Functional regular variation

Definition

Let S be a compact metric space, such as [0, 1]2for spatial applications. We writeF₊=

C {S, [0,∞)} for the closed subset of the Banach space of continuous functions x : S → R

endowed with the uniform normx_∞= sup_s∈S|x(s)|. A measurable closed subsetC ofF₊is called a cone if t x∈C for any x∈C and t > 0. In the study of extremes, the conesC= {0} orC= {x ∈F₊: infs∈Sx(s)= 0} are often excluded fromF+to avoid the

appearance of limiting measures with infinite masses at the origin or on the coordinate axes. Let M_F+\Cdenote the class of Borel measures onB(F+\C) for any coneC, and

say that a set A∈B(F₊\C) is bounded away fromCif d (A,C)= inf_x∈A,y∈Cx − y_∞> 0. A sequence of measures {Λn}∞₁ ⊂ MF₊\C is said to converge to a limitΛ ∈ MF₊\C,

writtenΛn

ˆ

w

−→ Λ (Hult and Lindskog, 2005), if Λn(A)→ Λ(A) as n → ∞, for all A ∈ B(F₊\C) bounded away fromCwithΛ(∂A) = 0, where ∂A denotes the boundary of A. For equivalent definitions of this so-called ˆw-convergence, see Lindskog et al. (2014,

Theorem 2.1).

A stochastic process X with sample paths inF₊\C is regularly varying (Hult and Lindskog, 2005) if there exist a sequence of strictly positive continuous functions {an}∞_n=1with an(s)→ ∞ as n → ∞ for all s ∈ S and a measure Λ ∈ MF₊\Csuch that

n Pr a_n−1X ∈ ·−→ Λ(·), n → ∞;wˆ (1.18) then we write X ∈ RV(F₊\C, an,Λ). The limiting measure Λ satisfies a homogeneity

property of order−1/ξ, i.e., for any t > 0,

Λ(t A) = t−1/ξΛ(A), A ∈B(F₊\C), (1.19) for some positiveξ called the tail index.

Mapping theorem

For the remainder of this section, X is a regularly varying stochastic process andΛn

refers to the sequence of measures n Pr a−1_n X ∈ ·defined in (1.18). As the notion of exceedance for functional peaks-over-threshold analysis is not unique, a means to switch between definitions is required. For this reason, we introduce the mapping theorem to link the different representations.

Theorem 1.5 (Mapping theorem, Lindskog et al. (2014)) Let F and F be measur- able and complete metric spaces with conesCandC respectively and let h be a measur- able mapping

h :F₊\C,→F₊ \C

such that h−1 A =x∈F₊\C: f (x)∈ A is bounded away fromCfor any A ∈B(F₊ \

C _{) h(F}

+\C) bounded away fromC . Then the mapping !h : MF+\C→ MF₊ \C defined by

!h(Λ) = Λ◦h−1

is continuous atΛ ∈ M_F+\C providedΛ(Dh)= 0, where Dhis the set of discontinuity

points of h.

For the study of extremes, we will be interested in the exclusion fromF₊of a specific family of cones that is tied to the definition of exceedances. First, we define a risk func-

tional r :F₊→ [0,∞) as a continuous functional satisfying a homogeneity property,

i.e., a functional for which there existsκ > 0 such that

r (ax)= aκr (x), x∈F₊, a> 0.

As r (·) could be replaced by r (·)1/κwithout loss of generality, below we assume that

κ = 1. For any risk functional r , the setCr = {x ∈F+: r (x)= 0} is a closed cone ofF+.

We say that an r -exceedance is an event of the form {r (X )un}, where the sequence

of thresholds un> 0 is chosen such that Pr{r (X )un}→ 0 as n → ∞.

Now we consider the mapping hr:F+\C→F+\C , withC =C∪Cr, defined as

hr(x)=



x, r (x)> 0,

0, r (x)= 0. (1.20)

Applying Theorem 2.1 in Lindskog et al. (2014) and Theorem 1.5 with the mapping (1.20) ensures the convergence of {Λn}∞₁ restricted to the space {x∈F+\C: r (x)> 0}.

Corollary 1.1 SupposeΛn→ Λ in M(F+\C) as n→ ∞. Then

Λn◦ h−1r → Λ ◦ h−1r , n→ ∞, (1.21)

in M {F₊\ (C∪Cr)}.

Corollary 1.1 implies that if X ∈ RV(F₊\C, an,Λ) then the stochastic process X is

also regularly varying onF₊\ (C∪Cr) with limit measureΛ ◦ h−1r , and the converse is

true only ifΛ{Cr\C}= 0 orC=Cr. The mapping theorem and Corollary 1.1 are key

components in describing the limit distribution of r -exceedances.

Pseudo-polar decomposition

Alternative representations of the limiting measureΛ are obtained from pseudo- polar transformations. For a norm · angonF+, called the angular norm, and a risk

functional r , a pseudo-polar transformation h_rppis a mapF₊\C→ [0,∞) ×Sr such

that hpp_r (x)=  r = r (x),w = x xang  , x∈F₊\C, (1.22) whereSr is the unit sphere {x∈F+\C: r (x)> 0,xang= 1}. If {x ∈F+\C: r (x)= 0} = ,

then hppr is a homeomorphism with inverse

hpp_r −1(r , w )= r × w

Theorem 1.6 combines the family of pseudo-polar mappings and the mapping theo- rem to factorize the limiting measureΛ.

Theorem 1.6 Suppose X ∈ RV(F₊\ {0}, an,Λ) and let r be a risk functional. Then there

existsξ > 0 such that nPr  X an∈ h_rpp−1[r ,∞),W−→ Λwˆ _ξ{[r ,∞)} × σr(W), n→ ∞, r > 0,W⊂Sr, in M (F₊\Cr), where Λξ{[r ,∞)} = (r )−1/ξΛ(Ar),

with Ar = {x ∈F+\C: r (x)1} andσr is the probability measure onB(Sr),

σr(·) =

Λx∈F₊\Cr : r (x)1, x/xang∈ (·)

 Λ(Ar)

.

The converse holds if there exist a family of risk functionals rl (l= 1,...,L) with L1

such that for every l , X ∈ RV F₊\Crl, an,Λ



and L_l=1Crl= {0}.

Theorem 1.6 is used in Section 1.4.2 to describe the asymptotic distribution of r - exceedances. For specific risk functionals such as sup_s∈Sx(s) or_Sx(s)d s for which

Cr= {0}, the pseudo-polar transformation is a homeomorphism and the convergence

of the factorized version of the measures is equivalent to the regular variation of the stochastic process X onF₊\{0}. Alternatively, let Sl⊂ S (l = 1,...,L) satisfy"L_l=1Sl= S,

and define the functions

rSl(x)=



S

1{s∈ Sl}x(s)d s, l= 1,...,L.

In this case, we see that {x ∈F₊: rSl(x)= 0} = {0} individually but the family rSl

satisfies L_l=1Cri = {0}, and then the conditions for equivalence in Theorem 1.6 are

met, ensuring regular variation of the stochastic process X onF₊\ {0}.

Theorem 1.6 implies that there is a pseudo-polar decomposition ofΛ for any valid risk functional, and for this reason we link pseudo-polar representations in Corollary 1.2.

Corollary 1.2 For an angular norm·angand 1-homogeneous risk functionals r1and r2withCr1=Cr2, the angular probability measuresσr1andσr2 are linked by

σr1(dw )= Λ(Ar2) Λ(Ar1)  r1(w ) r2(w ) 1/ξ σr2(dw ), dw∈Sr, (1.23) where Ari = {x ∈F+\Cri | ri(x)1}, i= 1,2.

Equation (2.3) can thus be used to obtain the probability measureσr2 whenσr1 is

known, as for certain combinations of norm and risk functional the mathematical expression forσr is much simpler.

Link with multivariate regular variation

All the previous definitions and results also hold for finite dimensions, i.e., for L- dimensional random vectors, by replacing ˆw-convergence by vague convergence

(Resnick, 2007, Section 3.3.5) on M_RL

+\CL, the class of Borel measures onB(R

L

+\CL)

endowed with the · _∞norm, whereCLdenotes a cone inRL₊(Opitz, 2013b).

The mapping theorem also makes it possible to describe the relation between the multivariate theory and functional regular variation. Let s1, . . . , sLbe L> 1 locations in

S, and consider the map hproj:F+\C→ RL₊\CLdefined as hproj(x)= {x(s1), . . . , x(sL)},

whereCL= hproj(C). We now use the mapping hprojto prove that multivariate regular

variation is embedded in the functional theory.

Corollary 1.3 If X ∈ RV(F₊\ {0}, an,Λ), then

Λn◦ h−1_proj→ Λ ◦ h−1_proj, n→ ∞,

in MRL₊\CL.

Corollary 1.3 shows how applications, which are by nature finite-dimensional, are linked to the functional regular variation model. Indeed, suppose that a physical process X , for instance rainfall or temperature, is produced by a continuous stochastic process over a region of interest. In practice, we observe the process at only a finite number of locations. Supposing functional regular variation for X implies, through Corollary 1.3, that the vector of sampled locations is also regularly varying with the same limiting measure as the full process. For parametric models, this means that the parameters estimated using station measurements equal those of the functional model.

However, multivariate regular variation does not in general imply functional regu- lar variation and the condition for equivalence is still an open question. Following Lindskog et al. (2014, Theorem 4.1), Theorem 1.7 gives a necessary condition for equivalence when replacingF₊byR∞₊.

Theorem 1.7 Suppose X ∈ RV R∞₊ \C, an,Λ



and for every L1, the closed setC⊂F₊

is such that hproj,L(C) is closed inRL₊and

{x(s1), . . . , x(sL)}∈ hproj,L(C)⇒ {x(s1), . . . , x(sL), 0∞}∈C, (1.24)

where⇒ means implication. Then Λn→ Λ in M

R∞

+ \C



if and only if for all L1

such thatRL₊\ hproj,L(C)= ,

Λn◦ h−1proj,L→ Λ ◦ hproj,L−1 , n→ ∞, in MRL₊\CL.

For the family of conesC_rL= {x ∈ RL₊: r (x)> 0} defined in Section 1.4.1, condition (1.24) is ensured by the homogeneity of the risk functional, and thus Theorem 1.7 holds.

In document Functional Peaks-Over-Threshold Analysis for Complex Extreme Events (Page 32-37)