Calculations in the Case of a Finite Number of Shape Functions

)) is independent of the eventZ(t)∈A_m(z) and the fact that we have a monotone limit.

By the second part of Theorem5.7the process Π₃|Z(t) =zcan be easily simulated by unconditionally simulating Π and restricting it to R^d×(0,∞)×G\(Kt,z∪Kt,z).

We close this section by noting that there exists a more general version of Theorem5.6 which we will need for simulation. LetB1, . . . , B_k,∈ B^d×((0,∞)∩ B)× G be pairwise disjoint withSk

j=1B_j =S. We introduce generalized “blurred” scenarios E^(m)

{n^(j)_A }(z) =E^(m)

{n^(j)_A A∈2^{1,...,n}\∅, j=1,...,k}(z)

={Π(I_A^(m)(z)∩B_j) =n^(j)_A , A∈2^{1,...,n}\ ∅, j = 1, . . . , k, Π(K_(t,z)^(m)) = 0} withn^(j)_A ∈N0 such thatPk

j=1

P

A:i∈An^(j)_A ≥1 fori∈ {1, . . . , n}. Analogously,generalized scenariosE

{n^(j)_A }(z) are defined. Then, in exactly the same way as Theorem5.6, the following theorem can be shown.

Theorem 5.8. With probability 1 we have P(E{n^(j)_A }(Z(t))|Z(t) =z) = lim

m→∞P(E^(m)

{n^(j)_A }(z)|Z(t)∈A_m(z)) for any scenario E

{n^(j)_A }(z) withPk j=1

P

A:i∈An^(j)_A ≥1 for all i∈ {1, . . . , n}.

The remainder of the chapter will address the problem of simulating Π₂ | Z(t) = z.

We propose a procedure consisting of two steps. First, we draw a scenario E_{nA}(Z(t)) conditional on Z(t) = z. Then, the points of Π2 corresponding to this scenario are simulated.

5.3 Calculations in the Case of a Finite Number of Shape Functions

As shown in Section 5.2, all we need for calculating P(E_{nA}(Z(t)) | Z(t) = z) is the exact asymptotic behaviour of Λ(I_A^(m)(z)). In particular, we have to analyse the behaviour of the intersection of two curvesK_ti,zi+δi∩K_tj,zj+δj for|δ_i|,|δ_j|small. Explicit calculations turn out to be quite involved. Therefore, we restrict ourselves to the case d= 1. Furthermore, the intersection depends on the derivative of the shape function.

We calculate the asymptotics of Λ(I_A^(m)(z)) for |A| = 1 (Proposition 5.13), |A| = 2 (Proposition 5.9) and |A| ≥ 3 (Proposition 5.11), see Figure 5.2. In the latter case, the rate of convergence of Λ(I_A^(m)) cannot be determined exactly. Nevertheless, the conditional probability of any scenario can be calculated (Theorem 5.16).

5. Conditional Sampling of Mixed Moving Maxima Processes

0.0 0.5 1.0 1.5 2.0

0.51.01.52.02.5

0.0 0.5 1.0 1.5 2.0

0.51.01.52.02.5

0.0 0.5 1.0 1.5 2.0

0.51.01.52.02.5

Figure 5.2: Blurred intersection sets for |A| = 1, |A| = 2 and |A| = 3. Black crosses:

data (ti, Z(ti)), dashed black lines: K_ti,Z(ti), black line: K_t,Z(t), grey area:

K_t,Z(t)^(m) , black area: I_A^(m)(Z(t)) with |A| = 1 (left), |A| = 2 (middle) and

|A|= 3 (right).

First we assume that G is a finite space of functions f : R → (0,∞) such that the intersections

M_c,t0 ={t∈R: f(t) =cf(t₀+t)} (5.10) are finite for all c >0, t₀ ∈R,f ∈G. This implies that each setI_A(z), A∈2^{1,...,n}\ ∅,

|A| ≥2,z>0, is finite. W.l.o.g. we assume thatPF({f})>0 for all f ∈G.

Proposition 5.9. Let t₁, t₂ ∈R, z₁, z₂ >0 such that I_{1,2}(z) ={(t0, y0, f)}.

Furthermore, letf be continuously differentiable in a neighbourhood of t_i−t₀ and t₂−t₀ with

z1f^′(t2−t0)6=z2f^′(t1−t0). (5.11) Then, we have

Λ(I_{^(m)_1,2}(z)) = 2^−2m

y₀²|z₁f^′(t₂−t₀)−z₂f^′(t₁−t₀)|P_F({f}) +o(2^−2m).

Proof. We note that (t0, y0, f) satisfies the equation f(t1−t0)

z₁ = f(t2−t0)

z₂ =y₀⁻¹. Let

H : (−z,∞)×R→R, (δ, t)7→ f(t₁−t)

z₁+δ₁ −f(t₂−t) z₂+δ₂ . Then,H(0, t₀) = 0 and

∂H

∂t (0, t₀) =−f^′(t₁−t₀) z1

+f^′(t₂−t₀) z2 6= 0

72

5.3. Calculations in the Case of a Finite Number of Shape Functions due to (5.11). The implicit function theorem yields the existence of a neighbourhood V of 0 and a continuously differentiable function h :V → R such that H(δ, h(δ)) = 0.

Using the notation (t_δ, y_δ, f) =I_{1,2}(z+δ) we geth(δ) =t_δ and the equality f(t₁−t_δ)

z₁+δ₁ = f(t₂−t_δ)

z₂+δ₂ =y_δ⁻¹.

Ash is continuously differentiable, we obtain t₀−t_δ∈ O(||δ||), and a Taylor expansion of f yields

f(t_i−t_δ) =f(t_i−t₀)−f^′(t_i−t₀)·(t_δ−t₀) +o(||δ||). (5.12) Letg(t) =f(t−t0). Thus, using (5.12),t_δ is given implicitly by

g(t₁)−g^′(t₁)·(t_δ−t₀)

z₁+δ₁ = g(t₂)−g^′(t₂)·(t_δ−t₀)

z₂+δ₂ +o(||δ||), which implies the explicit representation

t_δ=t₀+ δ₁g(t₂)−δ₂g(t₁)

z1g^′(t2)−z2g^′(t1) +o(||δ||). (5.13) Plugging in (5.13) into (5.12) yields

y_δ⁻¹= f(t₁−t_δ) z1+δ1

= g(t₁)

z1+δ1 −g^′(t₁)

z1 · δ₁g(t₂)−δ₂g(t₁)

z1g^′(t2)−z2g^′(t1) +o(||δ||).

As f and δ 7→ t_δ =h(δ) are C¹-functions, all the terms o(||δ||) are continuously differ-entiable for small||δ||. Therefore, the mapping

Φ :V →R×(0,∞), δ 7→(t_δ, y_δ⁻¹)

is continuously differentiable near the origin. Calculating the partial derivatives explic-itly, we obtain

det(DΦ(δ)) =− g²(t₁)

z²₁·(z₁g^′(t₂)−z₂g^′(t₁))+o(1). (5.14) As det(DΦ(0))6= 0, the inverse function theorem allows to regard Φ as a diffeomorphism restricted to an appropriate neighbourhood of 0. Thus, considering the transformed Poisson point process ˜Π =P

i∈Nδ_(S

i,U_i⁻¹) on R×(0,∞) whose intensity measure is the Lebesgue measure, we get

Λ({I_{1,2}(z+δ), z_i+δ_i ∈A_m(z_i), i= 1,2})

= Z

Φ((Am(z1)−z1)×(Am(z2)−z2))

PF({f}) d (t, y)

= Z

(Am(z1)−z1)×(Am(z2)−z2)|det(DΦ(δ))| ·PF({f}) dδ

= Z

(Am(z1)−z1)×(Am(z2)−z2)

1/y₀²+o(1)

|z₁g^′(t₂)−z₂g^′(t₁)|P_F({f}) dδ.

5. Conditional Sampling of Mixed Moving Maxima Processes

We note that the term o(1) is continuous w.r.t. δ and therefore the integrands can be locally bounded by

1/y₀²

|z₁g^′(t₂)−z₂g^′(t₁)|−ε_m

PF({f}) from below, and by

1/y₀²

|z₁g^′(t₂)−z₂g^′(t₁)|+ε_m

P_F({f})

from above for all (δ₁, δ₂)∈(A_m(z₁)−z₁)×(A_m(z₂)−z₂) with m large enough and an appropriate sequence (ε_m)_m∈Nwithε_m ց0. This implies that the integral has the form

2^−2m· 1/y²₀

|z₁g^′(t₂)−z₂g^′(t₁)|·P_F({f}) +o(2^−2m) which is the desired result.

Remark 5.10. 1. Using _f(t^z1

1−t0) = _f(t^z2

2−t0) =y0 we get that the equality z₁f^′(t₂−t₀) =z₂f^′(t₁−t₀)

holds if and only if

∂

∂t z₁ f(t₁−t)

t=t0

= ∂

∂t z₂ f(t₂−t)

t=t0

,

i.e. if and only if the two sets of admissible points, K_t1,z1 and K_t2,z2, are tangents to each other in (t₀, y₀, f) which is an event of probability zero by Assumption (5.10). Therefore, (5.11) is satisfied a.s.

2. IfI_{1,2}(z) consists of a finite number of points,

I_{1,2}(z₁, z₂) ={(t⁽¹⁾₀ , y₀⁽¹⁾, f₁), . . . ,(t^(k)₀ , y₀^(k), f_k)}, we get

Λ(I_{1,2}(z)) = 2^−2m· Xk j=1

P({f_j})

y²₀· |z₁f_j^′(t₂−t^(j)₀ )−z₂f_j^′(t₁−t^(j)₀ )|+o(2^−2m).

Proposition 5.11. Let t1, . . . , tl ∈R, z1, . . . , zl >0, l≥3 such that I_{1,...,l}(z) ={(t₀, y₀, f)}.

Let f :R→(0,∞) be continuously differentiable in a neighbourhood oft₁−t₀, . . . , t_l−t₀ with (5.11). Then, we have

Λ(I_{^(m)_1,...,l}(z))≤ 2^−2m

y₀²|z₁f^′(t₂−t₀)−z₂f^′(t₁−t₀)|P_F({f}) +o(2^−2m).

For any C >0, ε >0, there exists m_C,ε∈N such that Λ(I_{^(m)_1,...,l}(z))≥C2^−2m(1+ε) for allm≥m_C,ε.

74

5.3. Calculations in the Case of a Finite Number of Shape Functions Proof. The first assertion follows immediately from Proposition 5.9by the fact that

\l i=1

K_t^(m)_i,zi ⊂K_t^(m)_1,z1∩K_t^(m)_2,z2.

In order to verify the second assertion, we recall results from the proof of Proposition 5.9: we showed the existence of aC¹-function (δ1, δ2)7→t_δ1δ2 defined in an appropriate neighbourhood V of (0,0) such that

f(t1−t_δ1δ2)

z₁+δ₁ = f(t2−t_δ1δ2) z₂+δ₂ . Now, we consider the C¹-functions

H_i :V ×(−z_i,∞)→R, (δ₁, δ₂, δ_i)7→ f(t₁−t_δ1δ2)

z1+δ1 −f(t_i−t_δ1δ2) zi+δi

, i∈ {3, . . . , l}. AsH_i(0,0,0) = 0 and ^∂H_∂δⁱ

i(0,0,0) = ^f(ti−t0)

z_i² 6= 0, we get the existence of a continuously differentiable functionh_i defined on a neighbourhood of (0,0) such that

f(t₁−t_δ1δ2)

z₁+δ₁ = f(t_i−t_δ1δ2)

z_i+h_i(δ₁, δ₂). (5.15) Using Taylor expansions of g(·) = f(· −t₀) of first order, employing Equation (5.13), and solving Equation (5.15) yields

h_i(δ₁, δ₂) = g(t_i)

g(t₁)δ₁+z_ig^′(t₁)−z₁g^′(t_i) g(t₁)

g(t₂)δ₁−g(t₁)δ₂

z₁g^′(t₂)−z₂g^′(t₁) +o(|δ₁|) +o(|δ₂|). (5.16) So, there are constants c_1,i, c_2,i such that h_i(δ₁, δ₂) = c_1,iδ₁ +c_2,iδ₂ +o(|δ₁|) +o(|δ₂|).

Let A⁽ⁱ⁾_m = A_m(z_i)−z_i for i ∈ {1, . . . , n}. We are interested in those pairs (δ₁, δ₂) ∈ A⁽¹⁾_m ×A⁽²⁾_m with h_i(δ₁, δ₂) ∈ A⁽ⁱ⁾_m. By Lemma 5.3, for any C^′ > 0, ε > 0, we have that (−C^′2^−m(1+ε), C^′2^−m(1+ε)) ∈ A⁽ⁱ⁾_m, i = 1, . . . , n, for m large enough. Therefore, hi(δ1, δ2) ∈ A⁽ⁱ⁾m is guaranteed for |δ1| < ^C′2₃^−m(1+ε)_|c

1,i| and |δ2| < ^C′2₃^−m(1+ε)_|c

2,i| if m is suffi-ciently large.

By the same argumentation for alli∈ {3, . . . , l}, we get that the existence of allh_i(δ₁, δ₂) is ensured for

|δ₁|< C^′2^−m(1+ε)

3 max_i=3,...,l|c_1,i|, |δ₂|< C^′2^−m(1+ε)

3 max_i=3,...,l|c_2,i| (5.17) for m large enough. Furthermore, to ensure δ₁ ∈ A⁽¹⁾m , δ₂ ∈ A⁽²⁾m , we have to add the conditions|δ₁|,|δ₂|< C^′2^−m(1+ε). WithC_j = max{1,3 max_i=3,...,l|c_j,i|}forj= 1,2, this yields

Λ({I_{1,...,l}(z+δ), δi∈A⁽ⁱ⁾_m, i= 1, . . . , l})

≥Λ

t_δ1δ2, z1+δ1

f(t₁−t_δ1δ2), f

,|δj|< C^′

C_j2^−m(1+ε), j = 1,2

= (C^′)²2^−2m(1+ε)·PF({f})

y₀²C₁C₂|z₁f^′(t₂−t₀)−z₂f^′(t₁−t₀)|+o(2^−2m(1+ε))

where we use the same argumentation as in the proof of Proposition5.9. This completes the proof.

5. Conditional Sampling of Mixed Moving Maxima Processes

Remark 5.12. Note that Proposition5.11does not provide exact asymptotics which we need to determine P(Π({(t0, y0, f)}) = 1 |Π(I_{1,...,l}(z)) = 1, Z(t) =z) for (t0, y0, f) ∈ I_{1,...,l}(z) and |I_{1,...,l}(z)| > 1. However, we can get exact results by conditioning on Z(t_i) being in intervals of different size for eachi∈ {1, . . . , l} instead ofZ(t_i)∈A_m(z_i) for all i = 1, . . . , l. We will choose these intervals such that some restrictions on the intersection sets vanish asymptotically and we can resort to the results on the intersection of two curves.

The calculations in the proof of Proposition 5.11yield

|hi(δ1, δ2)| ≤(|c1,i|+o(1))· |δ1|+ (|c2,i|+o(1))· |δ2| ≤2^−m(|c1,i|+|c2,i|+o(1)) for (δ₁, δ₂)∈A⁽¹⁾m ×A⁽²⁾m . Thus, for anyε >0, we can replacemby⌊m(1−ε)⌋in Equation (5.17) and get that h_i(δ₁, δ₂) ∈ A⁽ⁱ⁾_{⌊m(1−ε)⌋} holds for |δ₁| < ^C′2−⌊m(1−ε)⌋(1+ε)

3|c1,i| ∼ 2^ε2m2^−m and |δ₂|< ^C′2−⌊m(1−ε)⌋(1+ε)

3|c_2,i| ∼2^ε2m2^−m form large enough. Therefore, h_i(δ₁, δ₂)⊂A⁽ⁱ⁾_{⌊m(1−ε)⌋}, i= 3, . . . , l

for allδ1 ∈A⁽¹⁾m ⊂(−2^−m,2^−m],δ2∈A⁽²⁾m ⊂(−2^−m,2^−m] if mis sufficiently large. This implies

nI_{1,...,l}(z+δ), δ1∈A⁽¹⁾_m , δ2 ∈A⁽²⁾_m , δi ∈A⁽ⁱ⁾_{⌊m(1−ε)⌋}, i= 3, . . . , lo

= n

I_{1,...,l}(z+δ), (δ₁, δ₂)∈A⁽¹⁾_m ×A⁽²⁾_m, h_i(δ₁, δ₂)∈A⁽ⁱ⁾_{⌊m(1−ε)⌋}, i= 3, . . . , lo

=

tδ1δ2, z₁+δ₁ f(t₁−t_δ1δ2), f

, δ1 ∈A⁽¹⁾_m , δ2∈A⁽²⁾_m

and, therefore Λn

I_{^(m)_1,...,l}(z+δ), δ₁ ∈A⁽¹⁾_m , δ₂∈A⁽²⁾_m , δ_i ∈A⁽ⁱ⁾_{⌊m(1−ε)⌋}, i= 3, . . . , lo

= 2^−2mPF({f})

y²₀|z₁f^′(t₂−t₀)−z₂f^′(t₁−t₀)|+o(2^−2m).

By conditioning on Z(t) ∈ A_m(z₁)×A_m(z₂) ×"i=3,...,lA_{⌊m(1−ε)⌋}(z_i), for I_{1,...,l}(z) = {(t⁽¹⁾₀ , y⁽¹⁾₀ , f₁), . . . ,(t^(k)₀ , y₀^(k), f_k)} for somel≥3, we can apply L´evy’s “Upward” Theo-rem (Rogers and Williams,2000, Thm. 50.3) and end up with

P(Π({(t₀, y₀, f)}) = 1|Π(I_{1,...,l}(Z(t))) = 1, Z(t) =z)

= lim

m→∞P Πn

(t_δ1δ2, y_δ, f) : δ₁ ∈A⁽¹⁾_m , δ₂ ∈A⁽²⁾_m , δ_i∈A⁽ⁱ⁾_{⌊m(1−ε)⌋}, i≥3o

= 1 Π(I_{1,...,l}(Z(t))) = 1, Z(t)∈Am(z1)×Am(z2)×"i=3,...,lA_{⌊m(1−ε)⌋}(zi)

= PF({f})

y²₀|z₁f^′(t₂−t₀)−z₂f^′(t₁−t₀)|·



 Xk j=1

PF({fj})

(y^(j)₀ )²|z₁f_j^′(t₂−t^(j)₀ )−z₂f_j^′(t₁−t^(j)₀ )|





−1

. (5.18)

76

5.3. Calculations in the Case of a Finite Number of Shape Functions Note that L´evy’s Upward Theorem implies that, with probability one, the right hand side of (5.18) does not depend on the choice of the labelling.

Thus, despite the lack of exact convergence rates of Λ(I_{^(m)_1,...,l}(z)), the distribution of Π(· ∩I_{1,...,l}(Z(t)))|Π(I_{1,...,l}(Z(t))) = 1 can be determined exactly.

Proposition 5.13. Let f ∈G, t∈Rⁿ, z>0 such thatf is continuously differentiable in a neighbourhood of t_i −t₀ for all (t₀, y₀, f) ∈ K_t,z ∩Sn

l=1l6=i(K_ti,zi ∩K_tl,zl), i.e. all (t₀, y₀, f)∈K_ti,zi involved in an intersection, for all i∈ {1, . . . , n}.

We denote the projection of the set I_{i}(z)∩(R×(0,∞)× {f}) onto its first component in R by

D^(f_i ⁾={t∈R: (t, y, f)∈I_{i}(z) for some y >0}. Then, we have

Λ

I_{^(m)_i} (z)∩(R×(0,∞)× {f})

= 2^−m·PF({f})· Z

D^(f)_i

f(ti−t)

z_i² dt+o(2^−m). (5.19) Proof. First, we note that by renumbering, it suffices to show the result fori= 1.

The idea of this proof is to assess the setD₁^(f) by the setsD^(m)_1,min from below andD^(m)_1,max from above. Here, D_1,min^(m) consists of all first components ofI_{^(m)_1}(z) which are not part of any intersections I_A^(m)(z), A ) {1} and D^(m)_1,max is the set of the first components of S

A⊃{1}I_A^(m)(z). Analogously, Λ

I_{^(m)_i} (z)∩(R×(0,∞)× {f})

can be bounded from below and above by replacingD^(f₁ ⁾in (5.19) by the setsD_1,min^(m) andD_1,max^(m) , respectively.

We will show that the difference, which consists of blurred intersectionsI_A^(m)(z),A){1}, vanishes asymptotically.

LetA⁽ⁱ⁾_m =A_m(z_i)−z_i. Then, for anyδ∈"ⁿ_i=1(−z_i,∞) we define D_1,δ^(f)=

t∈R:

t, z₁+δ₁ f(t1−t), f

∈I_{1}(z+δ)

=

t∈R: z1+δ1

f(t₁−t) < min

i=2,...,n

zi+δi

f(t_i−t)

. (5.20)

Thus, defining D_1,min^(m) =T

δ∈×ⁿ_i=1A⁽ⁱ⁾m D^(f)_1,δ andD^(m)_1,max =S

δ∈×ⁿ_i=1A⁽ⁱ⁾m D^(f_1,δ⁾, we get D_1,min^(m) ⊂D₁^(f) ⊂D_1,max^(m) .

On the other hand, we have

n(t, y, f)∈R×(0,∞)× {f}: t∈D_1,min^(f) , yf(t₁−t)∈A_m(z₁)o

⊂I_{^(m)_1}(z)⊂n

(t, y, f)∈R×(0,∞)× {f}: t∈D_1,max^(f) , yf(t₁−t)∈A_m(z₁)o

. (5.21) Now, lett∈D^(m)_1,max\D_1,min^(m) . Then, by definition of D_1,min^(f) andD^(f_1,max⁾ , there exist δ⁽¹⁾, δ⁽²⁾∈ ×ⁿi=1A⁽ⁱ⁾m such thatt∈D_1,δ^(f)₍₁₎, butt /∈D_1,δ^(f)₍₂₎. That is, by Equation (5.20),

z1+δ₁⁽¹⁾

f(t₁−t) < min

i=2,...,n

zi+δ_i⁽¹⁾

f(t_i−t) and z1+δ₁⁽²⁾

f(t₁−t) ≥ min

i=2,...,n

zi+δ⁽²⁾_i f(t_i−t).

5. Conditional Sampling of Mixed Moving Maxima Processes By continuity arguments, a δ∈ ×ⁿi=1A⁽ⁱ⁾m exists such that

z₁+δ₁

f(t₁−t) = min

i=2,...,n

z_i+δ_i f(t_i−t), i.e.t∈T₁^(m)={t∈R: (t, y, f)∈S

A:{1}(AI_A^(m)(z) for some y >0}. Thus,

D^(m)_1,max \D^(m)_1,min ⊂T₁^(m). (5.22) By definition,T₁^(m) denotes the set of first components involved in any blurred intersec-tion and we have T₁^(m) ց T₁ = {t ∈R : (t, y, f) ∈S

A:{1}(AI_A(z) for some y > 0} as m→ ∞andT1 is finite by Assumption (5.10). Therefore, dominated convergence yields

Z

T₁^(m)

f(t₁−t) dtց0, m→ ∞. (5.23)

Thus, by Equations (5.21) and (5.22) we get Λ

I_{^(m)_1}(z) ∆ n

(t, y, f)∈R×(0,∞)× {f}: t∈D₁^(f), yf(t1−t)∈Am(z1)o

≤Λ n

(t, y, f)∈R×(0,∞)× {f}: t∈D^(f)_1,max, yf(t₁−t)∈A_m(z₁)o / n(t, y, f)∈R×(0,∞)× {f}: t∈D^(f_1,min⁾ , yf(t₁−t)∈A_m(z₁)o

≤Λn

(t, y, f)∈R×(0,∞)× {f}: t∈T₁^(m), yf(t₁−t)∈A_m(z₁)o

=P_F({f}) Z

A⁽¹⁾m

Z

T₁^(m)

f(t₁−t)

(z₁+δ₁)²dtdδ₁=P_F({f}) Z

A⁽¹⁾m

o(1)

(z₁+δ₁)² dδ₁ ∈o(2^−m).

The last equality follows from Equation (5.23). Hence, we have Λ(I_{1}^(m)(z)) =PF({f})

Z

D^(f)₁

Z

A⁽¹⁾_m

f(t₁−t)

(z1+δ1)² dδ₁dt+o(2^−m)

= 2^−m·PF({f})· Z

D^(f)₁

f(t₁−t)

z₁² dt+o(1)

!

which completes the proof.

Now, we can use Theorem5.6 in order to compute the conditional probabilities P(E_{nA}(Z(t))|Z(t) =z) = lim

m→∞P(E_{^(m)_n

A}(z)|Z(t)∈A_m(z))

= lim

m→∞

P(E_{^(m)_n

A}(z)) P

{n˜A}∈N0P(E_{^(m)_n˜

A}(z)) (5.24) whereN₀ =

{n_A: A∈2^{1,...,n}\ ∅}: P

A:i∈An_A= 1, i= 1, . . . , n .

78

5.3. Calculations in the Case of a Finite Number of Shape Functions

Considering (5.24), we can restrict ourselves to those scenarios with the slowest rate of convergence to zero. Propositions 5.9, 5.11 and 5.13 yield that scenarios involving intersections of at least three sets are always of a dominating order. Therefore, the unknown terms from Proposition5.11 are cancelled out.

Example 5.14. Let F(x) = f(x) =: √¹ By the formulae from Propositions 5.9 and 5.13, we get

Λ(I_{^(m)_1}(a, a)) = 2^−m Note that the probability that the observations are generated by two different points of Π increases as a gets smaller. This is due to the fact that Π gets more intense as the second component decreases.

Using the formulae above, the limits of the conditional probabilities can always be calcu-lated explicitly except for those cases where two scenarios exist, both involving different

5. Conditional Sampling of Mixed Moving Maxima Processes

terms which cannot be determined exactly (cf. Proposition5.11). This may happen only if two setsA1 ={i1, . . . , ir}andA2={j1, . . . , js},r, s≥3,A1∩A2 6=∅, exist such that J_A1(Z(t))6=∅, J_A2(Z(t))6=∅and J_A1∪A2(Z(t)) =∅, (5.29) where

J_A(Z(t)) = [

B⊃A

I_B(Z(t)) =K_t,Z(t)∩ \

i∈A

K_ti,Z(ti), A∈2^{1,...,n}\ ∅.

In all other cases, the terms as in Proposition5.11are cancelled out. Note that we work with sets of the type J_A(Z(t)) in order to avoid case-by-case analysis for all the sets I_B(Z(t)) with B ⊃A.

Lemma 5.15. Let G consist of functions which are continuously differentiable a.e.

Then, for any fixed set{t₁, . . . , t_n} ⊂R we have

P(Z(t) satisfies(5.29)) = 0.

Proof. We proof that condition (5.29) has probability 0 for all fixed index sets A₁, A₂⊂ {1, . . . , n}. By renumbering, we may assumeA1 ={1, . . . , r}andA2 ={q, . . . , q+s−1} with q ≤ r. Assume that P(Z(t) satisfies (5.29))> 0. In a first step we only consider those realizations of Z(t₁), . . . , Z(t_r) with J_A1(Z(t₁), . . . , Z(t_r)) 6= ∅. Then, by the calculations in Propositions 5.9,5.11and 5.13, we get that

P(Π(I_A^(m)₁ (Z(t1), . . . , Z(tr))) = 1)∈ O/ (2^−2m(1+ε)) for any ε >0 and

P(Π(I_B^(m)₁ (Z(t1), . . . , Z(tr))) = 1,Π(I_B^(m)₂ (Z(t1), . . . , Z(tr))) = 1)∈ O(2^−3m) for any B₁, B₂ ⊂A₁,B₁∩B₂ =∅. This yields Π(J_A1(Z(t₁), . . . , Z(t_r))) = 1 a.s.

Similarly, for almost every Z_q, . . . , Z_q+s−1 such that J_A2(Z(t_q), . . . , Z(t_q+s−1)) 6=∅, we have Π(J_A2(Z(t_q), . . . , Z(t_q+s−1))) = 1. As

{ω : Z(t) satisfies (5.29)}

⊂ {ω : J_A1(Z(t₁), . . . , Z(t_r))6=∅} ∩ {ω : J_A2(Z(t_q), . . . , Z(t_q+s−1))6=∅}, we have Π(J_A1(Z(t₁), . . . , Z(t_r))) = 1 and Π(J_A2(Z(t_q), . . . , Z(t_q+s−1))) = 1 for Z(t) satisfying (5.29) almost surely. Therefore, we get P(Π(K_ti,Z(ti)) ≥ 2) > 0 for every i∈A₁∩A₂ sinceJ_A1∪A2(Z(t)) =∅. This is a contradiction to Proposition 5.5.

From the considerations above and Lemma 5.15 we immediately derive the following result.

Theorem 5.16. Let G consist of functions which are continuously differentiable a.e.

Then, with probability one,

P(E_{nA}(Z(t))|Z(t) =z) = lim

m→∞P(E_{^(m)_n

A}(z)|Z(t)∈Am(z))

can be calculated explicitly via the formulae given in Propositions 5.9, 5.11, 5.13 and Remark 5.12.

80

5.3. Calculations in the Case of a Finite Number of Shape Functions Remark 5.17. We may also consider the case that Gis countable. However, to trans-fer the results of the finite case, we have to ensure uniform convergence of the blurred intersection sets which is needed to computeP

nA:P

A:i∈AnA=1P(E_{^(m)_n

A}(z)) in the denom-inator of Equation (5.24). To this end, we have to impose some additional conditions.

For example, we could assume that, for almost everyz>0, there is only a finite number of shape functions involved in the intersection setsI_A(z),|A| ≥2.

We are still left with simulating Π₂ |Z(t) =zgiven the occurrence of a scenarioE_{nA}(z) withP

A:i∈An_A= 1 for all i∈ {1, . . . , n}, that is, we are interested in P



 \

A:n_A=1

{Π₂(C_A×(0,∞)× {f}) = 1}

E_{nA}(z)





forCA⊂R,f ∈G with (CA×(0,∞)× {f})∩IA(z)6=∅. Using Theorem 5.8with sets B_A=C_A×(0,∞)× {f},A∈2^{1,...,n}\ ∅, we get that

P



 \

A:n_A=1

{Π₂(I_A(z)∩B_A) = 1}

E_{nA}(z)



= lim

m→∞

Y

A:n_A=1

Λ(I_A^(m)(z)∩B_A) Λ(I_A^(m)(z)) . Thus, each random vector (T_A, F_A) ∈ R×G, A ∈ 2^{1,...,n} \ ∅, which is defined by Π2(({TA} ×(0,∞) × {FA})∩IA(Z(t))) = 1 if Π2(IA(Z(t))) = 1, can be simulated independently.

The distribution of (T_A, F_A) depends on the cardinal number ofA. IfA={i}for some i∈ {1, . . . , n}, we have

P(TA∈B, F_A=f) =

PF({f})R

D_i^(f)∩Bf(t_i−t) dt P

g∈GP_F({g})R

D^(g)_i g(ti−t) dt, B ∈ B. For|A| ≥2, let I_A(z) ={(t⁽¹⁾₀ , y₀⁽¹⁾, f₁), . . . ,(t^(k)₀ , y₀^(k), f_k)}. Then, we get

P(TA=t^(j)₀ , F_A=f_j) =

PF({fj})

(y^(j)₀ )²|z1f_j^′(t2−t^(j)₀ )−z2f_j^′(t1−t^(j)₀ )|

Pk j^′=1

PF({f_j′})

(y^(j₀^′))²|z1f_j^′_′(t2−t^(j₀^′))−z2f_j^′_′(t1−t^(j₀^′))|

, j= 1, . . . , k.

Thus, we end up with the following procedure for calculating the conditional distribution of Z(t₀) given Z(t) =z witht₀, t₁, . . . , t_n∈R,z>0.

1. Compute the conditional probabilities (5.24) for all the scenarios E_{nA}(z) and generate a random scenario following this distribution.

2. For a given scenarioE_{nA}(z), set Π₂=P

{A:nA=1}δ_(TA,UA,FA)}, where the law of (T_A, F_A) is given above andU_A= minⁿ_{i=1 F} ^zi

A(ti−TA).

3. Independently, sample points (Si, Ui, Fi)i∈N from Π3(·) = Π(· \(K_(t,z)∪K_(t,z))).

Then,Z(t₀) = max_{A:nA=1}(U_A·F_A(t₀−T_A))∨max_i∈N(U_i·F_i(t₀−S_i)).

5. Conditional Sampling of Mixed Moving Maxima Processes

In the next section, we will point out the capability of this exact approach by comparing it to other algorithms in the simple case of a deterministic shape function which is continuously differentiable. In Section5.5, we address the computational burden of this algorithm.

Im Dokument Spatial Interpolation and Prediction of Gaussian and Max-Stable Processes (Seite 79-90)