Abstract We consider the problem of numerical approximation of integrals ofrandom fields over a unit hypercube. We use a stratified Monte Carlo quadrature andmeasure the approximation performance by the mean squared error. The quadratureis defined by a finite number of stratified randomly chosen observations with thepartition generated by a rectangular grid (or design). We study the class of locallystationary random fields whose local behaviour is like a fractional Brownian fieldin the mean square sense and find the asymptotic approximation accuracy for asequence of designs for large number of the observations. For the Hölder classof random functions, we provide an upper bound for the approximation error.Additionally, for a certain class of isotropic random functions with an isolatedsingularity at the origin, we construct a sequence of designs eliminating the effectof the singularity point.
Keywords Numerical integration · Random field · Sampling design ·Stratified sampling · Monte Carlo methods
AMS 2000 Subject Classifications 60G60 · 65D30
1 Introduction
Let X(t), t ∈ [0, 1]d, d ≥ 1, be a continuous random field with finite second moment.We consider the problem of numerical approximation of the integral of X over the
K. Abramowicz (B) · O. SeleznjevDepartment of Mathematics and Mathematical Statistics,Umeå University, 90187 Umeå, Swedene-mail: [email protected]
unit hypercube using finite number of observations. The approximation accuracy ismeasured by the mean squared error. We use a stratified Monte Carlo quadrature(sMCQ) for the integral approximation introduced for deterministic functions byHaber (1966). The quadrature is defined by stratified random observations with thepartition generated by a rectangular grid (or design). We use cross regular sequencesof designs, generalizing the well known regular sequences pioneered by Sacks andYlvisaker (1966). We focus on random fields satisfying a local stationarity conditionproposed for stochastic processes by Berman (1974) and extended for random fieldsin Abramowicz and Seleznjev (2011a). Approximation of random functions fromthis class is studied in, e.g., Seleznjev (2000); Hüsler et al. (2003); Abramowiczand Seleznjev (2011a, b). For quadratic mean (q.m.) continuous locally stationaryrandom functions, we derive an exact asymptotic behavior of the approximationaccuracy. We propose a method for the asymptotically optimal sampling point dis-tribution between the mesh dimensions. We also study optimality of grid allocationalong coordinates and provide asymptotic optimality results in the one-dimensionalcase. For q.m. continuous fields satisfying a Hölder type condition, we determine anupper bound for the approximation accuracy. Furthermore, we investigate a certainclass of random fields with different q.m. smoothness at the origin (isolated singu-larity), and construct sequences of designs eliminating the effect of the singularitypoint.
Approximation of integrals of random functions is an important problem arisingin many research and applied areas, like environmental and geosciences (Ripley2004), communication theory and signal processing (Masry and Vadrevu 2009).Regular sampling designs for estimating integrals of stochastic processes are studiedin Benhenni and Cambanis (1992). Random designs of sampling points, includingstratified sampling for stochastic processes, are investigated in Cambanis and Masry(1992); Schoenfelder and Cambanis (1982). Multivariate numerical integration ofrandom fields satisfying Sacks-Ylvisaker conditions is studied in Ritter et al. (1995).Ritter (2000) contains a survey of various random function approximation andintegration problems.
The paper is organized as follows. First we introduce a basic notation. In Section 2,we consider a stratified Monte Carlo quadrature for continuous random fields whichlocal behavior is like a fractional Brownian field in the mean square sense. We derivean exact asymptotics and a formula for the optimal interdimensional sampling pointdistribution. Further, we provide an upper bound for the approximation accuracyfor q.m. continuous fields satisfying Hölder type conditions. In the second part ofthis section, we study random fields with an isolated singularity at the origin andconstruct sequences of designs eliminating the effect of the singularity. In Section 3,we present the results of numerical experiments, while Section 4 contains the proofsof the statements from Section 2.
1.1 Basic Notation
Let X = X(t), t ∈ D := [0, 1]d, d ≥ 1, be a random field defined on a probabilityspace (�,F , P). Assume that for every t, the random variable X(t) lies in thenormed linear space L2(�) = L2(�,F , P) of random variables with finite secondmoment and identified equivalent elements with respect to P. We set ||ξ || := (
Eξ 2)1/2
Methodol Comput Appl Probab
for all ξ ∈ L2(�). Let C(D) be the space of q.m. continuous random fields. We areinterested in a numerical approximation of
I(X) =∫
DX(t)dt
by a quadrature based on N observations for X ∈ C(D), in particular X is q.m.bounded, i.e., E(X2(t)) ≤ K, t ∈ D, and the integral I(X) exists.
For k ≤ d, let l = (l1, . . . , lk) be a vector of positive integers such that∑k
j=1 l j = d,
and let Li := ∑ij=1 l j, i = 1, . . . , k, L0 = 0, be the sequence of its cumulative sums.
Then the vector l defines the l-decomposition of D into D1 × . . . × Dk, with the l j-cube D j = [0, 1]l j , j = 1, . . . , k. For any s ∈ D, we denote by s j the coordinates vectorcorresponding to the j-th component of the decomposition, i.e.,
s j = s j(l) := (sL j−1+1, . . . , sL j) ∈ D j, j = 1, . . . , k.
For a vector α = (α1, . . . , αk), 0 < α j < 2, j = 1, . . . , k, and the decomposition vec-tor l = (l1, . . . , lk), let | s |α := ∑k
j=1
∣∣ s j∣∣α j for all s ∈ D with the Euclidean norms∣∣ s j
∣∣ , j = 1, . . . , k.
1.2 Classes of Random Fields
Now we introduce the classes of random fields considered in this paper.
Definition 1 Let X ∈ C(D). For fixed vectors α, l, and a positive constant C, wedefine the class Cα
l (D, C) of random fields satisfying Hölder condition, and say thatX ∈ Cα
l (D, C) if
|| X(t + s) − X(t) ||2 ≤ C | s |α for all t, t + s ∈ D. (1)
Definition 2 Let X ∈ C(D). For fixed vectors α, l, and a vector function c(t) =(c1(t), . . . , ck(t)), t ∈ D, with positive and continuous functions c1(·), . . . , ck(·), wedefine class Bα
l (D, c(·)) of locally stationary random fields, and write that X ∈Bα
l (D, c(·)) if
|| X(t + s) − X(t) ||2∑k
j=1 c j(t)∣∣ s j
∣∣α j→ 1 as s → 0 uniformly in t ∈ D. (2)
We assume additionally that for j = 1, . . . , k, the function c j(·) is invariant withrespect to permutations of coordinates within the j-th component.
For the classes Cαl and Bα
l , the withincomponent smoothness is defined by thevector α = (α1, . . . , αk). We denote the vector describing the smoothness for eachcoordinate by α∗ = (α∗
1 , . . . , α∗d), where α∗
i = α j, i = L j−1 + 1, . . . , L j, j = 1, . . . , k.Moreover, for one component fields, i.e., k = 1 and α = α, the corresponding Hölderand locally stationary classes are denoted by Cα
d and Bαd , respectively.
Example 1 Let m = (m1, . . . , mk) be a decomposition vector of [0, 1]m, and m =∑kj=1 m j. Denote by Bβ,m(t), t ∈ [0, 1]m, β = (β1, . . . , βk), 0 < β j < 2, j = 1, . . . ,
Methodol Comput Appl Probab
k, an m-dimensional fractional Brownian field with covariance function r(t, s) =12
(| t |β + | s |β − | t − s |β). Then Bβ,m has stationary increments,
∥∥Bβ,m(t + s) −Bβ,m(t)
∥∥2 = | s |β , t, t + s ∈ [0, 1]m, and therefore, Bβ,m ∈ Bβm(D, c(·)) with local sta-
tionarity functions c1(t) = . . . = ck(t) = 1, t ∈ [0, 1]m. In particular, if k = 1, thenBβ,m(t), t ∈ [0, 1]m, 0 < β < 2, m ∈ �, is an m-dimensional fractional Brownian fieldwith covariance function
r(t, s) = 12
(| t |β + | s |β − | t − s |β), t, t + s ∈ [0, 1]m. (3)
1.3 Cross Regular Designs
Let the hypercube D be partitioned into hyperrectangular strata by design (grid)points TN := {ti = (t1,i1 , . . . , td,id) : i = (i1, . . . , i j), 0 ≤ i j ≤ n∗
j , j = 1, . . . , d}, where
d∏
j=1
n∗j(N) = N. (4)
Since optimal designs for a fixed N are difficult to construct, for asymptoticalresults we develop the approach introduced by by Sacks and Ylvisaker (1966) forsome time series models (for approximation problems, see e.g., Seleznjev 2000).Let h∗
j(s), s ∈ [0, 1], j = 1, . . . , d, be positive and continuous density functions, calledwithindimensional densities, and
h∗(t) := (h∗1(t1), . . . , h∗
d(td)), t ∈ [0, 1]d.
We define the interdimensional knot distribution determined by a vector function
π∗(N) := (n∗1(N), . . . , n∗
d(N)), N ∈ �,
where n∗j(N) ∈ �, limN→∞ n∗
j(N) = ∞, j = 1, . . . , d, and Eq. 4 holds. We suppressthe argument N for the sampling grid sizes n∗
j = n∗j(N), j = 1, . . . , d, when doing so
causes no confusion.
Definition 3 For functions h∗(·) and π∗(·), the corresponding cross regular se-quence of sampling designs TN := {ti = (t1,i1 , . . . , td,id) : i = (i1, . . . , id), 0 ≤ i j ≤ n∗
j ,j = 1, . . . , d}, is generated by the equations
∫ t j,i
0h∗
j(v)dv = in∗
j, i = 0, 1, . . . , n∗
j , j = 1, . . . , d.
The introduced classes of random fields have the same smoothness and lo-cal behavior for each coordinate of the components generated by a decomposi-tion vector l. Therefore we use designs with the same within- and interdimen-sional grid distributions within the components. Formally, for the partition gen-erated by a vector l = (l1, . . . , lk), we consider cross regular designs TN , definedby functions h = (h1, . . . , hk) and π(N) = (n1(N), . . . , nk(N)), in the followingway: h∗
i (·) ≡ h j(·), n∗i = n j, i = L j−1 + 1, . . . , L j, j = 1, . . . , k. We call func-
tions h1(·), . . . , hk(·) and π(N) withincomponent densities and intercomponent griddistribution, respectively. The corresponding property of a design TN is denoted by:TN is cRS(h, π, l). If d = 1, then l = 1, π(N) = π1(N) = N, and the cross regular
Methodol Comput Appl Probab
sequences become regular sequences introduced by Sacks and Ylvisaker (1966). Wedenote such property of the design by: TN is RS(h).
For a given cross regular grid design, the hypercube D is partitioned into Ndisjoint hyperrectangular strata Di, i ∈ I, where I := {i = (i1, . . . , id), 0 ≤ ik ≤ n∗
k −1, k = 1, . . . , d}. Let 1d = (1, . . . , 1) and 0d = (0, . . . , 0) denote a d-dimensionalvectors of ones and zeros, respectively. The hyperrectangle Di is determinedby the vertex ti = (t1,i1 , . . . , td,id) and the main diagonal ri := ti+1d − ti, i.e., Di :={t : t = ti + ri ∗ s, s ∈ [0, 1]d
}, where ′∗′ denotes the coordinatewise multiplication,
i.e., for x = (x1, . . . , xd) and y = (y1, . . . , yd), x ∗ y := (x1 y1, . . . , xd yd).
1.4 Stratified Monte Carlo Quadrature
Let |Di| denote the volume of the hyperrectangle Di. For a random field X ∈ C(D),we define a stratif ied Monte Carlo quadrature (sMCQ) on a partition generatedby TN
IN(X, TN) := IN(X, TN(h, π, l)) =∑
i∈I
X(ηi)|Di|,
where ηi, i ∈ I, are independent random variables and independent of X(t), t ∈ D, ηiis uniformly distributed in the hyperrectangle Di, i ∈ I. Such defined quadrature is amodification of a well known midpoint quadrature.
We introduce some additional notation used throughout the paper. For sequencesof real numbers un and vn, we write un � vn if limn→∞ un/vn ≤ 1 and un ∼ vn iflimn→∞ un/vn = 1.
2 Results
Let Bβ,m(t), t ∈ Rm+ , 0 < β < 2, m ∈ N, denote an m-dimensional fractional Brownian
field with covariance function (3). For any u ∈ Rm+ , we denote
bβ,m(u) := 12
∫
[0,1]m
∫
[0,1]m| u ∗ (t − v) |β dtdv. (5)
In the following theorem, we provide an exact asymptotics for the accuracy of asMCQ for locally stationary random fields when cross regular sequences of griddesigns are used.
Theorem 1 Let X ∈ Bαl (D, c(·)) be a random f ield and let I(X) be approximated by
The next theorem presents an asymptotically optimal intercomponent grid distri-bution for a given total number of sampling points N. We define
ρ :=(
k∑
i=1
li
αi
)−1
=(
d∑
i=1
1α∗
i
)−1
, κ :=k∏
j=1
vl j/α j
j ,
where d·ρ is the harmonic mean of the smoothness parameters α∗j , j = 1, . . . , d.
Theorem 2 Let X ∈ Bαl (D, c(·)) be a random f ield and let I(X) be approximated by
sMCQ IN(X, TN), where TN is cRS(h, π, l). Then
|| I(X) − IN(X, TN) ||2 � kκρ
N1+ρas N → ∞. (6)
Moreover, for the asymptotically optimal intercomponent grid allocation,
n j,opt ∼ v1/α j
j
κρ/α jNρ/α j as N → ∞, j = 1, . . . , k, (7)
the equality in Eq. 6 is attained asymptotically.
In a general setting, numerical procedures can be used for finding optimaldensities. However, in the one dimensional case, we get the exact formula for thedensity minimizing the asymptotic constant, which is a straightforward consequenceof Theorem 1 and results from Seleznjev (2000).
Proposition 1 Let X ∈ Bα1 ([0, 1], c(·)) be a random process and let I(X) be approxi-
mated by sMCQ IN(X, TN), where TN is RS(h). Then
limN→∞
N1+α || I(X) − IN(X, TN) ||2 = aα
∫ 1
0c(t)h(t)−(1+α)dt.
The density minimizing the asymptotic constant is given by
hopt(t) = c(t)γ∫ 1
0 c(τ )γ dτ, t ∈ [0, 1], (8)
where aα := 1/((1 + α)(2 + α)) and γ := 1/(2 + α).
For random fields satisfying the introduced Hölder type condition, we providethe following proposition that gives an upper bound for the accuracy of sMCQ forHölder classes of continuous fields. In addition, we present the intercomponent griddistribution leading to an increased rate of the upper bound.
Proposition 2 Let X ∈ Cαl (D, C) be a random f ield and let I(X) be approximated by
sMCQ IN(X, TN), where TN is cRS(h, π, l). Then
|| I(X) − IN(X, TN) ||2 ≤ CN
k∑
j=1
d j
nα j(9)
for positive constants d1, . . . , dk. Moreover if n j ∼ Nρ/α j , j = 1, . . . , k, then
|| I(X) − IN(X, TN) ||2 = O(N−(1+ρ)
)as N → ∞.
Methodol Comput Appl Probab
The approximation rates obtained in the above proposition are optimal in acertain sense, i.e., the rate of convergence can not be improved in general forrandom functions satisfying Hölder type condition (see, e.g., Wasilkowski 1994). Therate of the upper bound corresponds to the optimal rate of Monte Carlo methodsfor the anisotropic Hölder-Nikolskii class, which is a deterministic analogue of theintroduced Hölder class (see, e.g., Peixin 2005).
2.1 Point Singularity at the Origin
In this subsection, we focus on one component random fields, i.e., k = 1, l = d, α = α,and consider the case of an isolated point singularity at the origin. More precisely,let a random function X(t), t ∈ [0, 1]d, satisfy the smoothness condition (1) withα = β, β ∈ (0, 2), for t ∈ [0, 1]d. In addition, let X be locally stationary, (Eq. 2), withparameter α > β, on any hyperrectangle A ⊂ [0, 1]d\{0d}. We construct sequences ofgrid designs with an asymptotic approximation rate N−(1+α/d).
The definition of cRS for k = 1 gives that n j = N1/d and h∗j(·) = h(·), j = 1, . . . , d,
for a positive and continuous density h(t), t ∈ [0, 1]. For the density h(·), we definethe related distribution functions
H(t) :=∫ t
0h(u)du, G(t) := H−1(t) =
∫ t
0g(v)dv, t ∈ [0, 1],
i.e., G(·) is a quantile function for the distribution H. Moreover, by
g(t) := G′(t) = 1/h(G(t)), t ∈ [0, 1], (10)
we denote the quantile density function. To formulate the forthcoming results, weintroduce additional classes of random functions.
Definition 4 Let X ∈ C(D). For fixed 0 < α < 2, a hyperrectangle A ⊂ D a positivecontinuous function V(t), t ∈ A, and, we define the class Cα
d (A, V(·)) of random fieldssatisfying local Hölder condition, and say that X ∈ Cα
for some t ∈ {t : t = t + s ∗ u, u ∈ [0, 1]d}.In particular, if V(t) = C, t ∈ A, where C is a positive constant, then X is Hölder
continuous;
Definition 5 Let X ∈ C(D). For fixed vector 0 < α < 2 and positive continu-ous functions c(t), V(t), t ∈ [0, 1]d\{0d}, we define the class of random fieldsCBα
d ([0, 1]d\{0d}, c(·), V(·)), and say that X ∈ CBαd ([0, 1]d\{0d}, c(·), V(·)) if X ∈
Cαd (A, V(·)) ∩ Bα
d (A, c(·)) for any hyperrectangle A ⊂ [0, 1]d\{0d}.By definition, we have that V(t) ≥ c(t), t ∈ [0, 1]d\{0d}.
Example 2 Consider a zero mean random field Xα(t), 0 < α < 2, t ∈ [0, 1]d, d ≥ 1,with covariance function r(t, s) = exp (− || t − s ||α). Let Yα,β(t) = || t ||β/2 Xα(t), t ∈[0, 1]d, where 0 < β < α. Then
∣∣∣∣ Yα,β(t + s) − Yα,β(t)
∣∣∣∣2
= (| t + s |β/2 − | t |β/2)2 + 2 | t |β/2 | t + s |β/2 (1 − e−| s |α )
Methodol Comput Appl Probab
and it follows by calculus that Yα,β ∈Cβ
d ([0, 1]d, M) ∩ CBαd ([0, 1]d\{0d},c(·),V(·)) with
M = 3, c(t) = 2 | t |β , and V(t) = β2/4 | t |β−2 + 2.
We say that a positive function f (t), t ∈ �d, satisfies a shifting condition if thereexist positive constants CL < CU , C, and a such that
f (s) ≤ Cf (v) for all s, v such that CL ≤ | s || v | ≤ CU , s, v ∈ [0, a]d\{0d}. (12)
An example of such function is f (t) = | t |α for any 0 < CL < CU < ∞ and α ∈ �. Inthe one-dimensional case, the condition (12) is satisfied, e.g., for any function f (·)which is regularly varying (on the right) at the origin (cf. Abramowicz and Seleznjev2011b).
Let X ∈ Cβ
d ([0, 1]d, M) ∩ CBαd ([0, 1]d\{0d}, c(·), V(·)), 0 < β < α < 2. For
β > α − d, we prove that under some condition on a local Hölder function V(·), thecross regular sequences attain the optimal approximation rate N−(1+α/d). Observethat β > α − d holds for all α, β ∈ (0, 2) if d ≥ 2 and for d = 1 if β > α − 1. DefineH(t) := (H(t1), . . . , H(td)), t ∈ [0, 1]d, and G(t) =: (G(t1), . . . , G(td)), t ∈ [0, 1]d. Weformulate the following condition:
(C) Let V(G(·)) be bounded from above by a function R(·) satisfying the shiftingcondition (12) with CL =1/
√3+d, CU =√
3+d, and such that∫[0,1]d R(H(t))dt<∞.
Theorem 3 Let X ∈ Cβ
d ([0, 1]d, M) ∩ CBαd ([0, 1]d\{0d}, c(·), V(·)), α − d < β < α, be
a random f ield and let I(X) be approximated by sMCQ IN(X, TN), where TN iscRS(h, π, d). If the local Hölder function V(·) satisf ies the condition (C), then
|| I(X) − IN(X, TN) ||2 ∼ 1N1+α/d
∫
Dc(t)bα,d(D1(t))
d∏
m=1
h(tm)−1dt (13)
as N → ∞, where D1(t) = (1/h(t1), . . . , 1/h(td)).
Remark 1 It is possible to show that expression (13) holds under slightly modifiedconditions in case of d = 1 and 0 < β ≤ α − 1, which is not included in the abovetheorem.
3 Numerical Experiments
In this section, we present some examples illustrating the obtained results. Forgiven withindimensional densities, interdimensional distributions, and covariancefunctions, we use numerical integration to evaluate the mean squared error.Denote by
the mean squared error of sMCQ IN(X) with strata generated by the grid TN .We write huni(·) to denote the vector of withincomponent uniform densities. Anal-ogously, by πuni(·) we denote the uniform interdimensional grid distribution, i.e.,n1 = . . . = nk.
Methodol Comput Appl Probab
5 6 7 8 9 10 11 12 13 14−18
−17
−16
−15
−14
−13
−12
−11
−10
−9
−8
−7
log(N)
log(
e N2)
πuniπ
opt
Fig. 1 The (fitted) plots of e2N(πuni) (dashed line) and e2
N(πopt) (solid line) versus N in a log-log scale
Example 3 Let D = [0, 1]3 and X(t) = Bα,l(t), t ∈ [0, 1]3, where α = (3/2, 1/2)
and l = (2, 1). Then X ∈ Bαl ([0, 1]3, c(·)), with c(t) = (1, 1), t ∈ [0, 1]3, k = 2, α∗ =
(3/2, 3/2, 1/2). We compare behavior of eN(πuni) = e2N(X, huni, πuni, l) and
eN(πopt) = e2N(X, huni, πopt, l), where the asymptotically optimal grid distribution
πopt is given by Theorem 2. Figure 1 shows the (fitted) plots of the mean squarederrors e2
N(πuni) (dashed line) and e2N(πopt) versus N (in a log-log scale). These plots
correspond to the following asymptotic behavior:
e2N(πuni) ∼ C1 N−7/6 + C2 N−3/2 ∼ C1 N−7/6,
e2N(πopt) ∼ C3 N−13/10 as N → ∞,
where C1 � 0.26, C2 � 0.20, and C3 � 0.48. Observe that utilizing the asymptoticallyoptimal intercomponent grid distribution leads to an increased rate of convergence.
log(N)
log(
e N2)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−8
−7
−6
−5
−4
−3
−2
−1
0
1
2α1 = 0.5
α2 = 1
α3 = 1.5
Fig. 2 The (fitted) plots of e2N(Xα j), j = 1, 2, 3, for α1 = 1/2 (solid line), α2 = 1 (dashed line), and
α3 = 3/2 (dotted) versus N in a log-log scale
Methodol Comput Appl Probab
Example 4 Consider a random field Xα(t) = √10Yα,1/5, t ∈ [0, 1]2, where Yα,β is
defined in the Example 2. We compare the behavior of the mean squared errorse2
N(Xα j) = e2N(Xα j, πuni, huni), j = 1, 2, 3, with α1 = 1/2, α2 = 1, and α3 = 3/2. The
local Hölder function V(t) = | t |−9/5 + 2, t ∈ [0, 1]2, satisfies the condition (C). Con-sequently by Theorem 3, the sMCQ with cross regular grid sequences attains theconvergence rate N−(1+α j/2), j = 1, 2, 3, respectively, despite the point singularity atorigin. Figure 2 shows the fitted plots of the mean squared errors e2
N(Xα j), j = 1, 2, 3versus N (in a log-log scale).
4 Proofs
Proof of Theorem 1 Let us recall the definitions:
I(X) =∫
DX(t)dt, IN(X) =
∑
i∈I
X(ηi)|Di|,
where ηi is uniformly distributed in the hyperrectangle Di, i ∈ I. Define the error ofnumerical integration
δN(X) := I(X) − IN(X)=∫
DX(t)dt −
∑
i∈I
X(ηi)|Di| =∑
i∈I
∫
Di
(X(t) − X(ηi))dt,
where EηδN(X) = 0. Denote by e2N = e2
N(X) := EδN(X)2 the corresponding meansquared error. Let dX(t, s) := ||X(t) − X(s)||2. By the uniformity and independenceof ηi, i ∈ I, we obtain the following expression for the MSE:
e2N = EδN(X)2 = EXEηδN(X)2 = EXVarη(δN(X))
= EX
(∑
i∈I
Varη
(∫
Di
(X(t) − X(ηi))dt))
= EX
(∑
i∈I
Eη
(∫
Di
(X(t) − X(ηi))dt)2
)
= EX
(∑
i∈I
Eη
(∫
Di
∫
Di
(X(t) − X(ηi))(X(s) − X(ηi))dtds))
= 12
∑
i∈I
Eη
(∫
Di
∫
Di
(dX(t, ηi) + dX(s, ηi) − dX(s, t)
)dtds
)
= 12
∑
i∈I
∫
Di
∫
Di
dX(t, v)dtdv. (14)
Now the local stationarity condition (2) implies that dX(t, v) =(∑kj=1 c j(v)
∣∣ t j − v j∣∣α j
)(1 + qN,i), t, v ∈ Di, and
e2N = 1
2
⎛
⎝∑
i∈I
k∑
j=1
c j(ti)
∫
Di
∫
Di
∣∣ t j − v j∣∣α j dtdv
⎞
⎠ (1 + εN), (15)
Methodol Comput Appl Probab
where by the positiveness and uniform continuity of local stationarity functions,we have that εN = max{|qN,i|, i ∈ I} = o(1) as N → ∞ (cf. Abramowicz and Se-leznjev 2011a). Recall that the hyperrectangle Di is determined by the vertexti = (t1,i1 , . . . , td,id) and the main diagonal ri = ti+1d − ti, i.e.,
Di := {t : t = ti + ri ∗ s, s ∈ [0, 1]d} .
It follows from the definition and the mean (integral) value theorem that
ri = (r1,i1 , . . . , rd,id) =(
1h∗
1(w1,i1)n∗1, . . . ,
1h∗
d(wd,id)n∗d
),
wm,im ∈ [tm,im , tm,im+1], m = 1, . . . , d. (16)
Denote by wi := (w1,i1 , . . . , wd,id). By the definition of cRS(h, π, l), we get
as N → ∞. Applying the uniform continuity of withincomponent densities, weobtain that
e2N = 1
2
( ∑
i∈I
|Di|2k∑
j=1
c j(ti)n−α j
j
∫
D j
∫
D j
∣∣∣ Dj(tji ) ∗ (t
j − v j)
∣∣∣α j
dtjdv j
)(1 + o(1))
=(∑
i∈I
|Di|2k∑
j=1
c j(ti)n−α j
j bα j,l j(Dj(ti))
)(1 + o(1)) as N → ∞,
where bα j,l j(·), j = 1, . . . , k, are defined by Eq. 5. By Eq 16, we have that
|Di| =d∏
m=1
1n∗
mh∗m(wm,im)
= 1N
d∏
m=1
1h∗
m(wm,im)
with wm,im ∈ [tm,im , tm,im+1], i ∈ I, m = 1, . . . , d. Furthermore, the uniform continuityof the withincomponent densities implies
e2N =
(∑
i∈I
|Di| 1N
d∏
m=1
1h∗
m(tm,im)
k∑
j=1
c j(ti)n−α j
j bα j,l j(D(tij))
)(1 + o(1))
=(
1N
k∑
j=1
n−α j
j
∑
i∈I
c j(ti)bα j,l j(D(tij))
d∏
m=1
h∗m(tm,im)
−1|Di|)
(1 + o(1))
Methodol Comput Appl Probab
as N → ∞. Finally, the Riemann integrability of function c j(t)bα j,l j(D(t))∏d
m=1 h∗m(tm)−1 gives
e2N =
(1N
k∑
j=1
n−α j
j
∫
Dc j(t)bα j,l j(D(t j))
d∏
m=1
h∗m(tm)
−1dt)
(1 + o(1))
=(
1N
k∑
j=1
v j
nα j
j
)(1 + o(1))
as N → ∞. This completes the proof. ��
Proof of Theorem 2 The proof is a direct consequence of the inequality for thearithmetic and geometric means (cf. Abramowicz and Seleznjev 2011a, proof ofTheorem 2). ��
Proof of Proposition 2 The main steps of the proof repeat those of Theorem 1, andthe assertion follows when after applying the Hölder conditions (1) to (14) (cf. theproof of Proposition 2, Abramowicz and Seleznjev 2011a), we get
e2N ≤ 1
2C
∑
i∈I
k∑
j=1
∫
Di
∫
Di
||t j − v j||α jdtdv
≤ 12
C∑
i∈I
k∑
j=1
lα j/2j
L j∑
m=L j−1+1
∫
Di
∫
Di
|tm − vm|α jdtdv,
where the second inequality follows from the fact that for any nonnegative numbersa1, . . . , ak and any α ∈ R+, the inequality
(k∑
i=1
ai
)α
≤ kα
k∑
i=1
aαi (17)
holds. ��
Proof of Theorem 3 Observe that, by the continuity of the withincomponent densi-ties and the mean value theorem, we have that
rm,im ≤ C∗m
n∗m
, i ∈ I, m = 1, . . . , d (18)
with C∗m = 1/ mins∈[0,1] h∗
m(s). The first steps of the proof repeat those of Theorem 1.Consider Eq. 14. The MSE can be decomposed as follows:
e2N =
∑
i∈I
(12
∫
Di
∫
Di
dX(t, v)dtdv)
=:∑
i∈I
e2i,N.
For a fixed δ > 0, we denote := [0, δ]d, and I := {i : Di ∩ �= ∅}. Consequently,
e2N =
∑
i∈I
e2i,N = e2
0d,N +∑
i∈I \{0d}e2
i,N +∑
i∈I\I
e2i,N = S1 + S2 + S3,
Methodol Comput Appl Probab
where S1 = S1(N) := e20d,N , S3 = S3(N) includes all terms ei,N such that Di ⊂ D\ ,
and S2 = S2(N) := e2N − S1 − S3. For S1, the Hölder condition and Eq. 18 imply that
e20d,N ≤ C
∫
D0d
∫
D0d
| t − v |β dtdv ≤ Cdβ/2d∑
m=1
∫
D0d
∫
D0d
|tm − vm|βdtdv
≤ C|D0d |2dβ/2aβ
d∑
m=1
rβ
m,0 ≤ C1d1+β/2aβ N−(2+β/d) (19)
for a positive constant C1, where the second inequality follows from Eq. 17. Hencee2
0d,N = o(N−(1+α/d)) for any β ∈ (0, 2), α ∈ (0, 2), if d ≥ 2, and for β > α − 1, if d = 1.For S2 by the local Hölder condition (11), we obtain the following upper bound
S2 =∑
i∈I \{0d}e2
i,N ≤∑
i∈I \{0d}
∫
Di
∫
Di
dX(t, s)dtds
≤∑
i∈I \{0d}V(vi)
∫
Di
∫
Di
| t − s |α dtds
for vi ∈ Di, i ∈ I \{0d}. The continuity of withincomponent grid generating densitiestogether and the definition of function G(·) and condition (C) give
S2 ≤ C1 N−(1+α/d)∑
i∈I \{0d}V(G(wi))|Di| ≤ C1 N−(1+α/d)
∑
i∈I \{0d}R(wi)|Di|,
where C1 is a positive constant and wi ∈ [i1/n∗1, (i1 + 1)/n∗
1] × . . . × [id/n∗d, (id +
1)/n∗d] =: D∗
i . The shifting property (Eq. 12) implies that for a positive constant C2,
S2 ≤ C2 N−(1+α/d)∑
i∈I \{0d}R(si)|Di|, (20)
where si = H(ui) ∈ D∗i is such that
R(si) = R(H(ui)) = minvi∈Di
R(H(vi)), i ∈ I \{0d}.
Consequently by Eq. 20 and condition (C), we have
S2 ≤ C2 N−(1+α/d)
∫
\D0d
R(H(t))dt.
Thus for any ε > 0 and sufficiently small δ by condition (C), we obtain that
N1+α/dS2 ≤ C∫
\D0d
R(H(t))dt < ε. (21)
For S3, similarly to Theorem 1, we get that
N1+α/dS3 =(∫
D\ c(t)bα,d(D1(t))
d∏
m=1
h(tm)−1dt
)
(1 + o(1)) := v1,δ(1 + o(1)) (22)
Methodol Comput Appl Probab
as N → ∞, where D1(t) = (1/h(t1), . . . , 1/h(td)). From the regularity of the within-component density and condition (C) it follows that for a positive constant C1,
∫
Dc(t)bα,d(D1(t))
d∏
m=1
h(tm)−1dt ≤ C1
∫
Dc(t)dt ≤ C1
∫
DV(t)dt
≤ C1
∫
DR(H(t))dt < ∞
and therefore, the monotone convergence gives
v1,δ ↑ v1 =∫
Dc(t)bα,d(D1(t))
d∏
m=1
h(tm)−1dt as δ → 0. (23)
So, for any ε > 0, first we select δ sufficiently small and apply Eqs. 21 and 23. Thenfor the selected δ and sufficiently large N, Eqs. 19 and 22 imply the assertion. Thiscompletes the proof. ��Acknowledgement The second author is partly supported by the Swedish Research Council grant2009-4489 and the project “Digital Zoo” funded by the European Regional Development Fund.
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