A Central Limit Theorem for the Volumes of High Excursions of Stationary Associated Random Fields

We prove that under certain conditions the excursion sets volumes of stationary positively associated random fields converge after rescaling to the normal distribution as the excursion level and the size of the observation window grow. In addition, we provide a number of examples.


Introduction
The study of excursion set properties plays an increasingly important role within modern theory of random fields.Such objects arise in connection to a wide range of stochastic models (see, e.g., [1,2]).Recently a number of works appeared ( [3,4,5]) focused on asymptotic properties of the excursion sets volumes corresponding to fixed excursion levels and sequences of growing (in a certain sense) observation windows.Spodarev gives in [6] an overview of the recent asymptotic results concerning the geometry of excursion sets of stationary associated random fields.
In this paper, we show that under certain conditions the excursion sets volumes of stationary positively associated random fields converge after rescaling to the normal distribution as both the excursion level and the size of the observation window grow.Note that a similar model was studied in the monograph of Ivanov and Leonenko [7] for the case of a Gaussian random field.For associated random fields on lattices the asymptotic behaviour of excursion sets cardinalities corresponding to a growing excursion level was examined in [8].
The present paper is organized as follows.In section 2 we provide the necessary definitions.Section 3 contains the main theorem.In section 4 we show how its conditions can be verified for Gaussian random fields, fields with regularly varying tails and shot-noise random fields.

Preliminaries
We assume that all random objects are defined on a complete probability space (Ω, F, P).
Let ∥ • ∥ and ∥ • ∥ ∞ be, respectively, the Euclidean norm and the maximum norm in R d and O(•) -the Landau notation.

Central Limit Theorem
Let X be a measurable strictly stationary positively associated random field on R d .We also assume that X is square integrable and its covariance function r(t) = Cov(X 0 , X t ), t ∈ R d , is continuous.Note that due to the latter condition X is also associated (see [10] for the definition of association and related dependence types).
For any u ∈ R and bounded measurable B, C ⊂ R d Fubini's theorem implies Consider an increasing sequence of excursion levels n ∈ N, and let σ 2 n = Var S n .Suppose that the random variable X 0 has a bounded density f (•) and set γ(x) = sup t x f (t), x ∈ R.

Theorem 1
Assume that the random field X satisfies the following conditions: where d −→ denotes convergence in distribution.
The following lemma can be proved in the same way as Lemma 7.3.4from [10].

Lemma 1
Under the conditions of Theorem 1 Proof of Theorem 1.One can find a sequence m n → ∞, n → ∞, such that m d n /σ n → 0 and To see this, consider Set Using ( 1), (A1) and ( 2), we also get where Approximating integrals with finite sums in L 1 (see [4]) it is not difficult to show that, since the random field X is positively associated and its covariance function is continuous, Before we continue with the proof of the theorem, we establish the following lemma.

Lemma 2
Under the conditions of Theorem 1 it holds that Proof.Due to (1), the variance of Z n is not greater than the variance of S n .Thus, it suffices to prove that Observe that for n > N 0 Below we show that Σ 1 and Σ 2 admit the following estimates Here, and in the following, C 1 , C 2 , C 3 , . . .are some positive real numbers which may only depend on d, r and µ.Applying ( 6) and ( 7), we have Due to (3) and ( 4), the latter inequality implies (5).Now let us obtain (6).It can be easily seen that (1) yields Thus, where Using ( 2) and (A1), we get Hence, the desired inequality holds for C 2 = 1/C 6 .To conclude the proof of the lemma, it remains to note that the method we used to estimate R 2 also yields (7).Lemma 2 is proved.

Random Fields with Regularly Varying Tails
Let X = {X t , t ∈ R d } be a measurable stationary square integrable positively associated random field.Assume that its covariance function r(•) belongs to the C 2 (R d ) class and fulfills (A1).We also assume that X 0 has a bounded density.Let F be the distribution function of X 0 .Suppose that it admits, for some α > 0, the following representation where L(•) is a slowly varying function on (0, ∞), i.e. 1 − F (•) is regularly varying in the sense of Karamata (see [11]).We will assume α > 2. Note that for every ε > 0 Since r(•) attains its maximum at zero, ∇r(0) = 0. Therefore, using Taylor's expansion, one can find b > 0 and δ ∈ (0, 1/2) such that We show that under certain restrictions on the growth rate of u n , n → ∞, X satisfies the conditions of Theorem 1.
We need to estimate the following covariance from below Clearly, Using Markov's inequality, (12) and the fact that 2(r(0) − r(t)) = E(X 0 − X t ) 2 , we obtain Applying (11), it is easy to show that for arbitrary ε > 0 Hence, for sufficiently large n and ∥t∥ u for any ν > 0 and some A = A(L, α, ν) > 0.
Therefore, for any ε, ν > 0 and large enough n, it holds that Consequently, Since the density of X 0 is bounded, we can always find c 0 such that We have where , where f is the density of X 0 and L 1 is a slowly varying function on (0, ∞), then c can be taken to be any nonnegative number less than α + 1.Consequently,

Shot-Noise Random Fields
In this section, the requirements of Theorem 1 are checked for certain shot-noise random fields and for the excursion level where β ∈ (0, 1).The following definition is taken from [12].Let ϕ be a stationary Poisson counting measure with intensity λ > 0. Also, let g : R → R be a deterministic function with ∫ R g(x)dx < ∞ and ∫ R g 2 (x)dx < ∞.The random field X = {X t , t ∈ R} defined by is called a shot-noise random field.The function g is called a response function.
Here we consider shot-noise random fields with the intensity λ > 1/2 and the response function In this case, X is square integrable and for any µ > 3 (A1) is fulfilled.The characteristic function φ X0 (•) of X 0 is (see [10,Lemma 1.3.7])given by It is not difficult to show that for λ > 1/2 φ X0 (•) is integrable and, consequently, X 0 has a bounded density.
In addition, one can show that the distribution of X 0 is infinitely divisible (see, e.g., [13]) with Lévy measure In order to find an approximation for the distribution of X 0 , we consider upper records.Let {Z n : n ∈ N} be a sequence of independent and identically distributed observations.The observation Z j , j ∈ N, is called an upper record, if its value exceeds all previous observations, that is Z j > max{Z 1 , Z 2 , . . ., Z j−1 }.Since the distribution of X 0 is infinitely divisible and the Lévy measure ψ has a density l on [0, ∞) with ∫ ∞ 0 l(y) dy = ∞, it follows from [14, Theorem 2] that (n + 1) − T n −→ X 0 a.s., where T n , n ∈ N, is the sum of the first (n + 1) upper records from a certain distribution H.Moreover, the density of H is known and, in our case, is given by Thus, H is the distribution function of a Beta-distributed random variable with parameters 2λ and 1.Such Beta records are considered in [15].There it is shown that where U j , j ∈ Z + , are independent random variables distributed uniformly on (0, 1).
Vervaat showed in Theorem 4.7.7 and Lemma 4.7.9] that the density f (x) of the random variable in the right-hand side of ( 14) (which is also the density of X 0 ) is nonincreasing, for sufficiently large x, and that To check the requirements of Theorem 1, the following result from [17] is used.
Let N = N(R) be the space of integer-valued σ-finite measures κ on R and let N be the smallest σ-algebra making the mappings κ → κ(B), κ ∈ N, measurable for all B ∈ B(R), where B(R d ) is the Borel σ-algebra of R d , d ∈ N.For a measurable ψ : N → R define D y ψ(κ) = ψ(κ + δ y ) − ψ(κ), κ ∈ N, where δ y is the Dirac measure at y ∈ R. For k ∈ N and (y 1 , . . ., y k ) ∈ R k set D k y1,...,y k ψ = D 1 y1 (D k−1 y2,...,y k ψ), where D 1 = D and D 0 ψ = ψ.Define R k ψ(y 1 , . . . ,y k ) = E D k y1,...,y k ψ(ϕ), if this expectation exists, where ϕ appears in (13).In [17,Theorem 1.1] it is shown that for any ψ, η ∈ L 2 (N, P ϕ ), where P ϕ is the distribution of ϕ in (N, N ), the following relation holds Here, v k is the Lebesgue measure on (R k , B(R k )), k ∈ N. Define ψ n (ϕ) = ∫ [0,n) 1I(X t u n ) dt.In order to establish (A2), it is sufficient to show that and Lemma 2 the terms Sn−E Sn √ Var Sn and Sn−E Sn √ Var Zn have the same limiting distribution as n → ∞.Thus, it suffices to show that Sn−E Sn √ Var Zn has the same limiting distribution as Zn−E Zn √ Var Zn and that Zn−E Zn √ Var Zn converges in distribution to the standard Gaussian law as n → ∞.Applying Newman's inequality (cf.[10, Corollary 1.5.5]),we get