2 Asymptotic normality of λ( S

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El e c t ro nic

Journ a l of

Pr

ob a b il i t y

Vol. 14 (2009), Paper no. 91, pages 2617–2635.

Journal URL

http://www.math.washington.edu/~ejpecp/

Asymptotic Normality in Density Support Estimation

Gérard Biau^∗ Benoît Cadre^† David M. Mason^‡ Bruno Pelletier^§

Abstract

Let X₁, . . . ,X_nbe nindependent observations drawn from a multivariate probability density f with compact supportS_f. This paper is devoted to the study of the estimatorSˆ_nofS_f defined as the union of balls centered at theX_iand with common radiusr_n. Using tools from Riemannian geometry, and under mild assumptions on f and the sequence (r_n), we prove a central limit theorem forλ(S_n∆S_f), whereλdenotes the Lebesgue measure onR^d and∆the symmetric difference operation.

Key words:Support estimation, Nonparametric statistics, Central limit theorem, Tubular neighborhood.

AMS 2000 Subject Classification:Primary 62G05, 62G20.

Submitted to EJP on April 29, 2009, final version accepted December 5, 2009.

∗LSTA & LPMA, Université Pierre et Marie Curie – Paris VI, Boîte 158, 175 rue du Chevaleret, 75013 Paris, France, [email protected]

†IRMAR, ENS Cachan Bretagne, CNRS, UEB, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France, [email protected]

‡University of Delaware, Food and Resource Economics, 206 Townsend Hall, Newark, DE 19717, USA, [email protected]

§IRMAR, Université Rennes 2, CNRS, UEB, Campus Villejean, Place du Recteur Henri Le Moal, CS 24307, 35043 Rennes Cedex, France,[email protected]

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1 Introduction

Let X₁, . . . ,X_n be independent and identically distributed observations drawn from an unknown probability density f defined onR^d. It is assumed thatd≥2 throughout this paper. We investigate the problem of estimating the support of f, i.e., the closed set

S_f ={x∈R^d :f(x)>0},

based on the sample X₁, . . . ,X_n. Here and elsewhere,Adenotes the closure of a Borel set A. This problem is of interest due to the broad scope of its practical applications in applied statistics. These include medical diagnosis, machine condition monitoring, marketing and econometrics. For a review and a large list of references, we refer the reader to Molchanov (1998), Baíllo, Cuevas, and Justel (2000), Biau, Cadre, and Pelletier (2008) and Mason and Polonik (2009).

Devroye and Wise (1980) introduced the following very simple and intuitive estimator ofS_f. It is defined as

S_n=

n

[

i=1

B(X_i,r_n), (1.1)

whereB(x,r)denotes the closed Euclidean ball centered at x and of radiusr>0, and where(rn) is an appropriately chosen sequence of positive smoothing parameters. For x∈R^d, let

f_n(x) = Xn

i=1

1_B(x,r

n)(X_i) be the (unnormalized) kernel density estimator of f. We see that

S_n={x ∈R^d: f_n(x)>0}.

In other words, S_n = S_f_n, i.e., it is just a plug-in-type kernel estimator with kernel having a ball-shaped support. Baíllo, Cuevas, and Justel (2000) argue that this estimator is a good generalist when no a priori information is available about S_f. Moreover, from a practical perspective, the relative simplicity of the estimation strategy (1.1) is a major advantage over competing multidi- mensional set estimation techniques, which are often faced with a heavy computational burden.

Biau, Cadre, and Pelletier (2008) proved, under mild regularity assumptions on f and the sequence (r_n), that for an explicit constantc>0,

Æ

nr_n^dEλ(S_n∆S_f)→c,

where4denotes the symmetric difference operation andλis the Lebesgue measure onR^d. In the present paper, we go one step further and establish the asymptotic normality ofλ(Sn4S_f). Precisely, our main Theorem 2.1 states, under appropriate regularity conditions on f and(r_n), that

n r_n^d

1/4

λ

S_n4S_f

−Eλ

S_n4S_f _D

→ N(0,σ²_f),

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for some explicit positive varianceσ²_f.

Denoting by∂S_f the boundary ofS_f, it turns out that, under our conditions,λ(∂S_f) =0 and f >0 on the interior ofS_f. Therefore, we have the equality

λ

S_f4¦

x∈R^d : f(x)>0©

=0.

Thus,λ(Sn4S_f)may be expressed more conveniently as λ(Sn4S_f) =

Z

R^d

1{f_n(x)>0} −1{f(x)>0} dx.

This quantity is related to the so-called vacancy V_n left by randomly distributed spheres (see Hall 1985, 1988), which in this notation is

V_n=λ

S_f −S_n

= Z

S_f

1{f_n(x)>0} −1{f(x)>0}

dx. (1.2)

Hall (1985) has proved a number of central limit theorems for V_n. One of them, his Theorem 1, states that if f has support in[0, 1]^d and is continuous then, as long asnr_n^d→awhere 0<a<∞, for some 0< σ²_a<∞,

pn V_n−EV_n D

→ N(0,σ²_a).

As pointed out in Hall’s paper, and to the best of our knowledge, the case when nr_n^d → ∞has not been examined, except for some restricted cases in dimension 1. It turns out that, by adapting our arguments to the vacancy problem, we are also able to prove a general central limit theorem for V_n when nr_n^d → ∞, thereby extending Hall’s results. For more about large sample properties of vacancy and their applications consult Chapter 3 of Hall (1988).

Another result closely related to ours is the following special case of the main theorem in Mason and Polonik (2009). For any 0 < c < sup{f(x) : x ∈R^d}, let C(c) = {x ∈R^d : f(x) > c} and Cˆ_n(c) ={x ∈R^d:fˆ_n(x)>c}, where fˆ_n denotes a kernel estimator of f. Then

λ

Cˆ_n(c)4C(c)

= Z

R^d

1{fˆ_n(x)>c} −1{f(x)>c} dx.

Mason and Polonik (2009) prove, subject to regularity conditions on f, as long as p

nr_n^d+² →γ, with 0≤γ <∞andnr_n^d/logn→ ∞, whereγ=0 in the cased=1, that for some 0< σ²_c<∞,

n r_n^d

1/4

λ

Cˆ_n(c)4C(c)

−Eλ

Cˆ_n(c)4C(c) _D

→ N(0,σ²_c).

The paper is organized as follows. In Section 2, we first set out notation and assumptions, and then state our main results. Section 3 is devoted to the proofs.

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2 Asymptotic normality of λ( S

_n

∆ S

_f

)

2.1 Notation and assumptions

Throughout the paper, we shall impose the following set of assumptions related to the support of f. To state some of them we shall require a few concepts and terms from Riemannian geometry. For a good introduction to the subject we refer the reader, for instance, to the book by Gallot, Hulin and Lafontaine (2004). We make use of the notationS^◦_f to denote the interior ofS_f.

Assumption Set 1

(a) The supportS_f of f is compact inR^d, with d≥2.

(b) f is of classC¹onR^d, and of classC² onT ∩S^◦_f, whereT is a tubular neighborhood ofS_f. (c) The boundary∂S_f ofS_f is a smooth submanifold ofR^d of codimension 1.

(d) The set{x ∈R^d:f(x)>0}is connected.

(e) f >0 onS^◦_f.

Roughly speaking, under Assumption 1-(c), the boundary∂S_f is a subset of dimension(d−1) of the ambient spaceR^d. For instance, consider a density supported on the unit ball ofR^d. In this case, the boundary is the unit sphere of dimension(d−1). Note also that one can relax Assumption 1-(d) to the case where the set{x ∈R^d: f(x)>0}has multiple connected components, see Remark 3.3 in Biau, Cadre, and Pelletier (2008).

More precisely, under Assumption 1-(c),∂S_f is a smooth Riemannian submanifold with Riemannian metric, denoted byσ, induced by the canonical embedding of ∂S_f inR^d. The volume measure on (∂S_f,σ)will be denoted by v_σ. Furthermore,(∂S_f,σ) is compact and without boundary. Then by the tubular neighborhood theorem (see e.g., Gray, 1990; Bredon, 1993, p. 93),∂S_f admits a tubular neighborhood of radiusρ >0,

V(∂S_f,ρ) =¦

x ∈R^d: dist(x,∂S_f)< ρ© ,

i.e., each point x ∈ V(∂S_f,ρ) projects uniquely onto ∂S_f. Let {e_p; p ∈ ∂S_f} be the unit-norm section of the normal bundle T∂S^⊥_f that is pointing inwards, i.e., for all p ∈ ∂S_f, e_p is the unit normal vector to∂S_f directed towards the interior ofS_f. Then each point x ∈ V(∂S_f,ρ) may be expressed as

x=p+ve_p, (2.1)

where p∈∂S_f, and where v∈R satisfies|v| ≤ρ. Moreover, given a Lebesgue integrable function

ϕonV(∂S_f,ρ), we may write

Z

V(∂S_f,ρ)

ϕ(x)dx =

Z

∂S_f

Z _ρ

−ρ

ϕ(p+ve_p)Θ(p,u)du v_σ(dp), (2.2)

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whereΘ is aC^∞ function satisfyingΘ(p, 0) =1 for all p∈∂S_f. (See Appendix B in Biau, Cadre, and Pelletier, 2008.)

Denote byD²_e

p the directional differentiation operator of order 2 onV(∂S_f,ρ)in the directione_p. It will be seen in the proofs in Section 3 that the variance in our central limit theorem is determined by the second order behavior of f near the boundary of its support. Therefore to derive this variance we shall need the following set of second order smoothness assumptions on f.

Assumption Set 2

(a) There existsρ >0 such that, for allp∈∂S_f, the mapu7→ f(p+ue_p)is of classC² on[0,ρ]. (b) There existsρ >0 such that

0< sup

p∈∂Sf

sup

0≤u≤ρD²_e

pf(p+ue_p)<∞. (c) There existsρ >0 such that

p∈∂infSf

0≤infu≤ρD²_e

pf(p+ue_p)>0.

For similar smoothness assumptions see Section 2.4 of Mason and Polonik (2009). The imposition of such conditions appears to be unavoidable to derive a central limit theorem. Note also that Assumption Sets 1 and 2 are the same as the ones used in Biau, Cadre, and Pelletier (2008). In particular, we assume throughout that the density f iscontinuouson R^d. Thus, we are in the case of a non-sharp boundary, i.e., f decreases continuously to zero at the boundary of its support. The case where f has sharp boundary requires a different approach (see for example Härdle, Park, and Tsybakov, 1995). The analytical assumptions on f (Assumption Set 2) are stipulations on the local behavior of f at the boundary of the support. In particular, the restrictions on f imply that inside the support and close to the boundary the mapsu7→ f(p+ue_p), with p∈∂S_f, are strictly convex (see the Appendix).

2.2 Main result Let

σ²_f =2^d Z

∂Sf

Z _∞

0

Z

B(0,1)

Φ(p,t,u)dudt v_σ(dp), (2.3) with

Φ(p,t,u) =exp

−ωdD_e²

pf(p)t²

exp

β(u)D²_e

pf(p)t² 2

−1

, ωd denoting the volume ofB(0, 1)and

β(u) =λ(B(0, 1)∩ B(2u, 1)).

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Remark 2.1. LetΓbe the Gamma function. We note thatβ(u)has the closed expression (Hall, 1988, p. 23)

β(u) =







2π^(d−¹^)/² Γ₁

2+ ^d₂ Z 1

|u|

(1− y²)^(d−¹^)/²dy, if0≤ |u| ≤1

0, if |u|>1,

which, in particular, gives

β(0) =ωd= π^d^/²

Γ

1+ ^d₂. We are now ready to state our main result.

Theorem 2.1. Suppose that both Assumption Sets 1 and 2 are satisfied. If (r.i) r_n → 0, (r.ii) nr_n^d/(lnn)^4/3→ ∞and (r.iii) nr_n^d+1→0, then

n r_n^d

1/4

λ

S_n4S_f

−Eλ

S_n4S_f _D

→ N(0,σ²_f), whereσ²_f >0is as in (2.3).

Remark 2.2. A referee pointed out that the methods in the paper may be applicable to obtain a central limit theorem for the histogram-based support estimator studied in Baíllo and Cuevas (2006). The reference Cuevas, Fraiman, and Rodríguez-Casal (2007) should be a starting point for such an investi- gation.

It is known from Cuevas and Rodríguez-Casal (2004) that the choice r_n=O((lnn/n)^1/d)gives the fastest convergence rate ofS_n towardsS_f for the Hausdorff metric, that is O((lnn/n)¹^/^d). For such a radius choice, the concentration speed ofλ(Sn∆Sf)around its expectation as given by Theorem 2.1 is O(p

n/(lnn)^1/4), close to the parametric rate.

Theorem 2.1 assumesd≥2 (Assumption 1-(a)). We restrict ourselves to the cased≥2 for the sake of technical simplicity. However, the cased =1 can be derived with minor adaptations, assuming r_n →0, nr_n/(lnn)⁴^/³ → ∞, andnr_n³^/² →0. In fact, the one-dimensional setting has already been explored in the related context of vacancy estimation (Hall, 1984).

As we mentioned in the introduction, the quantityλ(S_n4S_f) is closely related to the vacancy V_n (Hall 1985, 1988), which is defined as in (1.2). A close inspection of the proof of Theorem 2.1 reveals that taking intersection withS_f in the integrals does not effect things too much and, in fact, the asymptotic distributional behaviors ofλ(S_n4S_f)andV_n are nearly identical. As a consequence, we obtain the following result:

Theorem 2.2. Suppose that both Assumption Sets 1 and 2 are satisfied. If (r.i), (r.ii) and (r.iii) hold, then

n r_n^d

1/4

(Vn−EV_n)→ N^D (0,σ²_f), whereσ²_f >0is as in (2.3).

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Surprisingly, the limiting varianceσ²_f remains as in (2.3). Theorem 2.2 was motivated by a remark by Hall (1985), who pointed out that a central limit theorem for vacancy in the case nr_n^d → ∞ remained open.

3 Proof of Theorem 2.1

Our proof of Theorem 2.1 will borrow elements from Mason and Polonik (2009).

Set

"n= 1

(nr_n^d)¹^/⁴. (3.1)

Observe that, from (r.ii) and (r.iii), the sequence("n)satisfies (e.i)"n→0 and (e.ii)"n

pnr_n^d→ ∞. For future reference we note that from (r.i) and (r.iii), we get that

r_n

"n

→0. (3.2)

Set

En={x ∈R^d: f(x)≤"_n}. Furthermore, let

L_n("n) = Z

En

1{f_n(x)>0} −1{f(x)>0} dx and

L_n("n) = Z

En^c

1{f_n(x)>0} −1{f(x)>0} dx.

Noting that, under Assumption Set 1,λ(Sn∆Sf) =L_n("n) +L_n("n), our plan is to show that

n r_n^d

1/4

L_n("n)−EL_n("n) _D

→ N(0,σ²_f) (3.3)

and

n r_n^d

1/4

L_n("n)−EL_n("n)

→P 0, (3.4)

which together imply the statement of Theorem 2.1. To prove a central limit theorem for the random variableL_n "n

, it turns out to be more convenient to first establish one for the Poissonized version of it formed by replacing f_n(x)with

πn(x) =

N_n

X

i=1

1_B(_x,r

n)(Xi),

whereN_nis a meannPoisson random variable independent of the sampleX₁, . . . ,X_n. By convention, we setπn(x) =0 wheneverN_n=0. The Poissonized version ofL_n "n

is then defined by Πn("n) =

Z

En

1{πn(x)>0} −1{f(x)>0} dx.

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The proof of Theorem 2.1 is organized as follows. First (Subsection 3.1), we determine the exact asymptotic behavior of the variance of Πn "n

. Then (Subsection 3.2), we prove a central limit theorem forΠn "n

. By means of a de-Poissonization result (Subsection 3.3), we then infer (3.3).

In a final step (Subsection 3.4) we prove (3.4), which completes the proof of Theorem 2.1. This Poissonization/de-Poissonization methodology goes back to at least Beirlant, Györfi, and Lugosi (1994).

3.1 Exact asymptotic behavior of Var(Πn("n)) Let

∆n(x) =

1{πn(x)>0} −1{f(x)>0} .

In the sequel, the letterC will denote a positive constant, the value of which may vary from line to line.

Let("n)be the sequence of positive real numbers defined in (3.1). In this subsection, we intend to prove that, under the conditions of Theorem 2.1,

n→∞lim rn

r_n^dVar Πn("n)

=σ²_f, (3.5)

whereσ²_f is as in (2.3).

Towards this goal, observe first that Πn("n) = Z

E˜n

1{πn(x)>0} −1{f(x)>0} dx, where we set

E˜n=En∩S^r_fⁿ, with

S^r_fⁿ=¦

x∈R^d : dist(x,S_f)≤r_n© . Clearly,

Var(Π_n "_n ) =

Z

E˜n

Z

E˜n

C ∆_n(x),∆_n(y) dxdy,

where here and elsewhereCdenotes ‘covariance’. Since ∆n(x)and∆n(y)are independent when- everkx− yk>2r_n, we may write

Var Π_n("_n)

= Z

E˜n

Z

E˜n

1

kx−yk ≤2r_n C ∆_n(x),∆_n(y) dxdy.

Using the change of variable y =x+2r_nu, we obtain Var(Πn "n

)

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=2^dr_n^d Z

R^d

Z

R^d

1E˜n(x)1E˜n(x+2r_nu)1_B(0,1)(u)C ∆_n(x),∆_n(x+2r_nu) dxdu.

By construction, whenevernis large enough, ˜En is included in the tubular neighborhoodV(∂S_f,ρ) of∂S_f of radiusρ >0. In this case, each x ∈E˜n may be written as x = p+ve_p as described in (2.1). Hence, for all large enoughn, we obtain

Var(Πn "n

)

=2^dr_n^d Z

∂Sf

Z ρ

−rn

Z

B(0,1)

1E˜n(p+ve_p)1E˜n(p+ve_p+2r_nu)

×Θ(p,v)C

∆n(p+ve_p),∆n(p+ve_p+2r_nu)

dudv v_σ(dp).

For allp∈∂S_f, letκp("n)be the distance between pand the point x of the set{x ∈R^d:f(x) ="n} such that the vectorx−pis orthogonal to∂S_f. Using the change of variable v=t/p

nr_n^d, we may write

Var Πn("n)

= 2^dr_n^d pnr_n^d

Z

∂S_f

Z

pnr_n^dκp("n)

−p nr_n^d+2

Z

B(0,1)

1E˜n



p+ t pnr_n^d

e_p+2r_nu



Θ



p, t pnr_n^d





×C



∆_n



p+ t pnr_n^d

e_p



,∆_n



p+ t pnr_n^d

e_p+2r_nu







dudt v_σ(dp).

For a justification of this change of variable, refer to equation (2.2) and equation (4.2) in the Ap- pendix. By conditions (r.i) and (r.iii),nr_n^d+2→0. Consequently,

r n

r_n^d Var Πn("n)

=o(1) +2^d

Z

∂S_f

Z

pnr_n^dκp("n)

0

Z

B(0,1)

1_E_˜

n



p+ t pnr_n^d

se_p+2r_nu



Θ



p, t pnr_n^d





×C



∆n



p+ t pnr_n^d

e_p



,∆n



p+ t pnr_n^d

e_p+2r_nu







dudt v_σ(dp). (3.6) To get the limit asn→ ∞of the above integral, we will need the following lemma, whose proof is deferred to the end of the subsection.

Lemma 3.1. Let p∈∂S_f, t>0and u∈ B(0, 1)be fixed. Suppose that the conditions of Theorem 2.1 hold. Then

nlim→∞C



∆_n



p+ t pnr_n^d

e_p



,∆_n



p+ t pnr_n^d

e_p+2r_nu







= Φ(p,t,u), whereΦ(p,t,u)is defined in Theorem 2.1.

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Returning to the proof of (3.5), we notice that by (4.4) in the Appendix and (e.ii) we have pnr_n^dκ_p("_n) → ∞ as n → ∞ and Θ(p, 0) = 1. Therefore, using Lemma 3.1 and the fact that for allt>0 andu∈ B(0, 1)

1E˜n



p+ t

pnr_n^d +2r_nu



→1 asn→ ∞,

we conclude that the function inside the integral in (3.6) converges pointwise toΦ(p,t,u)asn→ ∞. We now proceed to sufficiently bound the function inside the integral in (3.6) to be able to apply the Lebesgue dominated convergence theorem. Towards this goal, fix p ∈ ∂S_f, u∈ B(0, 1) and 0< t ≤ p

nr_n^dκp("n). Since∆n(x) ≤ 1 for all x ∈R^d, using the inequality |C(Y1,Y₂)| ≤ 2E|Y₁| whenever|Y₂| ≤1, we have

C



∆_n



p+ t pnr_n^d

e_p



,∆_n



p+ t pnr_n^d

e_p+2r_nu









≤2E∆n



p+ t pnr_n^d

e_p



. (3.7)

By the bound in (4.3) in the Appendix, we see that sup

p∈∂Sf

κp("n)→0 as n→ ∞. (3.8) Then, since e_p is a normal vector to ∂S_f at p which is directed towards the interior ofS_f, there exists an integerN₀ independent of p, t andusuch that, for alln≥N₀, the pointp+ (t/p

nr_n^d)e_p belongs to the interior ofS_f. Therefore, f(p+ (t/p

nr_n^d)e_p)>0 and, letting ϕ_n(x) =P X ∈ B(x,r_n)

, we obtain

E∆n



p+ t pnr_n^d

e_p



=P



πn



p+ t pnr_n^d

e_p



=0





=E



P



∀i≤N_n : X_i∈ B/



p+ t pnr_n^d

e_p,r_n





N_n









=E



1−ϕn



p+ t pnr_n^d

e_p









N_n

=exp



−nϕn



p+ t pnr_n^d

e_p







, (3.9)

where we used the fact thatN_n is a meannPoisson distributed random variable independent of the sample. By a slight adaptation of the proof of Lemma A.1 in Biau, Cadre, and Pelletier (2008) and

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under Assumption 1-(b), one deduces that for all x ∈ V(∂S_f,ρ)∩S^◦_f, there exists a quantityK_n(x) such that

ϕn(x) =r_n^dωdf(x) +r_n^d⁺²K_n(x) and sup

n

sup

x∈V(∂S_f,ρ)∩S^◦_f

|K_n(x)|<∞. (3.10) For all x inV(∂S_f,ρ)written as x= p+ue_pwithp∈∂S_f and 0≤u≤ρ, a Taylor expansion of f atpgives the expression

f(x) = 1 2D²_e

pf(p+ξe_p)u², for some 0≤ξ≤usince, by Assumption 1-(b), D_e

pf(p) =0. Thus, in our context, expanding f at p, we may write

nϕn



p+ t pnr_n^d

e_p



=ωdD_e²

pf(p+ξep)t²

2 +nr_n^d+2R_n(p,t), for some 0≤ξ≤κp("n), and whereR_n(p,t)satisfies

sup

n

sup

§

R_n(p,t)

:p∈∂S_f and 0≤t ≤Æ

nr_n^dκ_p("_n)ª

<∞.

Furthermore, by (3.8), each pointp+ξep falls in the tubular neighborhoodV(∂S_f,ρ)for all large enough n. Consequently, by Assumption 2-(c) there existsα > 0 independent of nand N₁ ≥ N₀ independent ofp, t andusuch that, for alln≥N₁,

p∈∂infS_fD²_e

pf(p+ξe_p)>2α.

This, together with identity (3.9) and (r.i), (r.iii), which implynr_n^d+2→0, leads to

E∆_n



p+ t pnr_n^d

e_p



≤Cexp(−ω_dαt²) (3.11)

forn≥N₁ and all

0≤t≤Æ

nr_n^d sup

p∈∂Sf

κ_p("_n).

Thus, using inequality (3.11), we deduce that the function on the left hand side of (3.7) is dominated by an integrable function of(p,t,u), which is independent ofnprovidedn≥N₁. Finally, we are in a position to apply the Lebesgue dominated convergence theorem, to conclude that

n→∞lim r n

r_n^d Var Πn("n)

=2^d Z

∂Sf

Z _∞

0

Z

B(0,1)

Φ(p,t,u)dudt v_σ(dp) =σ²_f.

To be complete, it remains to prove Lemma 3.1.

Proof of Lemma 3.1 Let x_n = p+ (t/p

nr_n^d)ep. Since nr_n^d → ∞ and nr_n^d⁺² →0, both x_n and x_n+2r_nulie in the interior ofS_f for all large enough n. As a consequence, f(xn)>0 and f(xn+ 2r_nu)>0 for all large enoughn. Thus,

C ∆_n(x_n),∆_n(x_n+2r_nu)

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=C 1{πn(xn) =0},1{πn(xn+2r_nu) =0}

=P πn(x_n) =0,πn(x_n+2r_nu) =0

−P πn(x_n) =0

P πn(x_n+2r_nu) =0

=P ∀i≤N_n : X_i∈ B/ (x_n,r_n)∪ B(x_n+2r_nu,r_n)

−P ∀i≤N_n : X_i∈ B/ (x_n,r_n)

P ∀i≤N_n : X_i ∈ B(/ x_n+2r_nu,r_n)

=exp

−nµ B(x_n,r_n)∪ B(xn+2r_nu,r_n)

−exp

−nµ(B(x_n,r_n))−nµ(B(x_n+2r_nu,r_n)) ,

whereµdenotes the distribution ofX. LetB_n=B(x_n,r_n)∩ B(x_n+2r_nu,r_n). Using the equality µ B(x_n,r_n)∪ B(x_n+2r_nu,r_n)

=ϕ_n(x_n) +ϕ_n(x_n+2r_nu)−µ(B_n), we obtain

C ∆n(xn),∆n(xn+2r_nu)

(3.12)

=exp

−n ϕn(x_n) +ϕn(x_n+2r_nu) exp nµ(B_n)

−1 . Now,µ(B_n)may be expressed as

µ(Bn) = f(xn)λ(Bn) + Z

B_n

f(v)−f(xn) dv.

Since f is of classC¹onR^d, by developingf at x_n in the above integral, we obtain Z

B_n

f(v)−f(x_n)

dv=r_n^d⁺¹R_n, whereR_nsatisfies

R_n

≤Csup

K kgradfk,

and K is some compact subset of R^d containing ∂S_f and of nonempty interior. Next, note that λ(Bn) =r_n^dβ(u), where

β(u) =λ(B(0, 1)∩ B(2u, 1)).

Therefore, expanding f atpin the directione_p, we obtain µ(Bn) =β(u)t²

2nD²_e

pf

p+ξ t pnr_n^d

e_p

+r_n^d+1R_n,

whereξ∈(0, 1). Hence by (r.iii),

nlim→∞nµ(Bn) =β(u)D²_e

pf(p)t² 2.

The above limit, together with identity (3.12) and (3.10), leads to the desired result.

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3.2 Central limit theorem forΠn("n)

In this subsection we establish a central limit theorem forΠn("n). Set S_n("_n) = a_n Πn("n)−EΠn("n)

σn

, wherea_n= (n/r_n^d)^1/4 and

σ²_n=Var

a_n Π_n("_n)−EΠ_n("_n) . We shall verify that asn→ ∞

S_n("n)→ N^D (0, 1). (3.13) To show this we require the following special case of Theorem 1 of Shergin (1990).

Fact 3.1. Let(X_i,n :i∈Z^d) denote a triangular array of mean zero m-dependent random fields, and letJn⊂Z^d be such that

(i) Var P

i∈JnX_i,n

→1as n→ ∞, and (ii) For some2<s<3,P

i∈JnE|X_i,n|^s→0as n→ ∞.

Then X

i∈Jn

X_i,n→ N^D (0, 1).

We use Shergin’s result as follows. Recall the definition of"nin (3.1) and also that Var(Π_n "_n

) = Z

E˜n

Z

E˜n

C ∆_n(x),∆_n(y) dxdy, with

E˜n=En∩S^r_fⁿ. Next, consider the regular grid given by

A_i= (x_i₁,x_i₁₊₁]×. . .×(xi_d,x_i_d₊₁], where i=(i₁, . . . ,i_d),i₁, . . . ,i_d∈Zandx_i=i r_n fori∈Z. Define

R_i=A_i∩E˜n.

WithJn={i∈Z^d:A_i∩E˜n6=; }we see that{R_i: i∈ Jn}constitutes a partition of ˜En. Note that for eachi∈ Jn,

λ R_i

≤r_n^d. We claim that for all largen

Card(Jn)≤Cp"nr_n⁻^d. (3.14)

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To see this, we use the fact that, according to (4.3), there exists ¯ρ > 0 such that for all large n, E˜n⊂ V

∂S_f, ¯ρp"n

. Thus, sincer_n/p"n→0 by (3.2), [

i∈Jn

A_i⊂ V

∂S_f,(ρ¯+2)p"n

and, consequently,

r_n^dCard(Jn)≤λ V

∂S_f,(ρ¯+2)p"n

≤Cp"n.

Keeping in mind the fact that for any disjoint setsB₁, . . . ,B_kinR^d such that, for 1≤i6= j≤k, inf¦

>r_n,

then Z

B_i

∆n(x)dx,i=1, . . . ,k, are independent, we can easily infer that

X_i,n= a_n

Z

Ri

∆n(x)−E∆n(x) dx σn

, i∈ Jn, constitutes a 1-dependent random field onZ^d.

Recalling thata_n= (n/r_n^d)¹^/⁴ andσ²_n→σ²_f asn→ ∞by (3.5) we get, for alli∈ Jn,

X_i,n ≤ a_n

σn

λ(R_i)≤C(nr_n^3d)¹^/⁴. Hence, by (3.14),

X

i∈Jn

E|X_i,n|⁵^/²≤C Card(Jn)

(nr_n^3d)⁵^/⁸≤C(nr_n^3d/²)¹^/².

Clearly this bound when combined with (r.iii) andd≥2, gives asn→ ∞, X

i∈Jn

E|X_i,n|⁵^/²→0,

which by the Shergin Fact 3.1 (withs=5/2) yields S_n "n

= X

i∈Jn

X_i,n→ N^D (0, 1). Thus (3.13) holds.

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3.3 Central limit theorem for L_n("n)

Now we shall de-Poissonize the central limit forΠn("n)to obtain one forL_n "n

. Observe that (S_n("n)|N_n=n)=^D a_n L_n("n)−EΠn("n)

σn

. (3.15)

Our next goal is to apply the following version of a theorem in Beirlant and Mason (1995) (see also Polonik and Mason, 2009) to infer from (3.13) that

a_n L_n("n)−EΠn("n) σn

→ ND (0, 1). (3.16)

Fact 3.2. Let N_1,n and N_2,n be independent Poisson random variables with N_1,n being Poisson(nβn) and N_2,n being Poisson(n(1−βn))whereβn∈(0, 1). Denote N_n=N_1,n+N_2,n and set

U_n= N_1,n−nβn

pn and V_n= N_2,n−n(1−βn) pn . Let(S_n)be a sequence of real-valued random variables such that

(i) For each n≥1, the random vector(Sn,U_n)is independent of V_n. (ii) For someσ²<∞, S_n→^D σZ as n→ ∞.

(iii) β_n→0as n→ ∞. Then, for all x,

P(Sn≤x |N_n=n)→P(σZ≤x). Let

Dn={x ∈R^d: f(x)≤2"n}. We shall apply Fact 3.2 toS_n("_n)with

N_1,n=

N_n

X

i=1

1{X_i∈ Dn}, N_2,n=

N_n

X

i=1

1{X_i∈ D/ n} andβn=P(X ∈ Dn). Let

M= sup

x∈R^d

Xd

i=1

∂f (x)

∂x_i .

We see that for all large enoughn, wheneverx ∈ Enandy ∈ B x,r_n

, by the mean value theorem, f(y)≤ f(x) +M r_n≤"_n

1+ M r_n

"n

. This combined with (3.2) implies for all largen





 [

x∈En

B(x,r_n)





∩ D_n^c=∅.

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Therefore for all large enough n, the random variables S_n("n)and N_2,n are independent. Thus by (3.15) andβn→0, we can apply Fact 3.2 to conclude that (3.16) holds.

Next we proceed just as in Mason and Polonik (2009) to apply a moment bound given in Lemma 2.1 of Giné, Mason, and Zaitsev (2003) to show that

E a_n L_n("n)−EΠn("n) ²≤2σ²_n. Therefore, since by (3.5),

σ²_n→σ²_f <∞,

the sequence(a_n(L_n("n)−EΠn("n)))is uniformly integrable. Hence we get using (3.16) that a_n EL_n("_n)−EΠ_n("_n)

→0.

Thus, still by (3.16),

a_n L_n("n)−EL_n("n) σn

→ ND (0, 1). This in turn implies (3.3).

3.4 Completion of the proof of Theorem 2.1

It remains to verify (3.4). Observe that L_n("n) =

Z

E_n^c

1{f_n(x)>0} −1{f(x)>0} dx =

Z

E_n^c

1{f_n(x) =0}dx. To begin with note that for allx ∈ E_n^c,

P(f_n(x) =0) = 1−ϕn(x)n

≤exp −nϕn(x) , where we recall that

ϕn(x) =P X ∈ B(x,r_n) . Since f is of classC¹, we have, for allt∈ B(x,r_n),

|f(t)−f(x)| ≤κ1r_n,

where κ1 > 0 is independent of x. Therefore, using the properties f(x) ≥ "n and r_n/"n →0 by (3.2), we obtain

ϕn(x) =P X∈ B(x,r_n)

= f(x)ωdr_n^d+ Z

B(x,r_n)

f(t)−f(x) dt

≥"nr_n^d

ωd−κ1

r_n

"_n

≥κ2"nr_n^d,