so that together with (4.2), we deduce that γ4 is at most of order q

V (W )/Vd(W ). Jensen’s inequality and Lemma 4.3 lead to

E kD(x,K)V(Z ∩ W )k⁶≤ (d + 1)³C⁶V ((x + K) ∩ W )⁶ (4.3) forx ∈ R^dandK ∈ K^dand

E kD²(x₁,K₁),(x₂,K₂)V(Z ∩ W )k⁶≤ (d + 1)³C⁶V ((x₁+ K₁) ∩ (x₂+ K₂) ∩ W )⁶

≤ (d + 1)³C⁶V ((x1+ K1) ∩ W )^{6 (4.4)} for x1, x2 ∈ R^{d and} K1, K2 ∈ K^d. This implies that, for x, y ∈ R^d, K, L ∈ K^d, and

` ∈ {1, 2},

E kD(x,K)V(Z ∩ W )k^{`</sup>+ kD<sup>2</sup>(x,K),(y,L)V(Z ∩ W )k<sup>`
3}

≤ 4E kD(x,K)V(Z ∩ W )k^{3`</sup>+ 4E kD(x,K),(y,L)<sup>2</sup> V(Z ∩ W )k<sup>3`}

≤ 4 E kD(x,K)V(Z ∩ W )k⁶
`/2

+ 4 E kD²(x,K),(y,L)V(Z ∩ W )k⁶
`/2

≤ 8(d + 1)^{3`/2</sup>C<sup>3`}V ((x + K) ∩ W )^3`.

From [12, Lemma 3.2] or [17, Lemma 22.6] it follows that, fori ∈ {0, . . . , d},x1, x2∈ R^d, andK1, K2∈ K^d,

D_(x²

1,K1),(x2,K2)V_i(Z ∩W ) = V_i(Z ∩(x₁+K₁)∩(x₂+K₂)∩W )−V_i((x₁+K₁)∩(x₂+K₂)∩W ).

Consequently, we have that

1{D_(x²

1,K₁),(x₂,K₂)V(Z ∩ W ) 6= 0} ≤ 1{(x₁+ K₁) ∩ (x₂+ K₂) ∩ W 6= ∅}

≤ V ((x₁+ K₁) ∩ (x₂+ K₂) ∩ W ).

From Hölder’s inequality, (4.3), and (4.4), we obtain that, forx₁, x₂, y ∈ R^d,K₁, K₂, L ∈ K^d, and` ∈ {1, 2},

E 1{D²(x₁,K₁),(y,L)V(Z ∩ W ) 6= 0, D²_(x

2,K₂),(y,L)V(Z ∩ W ) 6= 0}

× kD_(x1,K1)V(Z ∩ W )k^{`</sup>+ kD<sup>2</sup><sub>(x1,K1),(y,L)</sub>V(Z ∩ W )k<sup>`}
3/4

× kD_(x2,K2)V(Z ∩ W )k^`+ kD²_(x

2,K₂),(y,L)V(Z ∩ W )k^`
3/4

× |D_(x1,K1)Vi(Z ∩ W )|^3/2|D_(x2,K2)Vi(Z ∩ W )|^3/2

≤ 1{(x₁+ K₁) ∩ (y + L) ∩ W 6= ∅}1{(x₂+ K₂) ∩ (y + L) ∩ W 6= ∅}

× E kD(x1,K1)V(Z ∩ W )k^3`/4+ kD_(x²

1,K₁),(y,L)V(Z ∩ W )k^3`/4

× kD(x₂,K₂)V(Z ∩ W )k^3`/4+ kD²_(x

2,K2),(y,L)V(Z ∩ W )k^3`/4

× |D_(x1,K1)V_i(Z ∩ W )|^3/2|D_(x2,K2)V_i(Z ∩ W )|^3/2

≤ 1{(x1+ K₁) ∩ (y + L) ∩ W 6= ∅}1{(x₂+ K₂) ∩ (y + L) ∩ W 6= ∅}

× E kD(x1,K1)V(Z ∩ W )k^6
`/8

+ E kD(x²₁,K₁),(y,L)V(Z ∩ W )k^6
`/8

× E kD(x₂,K₂)V(Z ∩ W )k⁶
`/8

+ E kD(x²2,K2),(y,L)V(Z ∩ W )k⁶
`/8

× E |D(x1,K1)Vi(Z ∩ W )|³|D_(x2,K2)Vi(Z ∩ W )|^3
1/2

≤ 1{(x₁+ K₁) ∩ (y + L) ∩ W 6= ∅}1{(x₂+ K₂) ∩ (y + L) ∩ W 6= ∅}

× 4(d + 1)^{3`/4</sup>C<sup>3`/2}V ((x1+ K1) ∩ W )^{3`/4</sup>V ((x2+ K2) ∩ W )<sup>3`/4}

× C³V ((x₁+ K₁) ∩ W )^3/2V ((x₂+ K₂) ∩ W )^3/2

= 1{(x1+ K1) ∩ (y + L) ∩ W 6= ∅}1{(x2+ K2) ∩ (y + L) ∩ W 6= ∅}

× 4(d + 1)^{3`/4</sup>C<sup>3+3`/2}V ((x₁+ K₁) ∩ W )^{3/2+3`/4</sup>V ((x<sub>2</sub>+ K<sub>2</sub>) ∩ W )<sup>3/2+3`/4}.

Combining the previous estimates with Lemma 4.3 yields

˜

γ₅³≤ 3(d + 1)²E Z

(R^d)³

V ((x1+ Z1) ∩ (x3+ Z3) ∩ W )V ((x2+ Z2) ∩ (x3+ Z3) ∩ W )

× 4^2/3(d + 1)^1/2C³V ((x1+ Z1) ∩ W )^3/2V ((x2+ Z2) ∩ W )^3/2

× C²V ((x1+ Z1) ∩ W )V ((x2+ Z2) ∩ W )d(x1, x2, x3) + 27(d + 1)²E

Z

(R^d)³

2(d + 1)^1/2CV ((x1+ Z1) ∩ W )^1/2V ((x2+ Z2) ∩ W )^1/2

× C²V ((x₁+ Z₁) ∩ (x₃+ Z₃) ∩ W )V ((x₂+ Z₂) ∩ (x₃+ Z₃) ∩ W )

× C²V ((x1+ Z1) ∩ W )V ((x2+ Z2) ∩ W )d(x1, x2, x3)

and

˜

γ₆⁴≤ 3(d + 1)²E Z

(R^d)³

V ((x1+ Z1) ∩ (x3+ Z3) ∩ W )V ((x2+ Z2) ∩ (x3+ Z3) ∩ W )

× 4^2/3(d + 1)C⁴V ((x₁+ Z₁) ∩ W )²V ((x₂+ Z₂) ∩ W )²

× C²V ((x1+ Z1) ∩ W )V ((x2+ Z2) ∩ W )d(x1, x2, x3) +81

4 (d + 1)²E Z

(R^d)³

2(d + 1)C²V ((x1+ Z1) ∩ W )V ((x2+ Z2) ∩ W )

× C²V ((x₁+ Z₁) ∩ (x₃+ Z₃) ∩ W )V ((x₂+ Z₂) ∩ (x₃+ Z₃) ∩ W )

× C²V ((x₁+ Z₁) ∩ W )V ((x₂+ Z₂) ∩ W )d(x₁, x₂, x₃).

Monotonicity and translation invariance of the Wills functional and Lemma 4.4 imply

˜

γ₅³≤ 3 · 4^2/3(d + 1)^5/2C⁵E V (Z¹)^7/2V (Z2)^7/2V (Z3)²V (W ) + 54(d + 1)^5/2C⁵E V (Z¹)^5/2V (Z2)^5/2V (Z3)²V (W ) and

˜

γ⁴₆≤ 3 · 4^2/3(d + 1)³C⁶E V (Z1)⁴V (Z₂)⁴V (Z₃)²V (W ) +81

2 (d + 1)³C⁶E V (Z¹)³V (Z2)³V (Z3)²V (W ).

Thus, γ₅ and γ₆ are at most of the ordersV (W )^1/3/V_d(W )^5/6 andV (W )^1/4/V_d(W )^3/4, respectively. By [12, Lemma 3.7], there exists a dimension dependent constantC_d ∈ (0, ∞)such that

V (W )

V_d(W ) ≤ C_d for all W ∈ K^d with r(W ) ≥ 1.

This implies thatγ₁,γ₂,γ₃,γ₄,γ₅, andγ₆have at most the order1/pV_d(W ). It is known [12, Theorem 3.1] that there exists a constantCΣ∈ (0, ∞)only depending ond,γ, andQ such that

   

Cov(Vi(Z ∩ W ), Vj(Z ∩ W )) Vd(W ) − σ_i,j

   

≤ C_Σ 1 r(W )

fori, j ∈ {0, . . . , d} andW ∈ K^d with r(W ) ≥ 1. Now Theorem 1.1 and Theorem 1.2 complete the proof.

4.3 Multivariate normal approximation for functionals of marked Poisson

In document Multivariate second order Poincaré inequalities for Poisson functionals (Page 30-35)