Extracting the incremental process

generalized POT methods instead of aggregated data. In contrast to the ML approach, here, we assume thatζ admits one of the two representations (7.3) and (7.5) and we aim at extracting realizations of the processesW and F, respectively, from single extreme events.

Here, the specification of a single extreme event will depend on the respective representation.

InEngelkeet al.(2012a), this concept is applied to derive estimators for the class of Brown-Resnick processes (Brown & Brown-Resnick,1977;Kabluchko et al.,2009), which have the form (7.3) by construction. With a(n) being a sequence with limn→∞a(n) =∞, the convergence

in distribution

η(t₁)

η(t0), . . . ,η(t_k) η(t0)

η(t₀)> a(n)

!

⇒ W(t₁), . . . , W(t_k), (7.6) fort₀,t₁, . . . ,t_k∈T, k∈N, is established forη being in the MDA of a Brown-Resnick process and withW being the corresponding log-Gaussian random field.

Basrak & Segers(2009) and Meinguet & Segers (2010) consider multivariate time series (X_t)_t∈Z and time series in general Banach spaces, respectively, rescaled and conditioned on kX₀kbeing large. They provide equivalent conditions for the existence of the corresponding tail processes and its spectral decomposition, whereas here, we explicitly calculate the limiting processes under more specific assumptions.

In this direction, we generalize the convergence result (7.6) in two different aspects.

Arbitrary non-negative processes{W(t) :t∈T}with EW(t) = 1,t∈T, are considered, and convergence of the conditional increments ofηin the sense of finite-dimensional distributions as well as weak convergence in continuous function spaces is shown (Theorems7.2.1 and 7.2.10). Since also M3 processes might admit an incremental representations (7.3) we provide formulae for switching between the two representations in Section 7.3. Section7.4gives an exemplary outlook on how our results can be applied for statistical inference.

7.2 Extracting the incremental process

Throughout this section, we consider a slightly more general version of representation 7.3, which does not require that W(t0) = 1 a.s. for some t0 ∈ T. In fact, we suppose that {ζ(t) : t ∈T}, T an arbitrary index set, is normalized to standard Fréchet margins and admits a representation

ζ(t) = max

i∈N

U_iV_i(t), t∈T. (7.7)

Here, ^P_i∈

NδUi is a PPP on (0,∞) with intensity u⁻²du, which we callFréchet point process in the following. The{V_i}_i∈N are independent copies of a non-negative stochastic process {V(t) : t∈T}with EV(t) = 1,t∈T. For any fixed t₀∈T, we have

ζ(t)= max^d

i∈N

U_i1_Pi=0V_i⁽¹⁾(t) +1_Pi=1V_i⁽²⁾(t), t∈T, (7.8)

where{P_i}_i∈N are i.i.d. Bernoulli variables with parameterp=P(V(t0) = 0) and the V_i⁽¹⁾ andV_i⁽²⁾ are independent copies of the process {V(t) : t∈T}, conditioned on the events {V(t0)>0}and {V(t0) = 0}, respectively.

Note that fork∈N,t₀, . . . , t_k ∈T, the vectorζ = (ζ(t₀), . . . ,ζ(t_k)) follows a (k+ 1)-variate extreme-value distribution and its distribution functionG can therefore be written as

G(x) = exp(−µ([0,x]^C)), x∈R^k+1, (7.9) where µis a measure on E= [0,∞)^k+1\ {0}, the so-called exponent measure ofG(Resnick, 2008, Prop. 5.8), and [0,x]^C =E\[0,x].

The following convergence result provides the theoretical foundation for statistical inference based on the incremental processV.

Theorem 7.2.1. Let {η(t) : t∈T} be non-negative and in the MDA of some max-stable processζ that admits a representation (7.7) and suppose that η is normalized such that (7.1) holds withcn(t) = 1/n and bn(t) = 0 for n∈N and t∈T. Let a(n)→ ∞as n→ ∞. For k∈Nand t₀, . . . ,t_k∈T we have the convergence in distribution on R^k+1

η(t0) a(n),η(t1)

η(t₀), . . . ,η(tk) η(t₀)

η(t0)> a(n)

!

⇒Z, ∆V˜⁽¹⁾, n→ ∞, where the distribution of ∆V˜⁽¹⁾ is given by

P(∆V˜⁽¹⁾∈dz) = (1−p)P(∆V⁽¹⁾ ∈dz)E V⁽¹⁾(t0)∆V⁽¹⁾=z, (7.10) z≥0. Here, ∆V⁽¹⁾ denotes the vector of increments ^V(1)(t1)

V⁽¹⁾(t0), . . . ,^V_V⁽¹⁾₍₁₎^(t_(t^k)

0)

with respect to t0, andZ is an independent Pareto variable.

Remark 7.2.2. Note that, if ζ has standard Fréchet margins, any process η satisfying the convergence in (7.1) can be normalized such that the norming functions in (7.1) are c_n(t) = 1/n andb_n(t) = 0, n∈N, t∈T (Resnick, 2008, Prop. 5.10).

Proof of Theorem 7.2.1. For X = (η(t0), . . . ,η(t_k)), which is in the MDA of the random vectorζ = (ζ(t₀), . . . ,ζ(t_k)), it follows fromResnick (2008, Prop. 5.17) that

m→∞lim mP(X/m∈B) =µ(B), (7.11) for all elementsB of the Borelσ-algebraB(E) ofE bounded away from{0}withµ(∂B) = 0, where µis defined by (7.9). For s0>0 ands= (s1, . . . , s_k)∈[0,∞)^k, we consider the sets A_s0 = (s₀,∞)×[0,∞)^k, A = A₁ and B_s = {x ∈ [0,∞)^k+1 : (x⁽¹⁾, . . . ,x^(k)) ≤x⁽⁰⁾s} for s satisfyingP(∆V˜⁽¹⁾∈∂[0,s]) = 0. Since Bs satisfies Bs=cBs for anyc >0, we obtain

P

η(t₀)> s₀a(n),(η(t₁)/η(t₀), . . . , η(t_k)/η(t₀))≤sη(t₀)> a(n)

= a(n)P(X/a(n)∈B_s∩A∩A_s0)

a(n)P(X/a(n)∈A) −→ µ(B_s∩A∩A_s0)

µ(A) , (n→ ∞), (7.12)

7.2 Extracting the incremental process 119

1 s₀

s B_s

A_s0

B_s∩^As0

Figure 7.1: The setsA,Bs andBs∩As₀ fork= 1.

where the convergence follows from (7.11), if µ{∂(B_s∩A∩A_s0)}= 0. Let ζ⁽¹⁾(t) = max

i∈N

U_i⁽¹⁾V_i⁽¹⁾(t), t∈T, (7.13) where ^P_i∈

Nδ_U(1) i

is a Poisson point process with intensity (1−p)u⁻²du, and let µ⁽¹⁾ be the exponent measure of the associated max-stable random vector (ζ⁽¹⁾(t₀), . . . , ζ⁽¹⁾(t_k)).

Then the choice A= (1,∞)×[0,∞)^k guarantees thatµ(· ∩A) =µ⁽¹⁾(· ∩A). Comparing the construction ofζ⁽¹⁾ in (7.13) with the definition of the exponent measure, we see that µ⁽¹⁾ is the intensity measure of the Poisson point process^P_i∈

Nδ_(U(1)

i V_i⁽¹⁾(t0), ..., U_i⁽¹⁾V_i⁽¹⁾(t_k)) onE.

Hence,

µ(A) = Z ∞

0

(1−p)u⁻² Z

[u⁻¹,∞)P(V⁽¹⁾(t0)∈dy)du

= (1−p) Z ∞

0

yP(V⁽¹⁾(t₀)∈dy) = (1−p)EV⁽¹⁾(t₀) = 1, (7.14) where the last equality follows from EV⁽¹⁾(t₀) =EV(t₀)/(1−p). Furthermore, for s₀ ≥1 and s∈[0,∞)^k with P(∆V˜⁽¹⁾∈∂[0,s]) = 0,

µ(B_s∩A∩A_s0)/((1−p)µ(A))

= Z ∞

0

u⁻² Z

[s0u⁻¹,∞)P

V⁽¹⁾(t₀)∈dy∆V⁽¹⁾ ≤sP(∆V⁽¹⁾ ≤s)du

= Z

[0,s]

Z

[0,∞)

ys⁻¹₀ ·P

V⁽¹⁾(t₀)∈dy∆V⁽¹⁾ =zP(∆V⁽¹⁾∈dz)

=s⁻¹₀ Z

[0,s]E

V⁽¹⁾(t₀)∆V⁽¹⁾=zP(∆V⁽¹⁾∈dz). (7.15)

Equation (7.15) shows that the convergence in (7.12) holds for all continuity pointss∈[0,∞)^k of the distribution function of∆V⁽¹⁾.

Remark 7.2.3. 1. If V⁽¹⁾(t0) is stochastically independent of the increments ∆V⁽¹⁾, we simply have P(∆V˜⁽¹⁾∈dz) =P(∆V⁽¹⁾∈dz). This is particularly the case ifζ admits a representation (7.3), which shows that(7.6)is indeed a special case of Theorem7.2.1.

2. If p=P(V(t₀) = 0) = 0, the exponent measure µof any finite-dimensional vector ζ= (ζ(t₀), . . . , ζ(t_k)), t₀, . . . , t_k∈T, k∈N, satisfies the condition µ{0} ×[0,∞)^k= 0, and following Proposition 7.2.7, the incremental representation of ζ according to (7.3) is given by ζ = max_i∈NUi·(1, ∆V˜_i)^>, where ∆V˜_i, i∈N, are independent copies of

∆V˜ =∆V˜⁽¹⁾.

Remark 7.2.4. In the above theorem, the thresholdsa(n)tend to∞ to make{η(t₀)> a(n)}

a rare event. For statistical applications,a(n) should also be chosen such that the number of exceedancesN(n) =^Pn_i=11{η_i(t₀)> a(n)} converges to ∞ almost surely, where η_i, i∈N, are independent copies ofη. By the Poisson limit theorem, this is equivalent to the additional assumption thatlim_n→∞a(n)/n= 0, since then nP(η(t₀)> a(n)) =n/a(n)→ ∞.

Remark 7.2.5. Engelke et al.(2012a) consider Hüsler-Reiss distributions (Hüsler & Reiss, 1989; Kabluchko, 2011) and obtain their limiting results by conditioning on certain extremal

eventsA⊂E. They show that various choices of A are sensible in the Hüsler-Reiss case, leading to different limiting distributions of the increments ofη. In case thatζ is a Brown-Resnick process and A= (1,∞)×[0,∞)^k, the assertions of Theorem 7.2.1 and Engelke et al.

(2012a, Thm. 3.3) coincide.

Example 7.2.6 (Extremal Gaussian process (Schlather,2002)). A commonly used class of stationary yet non-ergodic max-stable processes on R^d is defined by

ζ(t) = max

i∈N

U_iY_i(t), t∈R^d, (7.16)

where^P_i∈

NδUi is a Fréchet point process, Yi(t) = max(0,Y˜i(t)), i∈N, and the Y˜i are i.i.d.

stationary, centered Gaussian processes with E(max(0,Y˜i(t))) = 1 for all t∈R^d (Schlather, 2002;Blanchet & Davison,2011). Note that in general, at₀ ∈R^ds.t.Y_i(t₀) = 1a.s. does not exist, i.e., the process admits representation (7.7) but not representation (7.3). In particular, for the extremal Gaussian process we havep=P(V(t0) = 0) = 1/2 and the distribution of the increments in (7.10) becomes

P(∆V˜⁽¹⁾∈dz) = 1 2E

h

Y(t0)(Y(t1)/Y(t0), . . . , Y(tk)/Y(t0)) =z, Y(t0)>0ⁱ

·P

Y(t1)/Y(t0), . . . , Y(tk)/Y(t0)∈dzY(t0)>0.

While the Hüsler-Reiss distribution is already given by the incremental representation (7.3), cf.Kabluchko (2011), other distributions can be suitably rewritten, provided that the

respective exponent measureµis known.

7.2 Extracting the incremental process 121

Proposition 7.2.7. Let ζ = (ζ(t₀), . . . ,ζ(t_k)) be a max-stable process on T ={t₀, . . . , t_k} with standard Fréchet margins and suppose that its exponent measure µis concentrated on (0,∞)×[0,∞)^k. Define a random vector W via

P(W≤s) =µ(Bs∩A), s∈[0,∞)^k, (7.17) where A= (1,∞)×[0,∞)^k and Bs={x∈[0,∞)^k+1: (x⁽¹⁾, . . . ,x^(k)) ≤x⁽⁰⁾s}. Then W is the incremental process of ζ in (7.3).

Proof. First, we note that (7.17) indeed defines a valid cumulative distribution function. To this end, consider the measurable transformation

T : (0,∞)×[0,∞)^k→(0,∞)×[0,∞)^k, (x₀, . . . , x_k)7→

x₀,x1

x₀, . . . ,x_k x₀

.

Then,T(Bs∩A) = (1,∞)×[0,s] and the measureµ^T(·) =µ(T⁻¹((1,∞)× ·)) is a probability measure on [0,∞)^k. Since

µ(Bs∩A) =µ(T⁻¹((1,∞)×[0,s])) =µ^T([0,s]), the random vectorW is well-defined and has law µ^T.

By definition of the exponent measure, we have ζ = max^d _i∈NX_i, where Π = ^P_i∈

Nδ_Xi is a PPP on E with intensity measure µ. Then, the transformed point process T Π = P

i∈Nδ

(X_i⁽⁰⁾, X_i⁽¹⁾/X_i⁽⁰⁾, ..., X_i^(k)/X_i⁽⁰⁾) has intensity measure µ((c,∞)˜ ×[0,s]) =µT⁻¹((c,∞)×[0,s])

=µ(Bs∩((c,∞)×[0,∞)^k)) =c⁻¹µ(Bs∩A)

for any c >0, s ∈ [0,∞)^k, where we use the homogeneity propertyc⁻¹µ(dx) = µ(d(cx)).

Thus,T Π has the same intensity as^P_i∈

Nδ_(Ui,Wi), where^P_i∈

Nδ_Ui is a Fréchet point process and W_i,i∈N, are i.i.d. vectors with lawµ^T. Hence,

ζ= max^d

i∈N

T⁻¹ X_i⁽⁰⁾, X_i⁽¹⁾/X_i⁽⁰⁾, . . . , X_i^(k)/X_i⁽⁰⁾

= maxd i∈N

T⁻¹ U_i,W_i = max

i∈N

U_iW_i.

Example 7.2.8 (Bivariate Pareto-based distribution, cf. Resnick(2008, Ex. 5.16) and the references therein). For T ={t₀,t₁}, the extreme value distribution

P(ζ(t0)≤x, ζ(t1)≤y) = exp(−x⁻¹−y⁻¹+ (x+y)⁻¹), x,y >0,

is the max-limit of a bivariate Pareto distribution. Using the density of the exponent measure, µ(dx,dy) = 2(x+y)⁻³dxdy,

we get fors≥0

µ(Bs∩A) = Z ∞

1

Z sx 0

µ(dx,dy)

= Z ∞

1

x⁻¹1−(1 +s)⁻²dx= 1−(1 +s)⁻². Thus, for W in the representation (7.3), we have

P(W(t₁)≤s) = 1−(1 +s)⁻².

Example 7.2.9 (Symmetric logistic distribution, cf. Gumbel(1960)). Choosing the index setT ={t₀, . . . ,t_k}, the symmetric logistic distribution is given by

P(ζ(t₀)≤x₀, . . . , ζ(t_k)≤x_k) = exp

−x^−q₀ +. . .+x^−q_k ^1/q

, (7.18)

for x₀, . . . ,x_k>0 andq >1. Hence, the density of the exponent measure is

µ(dx₀, . . . ,dx_k) =^Pk_i=0x^−q_i ^{1/q−k−1Qk}_i=1(iq−1)^Qk_i=0x^−q−1_i dx₀. . . dx_k. Applying Proposition 7.2.7, the incremental process W in (7.3) is given by

P(W(t1)≤s1, . . . W(tk)≤sk) =1 +^Pk_i=1s^−q_i ^1/q−1. 7.2.1 Continuous sample paths

In this subsection, we provide an analog result to Theorem7.2.1, replacing convergence in the sense of finite-dimensional distributions by weak convergence on function spaces. For a Borel setU ⊂R^d, we denote byC(U) andC⁺(U) the space of non-negative and strictly positive continuous functions on U, respectively, equipped with the topology of uniform convergence on compact sets.

Theorem 7.2.10. LetK⊂R^d be compact and{η(t) :t∈K}aC⁺(K)-valued process in the MDA of a max-stable process {ζ(t) :t∈K} as in (7.3) in the sense of weak convergence on C(K). W.l.o.g., assume that n⁻¹maxⁿ_i=1η_i(·)⇒ ζ(·) as n→ ∞. Let W be the incremental process from (7.3) and Z a Pareto random variable, independent of W. Then, for any sequencea(n)→ ∞, as n→ ∞, we have the weak convergence on (0,∞)×C(K)

η(t₀) a(n), η(·)

η(t₀)

η(t₀)> a(n)

⇒(Z, W(·)).

Remark 7.2.11. Analogously to Whitt (1970, Thm. 5), weak convergence of a sequence of probability measuresP_n,n∈N, to some probability measureP onC(R^d)is equivalent to weak convergence of P_nr_j⁻¹ to P r⁻¹_j on C([−j, j]^d) for all j ≥1, where r_j :C(R^d) →C([−j,j]^d) denotes the restriction to the cube [−j, j]^d. Hence, Theorem 7.2.10 remains valid if the compact setK is replaced by R^d.

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