Approximation error for general ψ

In this subsection, be ψ a function which fulfills the assumptions of Theorem 3.10, i.e. it is a function in the representation (2.2) with linear independent vectors θ₁, . . . , θ_d ∈ R^d for which ||θ₁||₂ = . . . = ||θ_d||₂ = 1, and 0 < λ₁ ≤ . . . ≤ λ_d. In order to be able to use Corollary 2.6, we also have to assume that 0 < ρ < 2λ₁. Be also assumed that for the parameters A, B and D the inequalities 0 < D ≤ B ≤ ^√1

d and 1 < A hold, and that N := ^B_D and M := ^A_D are integers. As already defined in Lemma 3.9, be also in this subsection C_min := min{C₁, . . . , C_d}, C_max := max{C₁, . . . , C_d}, S₂ := {ξ ∈ R^d: ||ξ||₂ = 1} and θ_ξ := _||ξ||^ξ

2 ∈ S₂. As in Theorem 3.10, be ˜I a short notation for the integral R

S2

Pd

k=1|< θ_ξ, θ_k>|⁻

ρ λ1

^−Hα−q_ρ dθ_ξ.

Remark 3.24. Obviously the first version of discretisation is a special case of the second one, which is obtained by setting N = 1. In this case, the value of B is B = N · D = D.

Therefore, results for the first version can be derived from the results for the second version by setting N = 1 and B = D. Thus, in this subsection the results are calculated for the second version of discretisation, and then the corresponding results for the first version are derived by interpreting this discretisation as a special case of the second one.

Second version of the discretisation

Remark 3.25. Be X_ψ^A,M,N an approximation of X_ψ^A which uses the second version of discretisation (see (3.18)). According to the lemmas 3.14 and 3.19, the error of approxi-mation can be estimated by

||X_ψ^A(x) − X_ψ^A,M,N(x)||^α_α

≤ X

~k∈{−M,...,M −1}^d

Z

∆_~k

 

 ei<x,ξ>− 1
ψ(ξ)^−H−αq − g_~k  

α

dξ

≤I0+ I1 + I2

with

I₀ = ||x||^α₂ · Z

[−B,B]^d

||ξ||^α₂ · ψ(ξ)^−αH−q dξ I1 = 2 · ||x||^α₂ · D^αd^α2 ·

Z

[−A,A]^d\[−B,B]^d

ψ(ξ)^−αH−q dξ I₂ = 2^1+α· X

~k∈JN

Z

∆_~k

 

ψ(ξ)^−H−αq − ψ(ξ_~k)^−H−αq   

α

dξ

Lemma 3.26. Under the given assumptions, the term I₀ can be estimated by Proof. From the assumption B ≤ ^√1

d follows ||ξ||2 ≤ 1 for all ξ ∈ [−B, B]^d. According

holds for these ξ. Using this inequality, we estimate I₀ = ||x||^α₂ ·

Lemma 3.27. Under the given assumptions, and the condition that αH + q − dλ_d6= 0, the term I₁ can be estimated by

I1 ≤ eC1,A,B,x· D^α

for ξ from the latter part. These inequalities are applied in the following calculation:

I1 = 2 · ||x||^α₂ · D^α· d^α2 ·

≤ 2 · ||x||^α₂ · D^α· d^α2 · C

and under the assumption αH + q − dλ_d 6= 0, the exponent of the first integral is

−αH−q+dλ_d

λd − 1 6= −1, so that this integral can be calculated analogously:

Z A√

Remark 3.28. If λ₁ = . . . = λ_d, then dλ₁ = dλ_d= q and I₁ ≤ eC_1,A,B,x· D^α with

Ce_1,A,B,x = 2 · ||x||^α₂ · d^α2 · C

−αH−dλ1 ρ

min · ˜I · λ₁ αH ·

 B

−αH λ1 −

A ·√

d^−αH_λ1  .

Before estimating I₂, some lemmas are shown first which are used to estimate I₂. In the following, the variables J_N,1 and J_N,2 refer to two subsets of the set

J_N = {−M, . . . , M − 1}^d\{−N, . . . , N − 1}^d:

J_N,1:= {~k ∈ J_N : ∆_~k∩ [−1, 1)^d 6= ∅}, J_N,2:= {~k ∈ J_N : ∆_~k∩ [−1, 1)^d = ∅}.

Then J_N,1, J_N,2 are disjoint, J_N,1∪ J_N,2= J_N and

~k ∈ J_N,1 ⇒ ∃ ξ ∈ ∆_~k: ||ξ||∞ ≤ 1

~k ∈ J_N,1 ⇒ ∀ ξ ∈ ∆_~k: ||ξ||∞ ≤ 1 + D

~k ∈ J_N,2 ⇒ ∀ ξ ∈ ∆_~k: ||ξ||∞ ≥ 1

Lemma 3.29. Be

c₁ := C

1 ρ

min· min

θξ∈S2

d

X

k=1

|< θ_ξ, θ_k >|

ρ λ1

!¹_ρ . Then

(a) c1 > 0.

(b) If ||ξ||2 ≥ 1 then ψ(ξ) ≥ c1· ||ξ||

1

∞λd. (c) If ||ξ||₂ ≤ 1 then ψ(ξ) ≥ c₁· ||ξ||

1

∞λ1.

(d) ψ(ξ) ≥ c1· B^λ11 for all ξ ∈ [−A, A]^d\[−B, B]^d. Proof.

(a) The function ˜ψ(x) =  Pd

k=1|< θ_ξ, θ_k >|

ρ λ1

_ρ¹

is a continuous E-homogeneous function (according to Corollary 2.6), and this implies min_θξ∈S2 ˜ψ(ξ)

> 0 (ac-cording to Remark 2.4). Because C₁, . . . , C_d are assumed to be positive, C_min :=

min{C₁, . . . , C_d} is also positive. Thus, c₁ > 0.

(b) If ||ξ||₂ ≥ 1, then ψ(ξ) ≥ C_min^1/ρ · ||ξ||^1/λ₂ ^d· Pd

k=1|< θ_ξ, θ_k >|^ρ/λ11/ρ

(see Lemma 3.9). This implies ψ(ξ) ≥ c1· ||ξ||^1/λ₂ ^d ≥ c1· ||ξ||^1/λ∞ ^d. (c) If ||ξ||₂ ≤ 1, then ψ(ξ) ≥ C_min^1/ρ · ||ξ||^1/λ₂ ¹·

Pd

k=1|< θ_ξ, θ_k>|^ρ/λ11/ρ

(see Lemma 3.9). This implies ψ(ξ) ≥ c₁· ||ξ||^1/λ₂ ¹ ≥ c₁· ||ξ||^1/λ∞ ¹. (d) Be ξ ∈ [−A, A]^d\[−B, B]^d. Then ||ξ||∞≥ B.

If ||ξ||₂ ≤ 1, then ψ(ξ) ≥ c₁· ||ξ||^1/λ∞ ¹ ≥ c₁· B^1/λ1,

and if ||ξ||₂ > 1, then ψ(ξ) ≥ c₁· ||ξ||^1/λ₂ ^d ≥ c₁· 1 ≥ c₁· B^1/λ1.

Lemma 3.30. There is a c₄ > 0 and a δ > 0 (depending on ψ) such that for all ~k ∈ J_N, for all ξ ∈ ∆~k

|ψ(ξ_~k) − ψ(ξ)| ≤ c₄D^{(1/λd−δ)β}

Proof. According to Corollary 2.6, ψ is (β, E)-admissible for a matrix E which fulfills the equation E^Tθ_j = λ_jθ_j for all 1 ≤ j ≤ d (in other words: a matrix E for which the transpose E^T has the eigenvectors θ_j and the corresponding eigenvalues λ_j) and for a real number β with 0 < β < min

λ₁, ρ^λ_λ¹

d



if λ₁ ≤ ρ and β = ρ if λ₁ > ρ. With Definition 2.5, this implies that for any 0 < ˜A < ˜B there exists a ˜C > 0 so that for all x, y ∈ R^d with A ≤ ||y|| ≤ B and τ (x) ≤ 1, the inequality |ψ(x + y) − ψ(y)| ≤ Cτ (x)^β holds. As ξ_~k, ξ ∈ ∆_~k ⇒ ||ξ_~k− ξ||∞ ≤ D and as we are interested in results for small D > 0, it may be assumed that ξ~k and ξ are sufficiently close to each other, and especially that the inequality τ (ξ~k− ξ) < 1 holds. With this assumption, it follows that

|ψ(ξ~k) − ψ(ξ)| = |ψ(ξ + (ξ~k− ξ)) − ψ(ξ)| ≤ ˜Cτ (ξ~k− ξ)^β.

From Lemma 2.1 it follows that there are a norm || · ||0 and constants c2, δ > 0 so that τ (ξ_~k− ξ) ≤ c₂||ξ_~k− ξ||

1 λd−δ

0 . Because all norms are equivalent in R^d, there exists a

constant c₃ > 0 for which ||x||₀ ≤ c₃||x||∞ for all x ∈ R^d. Therefore,

In the following two lemmas, this lemma is used to find estimations for the expression

|ψ(ξ)^−H−qα − ψ(ξ_~k)^−H−αq| in the two cases ~k ∈ J_N,1 and ~k ∈ J_N,2:

(where t(~k) is defined as in the proof of Theorem 3.16).

Proof. Be ~k ∈ J_N,2. Then ||ξ||∞ ≥ 1 ⇒ ||ξ||₂ ≥ 1, and with Lemma 3.29 follows ψ(ξ) ≥ c₁· ||ξ||

1

∞λd (for all ξ ∈ ∆_~k). The existence of c⁰⁰₄, δ > 0 with |ψ(ξ_~k) − ψ(ξ)| ≤ c⁰⁰₄· D^{(1/λd−δ)β} (according to Lemma 3.30) is also used:

|ψ(ξ)^−H−αq − ψ(ξ~k)^−H−αq| (3.14) in the proof of Theorem 3.16.

The preceding lemmas are now used to estimate I₂:

Lemma 3.33. Under the assumptions stated at the beginning of this subsection, and the conditions that −αH − α − q + dλ_d 6= 0 and αH + α + q − dλ_d+ λ_d ≥ 0, there exist constants (i.e. values which are independent of D) eC_2,A,B, δ⁰ > 0 such that

I₂ < eC_2,A,B· D^{αβλd−δ0} · (1 + D)^d.

Proof. According to Remark 3.25 and Lemma 3.19, I₂ = 2^1+α· X

3.31 and 3.32, this term can be transformed as follows:

Analogous to equation (3.16), the sum can be transformed to X

Therefore

(obviously, the L_m are pairwise dis-joint).

Thus the remaining sum in (3.19) can be estimated as follows(similar to the estimation of the sum on page 36,in the proof of Theorem 3.20):

X

=d ·

. In both cases follows

λ_d

≤ 2^1+α+d· c^α₅ · D^{αβλd−δ0} · B^{−αH−q−αλ1} · (1 + D)^d

Theorem 3.34. Be X_ψ^A the approximation of a harmonizable OSSRF by truncation of the domain of integration, and X_ψ^A,M,N a discretisation of this approximation which uses the second type of discretisation (see (3.18)). If the function ψ and the other parameters fulfill the assumptions stated at the beginning of this subsection, as well as also the conditions αλ1 − αH − q + dλ1 > 0, −αH − q + dλd 6= 0, −αH − α − q + dλd 6= 0 and αH + α + q − dλ_d + λ_d ≥ 0, then exist constants δ⁰, eC_0,x (both independent from the parameters A, B and D) and eC_1,A,B,x, eC_2,A,B (independent from D), so that the approximation error due to the discretisation can be estimated by



Proof. According to the lemmas 3.14 and 3.19, the error of approximation can be

esti-mated by

In the lemmas 3.26, 3.27 and 3.33 it has been shown that these terms can be estimated by I₀ ≤ eC_0,x· Bαλ1−αH−q+dλ1

λ1 , I₁ ≤ eC_1,A,B,x· D^α and I₂ < eC_2,A,B· D^{αβλd−δ0} · (1 + D)^d. Similar to Corollary 3.21, the results for ||X_ψ(x) − X_ψ^A(x)||^α_α and ||X_ψ^A(x) − X_ψ^A,M,N(x)||^α_α can be combined also in this more general case in order to give an estimation of ||X_ψ(x)−

X_ψ^A,M,N(x)||^α_α:

Corollary 3.35. Be X_ψ a harmonizable OSSRF, and X_ψ^A,M,N an approximation of this OSSRF which uses the second type of discretisation (see (3.18)). If the function ψ and the other parameters fulfill the assumptions of Theorem 3.10 and Theorem 3.34, then exist constants δ⁰, eC, eC_0,x (independent from the parameters A, B and D) and eC_1,A,B,x, eC_2,A,B (independent from D), so that the approximation error (due to the truncation and the discretisation) can be estimated by

 Proof. In the proof of Corollary 3.21 it has been shown that



which implies

According to the proof of Theorem 3.10 Z

and according to the proof of Theorem 3.34 X

Remark 3.36. If all assumptions on the parameter values are fulfilled, then the exponent of A in this estimation is negative and the exponents of B and D are positive. Therefore, the error estimate decreases with increasing values of A and decreasing values of B and D. choosing suitable values for the approximation parameters A, B and D:

(a) First choose a sufficiently large A > 0 and a sufficiently small B > 0 so that C · Ae ^{−αH−q+dλdλd} < ₄^ε and eC_0,x · Bαλ1−αH−q+dλ1

λ1 < ^ε₄. This is possible because the exponent of A is positive and the one of B is negative.

(b) Then choose a sufficiently small value for D (depending on the values of A and B which have been selected in the previous step), so that eC_1,A,B,x· D^α < ^ε₄ and Ce_2,A,B· D^{αβλd−δ0}· (1 + D)^d< ^ε₄. This can be done because α > 0, ^αβ_λ

d − δ⁰ is supposed to be positive, too, and the factor (1+D)^dis bounded by a constant: (1+D)^d≤ 2^d. If the parameters are chosen so that these inequalities are fulfilled, then the previous corollary implies that

First version of the discretisation

Corollary 3.37. Be X_ψ^A the approximation of a harmonizable OSSRF by truncation of the domain of integration, and X_ψ^A,M a discretisation of this approximation which uses the first type of discretisation (see (3.7)). If the function ψ and the other parameters fulfill the assumptions stated at the beginning of this subsection, as well as also the conditions αλ₁− αH − q + dλ₁ > 0, −αH − q + dλ_d 6= 0, −αH − α − q + dλ_d6= 0 and αH +α+q−dλ_d+λ_d ≥ 0, then exist constants eC_0,x, eC_1,x, eC_1,A,x, eC₂ and eC_2,A (independent from D), so that the approximation error due to the discretisation can be estimated by

 Proof. The first version of the discretisation can be interpreted as a special case of the second version with parameter value N = 1 (which implies B = D). Therefore, we can conclude from the proof of Theorem 3.34 by replacing N by 1 and B by D, that the error of approximation can be estimated by

 following estimations of I₀, I₁ and I₂ can be obtained:

From the proof of Lemma 3.26 follows

and from the proof of Lemma 3.27 follows:

I₁ ≤ 2 · ||x||^α₂ · d^α2 · C

Finally, the proof of Lemma 3.33 implies:

I₂ ≤ D^{αβλd−δ0} · (1 + D)^d·

with

Ce₂ = 2^1+α+d· c^α₅ and

Ce_2,A = 2^1+α+d· c^α₆ · d ·



1 + λ_d

−αH − α − q + dλ_d



A−αH−α−q+dλd

λd − 1



.

Remark 3.38. Like already in the case of a ρ-norm as E-homogeneous function ψ, no convergence of the error estimate to 0 for D → 0 can be shown when using the first version of discretization, because the exponent ^{−αH−q−α}_λ

1 + ^αβ_λ

d − δ⁰ can’t be assumed to be positive (compare Remark 3.18).

Approximation of OSSRFs in moving

In document Simulation and estimation of operator scaling stable random fields (Page 53-71)