ERGODIC DECOMPOSITIONS OF STATIONARY MAX-STABLE PROCESSES IN TERMS OF THEIR SPECTRAL FUNCTIONS 6 1 CLÉMENTDOMBRYANDZAKHARKABLUCHKO 0 2 p Abstract. We revisit conservative/dissipative and positive/null decomposi- e tionsofstationarymax-stableprocesses. Originally,bothdecompositionswere S definedinanabstractwaybasedontheunderlyingnon-singularflowrepresen- tation. Weprovidesimplecriteriawhichallowtotellwhetheragivenspectral 0 functionbelongstotheconservative/dissipativeorpositive/nullpartofthede 3 Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to 0 in the Cesàro sense. For processes with ] locally bounded sample paths we show that a spectral function is dissipative R iff it converges to 0. Surprisingly, for such processes a spectral function is P integrable a.s. iff it converges to 0 a.s. Based on these results, we provide . newcriteriaforergodicity,mixing,andexistenceofamixedmovingmaximum h representationofastationarymax-stableprocessintermsofitsspectralfunc- t a tions. Inparticular, westudy adecomposition ofmax-stableprocesses which m characterizes themixingproperty. [ 2 v 1. Statement of main results 2 9 1.1. Introduction. Astochasticprocess(η(x)) on =Zd or =Rd iscalled x∈X 7 X X max-stable if 0 n 0 1 η f.=d.d.η for all n 1, i . n ≥ 1 i=1 _ 0 f.d.d. where η ,...,η are i.i.d. copies of η, is the pointwise maximum, and = 6 1 n denotes the equality of finite-dimensional distributions. Max-stable processes arise 1 W : naturallywhenconsideringlimits for normalizedpointwisemaxima ofindependent v and identically distributed (i.i.d.) stochastic processes and hence play a major role i X in spatial extreme value theory; see, e.g., de Haan and Ferreira [4]. We restrict r ourattentiontoprocesseswithnon-degenerate(non-constant)margins. Theabove a definition implies that the marginal distributions of η are 1–Fréchet, that is P[η(x) z]=e−c(x)/z for all z >0, ≤ where c(x)>0 is a scale parameter. A fundamental representation theorem by de Haan [3] states that any stochas- tically continuous max-stable process η can be represented (in distribution) as (1) η(x)= U Y (x), x , i i ∈X i≥1 _ 2010MathematicsSubjectClassification. Primary: 60G70;Secondary: 60G52,60G60,60G55, 60G10,37A10,37A25. Key words and phrases. max-stable random process, de Haan representation, non-singular flow,conservative/dissipativedecomposition,positive/nulldecomposition,ergodicprocess,mixing process,mixedmovingmaximumprocess. 1 2 CLÉMENTDOMBRYANDZAKHARKABLUCHKO where - (U ) isadecreasingenumerationofthepointsofaPoissonpointprocess i i≥1 on (0,+ ) with intensity measure u−2du, ∞ - (Y ) , which are called the spectral functions, are i.i.d. copies of a non- i i≥1 negative process (Y(x)) such that E[Y(x)]<+ for all x , x∈X ∞ ∈X - the sequences (U ) and (Y ) are independent. i i≥1 i i≥1 Inthispaper,wefocusonstationary max-stableprocessesthatplayanimportant role for modelling purposes; see, e.g., Schlather [21]. The structure of stationary max-stableprocesseswasfirstinvestigatedbydeHaanandPickands[5]whorelated them to non-singular flows (which are referred to as “pistons” in [5]). Using the analogy between max-stable and sum-stable processes and the works of Rosiński [13,14],RosińskiandSamorodnitsky[15]andSamorodnitsky[19,20]onsum-stable processes, the representation theory of stationary max-stable processes via non- singular flows was developed by Kabluchko [7], Wang and Stoev [26, 25], Wang et al.[24]. Inthesepapers,theconservative/dissipative(orHopf)andpositive/null(or Neveu)decompositionsfromnon-singularergodictheorywereusedtointroducethe correspondingdecompositionsη =η η andη =η η ofthestationarymax- C D P N ∨ ∨ stableprocess. Thesedefinitionswereratherabstract(seeSections3and4wherewe shallrecallthem)anddidnotallowtodistinguishbetweenconservative/dissipative orpositive/nullcasesby lookingjustatthe spectralfunctions Y fromthe de Haan i representation(1). Thepurposeofthispaperistoprovideaconstructive definition of these decompositions. Our main results in this direction can be summarized as follows. In Section 3 we shall prove that in the case when the sample paths of η are a.s. locally bounded, a spectral function Y belongs to the dissipative (=mixed i moving maximum) part of the process if and only if lim Y (x) = 0. The class x→∞ i of locally bounded processes is sufficiently general for applications. On the other hand,the assumptionoflocalboundednesscannotbe removed;seeExample 11. In Section 4 we shall prove that a spectral function Y belongs to the null (=ergodic) i part if and only if it converges to 0 in the Cesàro sense. In Section 5, we shall introduce one more decomposition which characterizes mixing. 1.2. Ergodic properties of max-stable processes. Our results can be used to give new criteria for ergodicity, mixing, and existence of mixed moving maximum representation of max-stable processes. These criteria extend and simplify the results of Stoev [22], Kabluchko and Schlather [8] and Wang et al. [24]. In the following,(η(x)) denotes a stationary,stochastically continuousmax- x∈X stable processon =Zd or Rd with de Haanrepresentation(1). In the casewhen =Rd,theproceXssY iscontinuousinL1byLemma2in[3]. SincecontinuityinL1 X implies stochastic continuity and since every stochastically continuous process has a measurable and separable version, we shall tacitly assume throughout the paper thatboth η andY aremeasurableand separableprocesses. These assumptions (as well as the assumption of stochastic continuity) are empty (and can be ignored)in the discrete case =Zd. X Our first result is a characterization of ergodicity. Let λ(dx) be the counting measure on Zd (in the discrete-time case) or the Lebesgue measure on Rd (in the continuous-time case), respectively. For r >0, write B =[ r,r]d . r − ∩X Theorem 1. For a stationary, stochastically continuous max-stable process η the following conditions are equivalent: ERGODIC DECOMPOSITIONS OF STATIONARY MAX-STABLE PROCESSES 3 (a) η is ergodic; (b) η is weakly mixing; (c) η has no positive recurrent component in its spectral representation, that is η =0; P (d) lim 1 E[Y(x) Y(0)]λ(dx)=0; r→∞ λ(Br) Br ∧ (e) lim 1 Y(x)λ(dx)=0 in probability; r→∞ λ(Br)RBr (f) liminf 1 Y(x)λ(dx)=0 almost surely. r→∞ λ(BRr) Br The equivalence of (a)R, (b), (c), (d) in Theorem 1 was known before (see The- orem 3.2 in [8] for the equivalence of (a), (b), (d) in the case d = 1, Theorem 8 in [7] for the equivalence of (a) and (c) in the case d=1, and Theorem 5.3 in [24] for an extension to the d-dimensional case). We shall prove in Section 3 that (c), (e), (f) are equivalent by exploiting a new characterization of the positive/null decomposition. The next theorem characterizes mixing (which is a stronger property than er- godicity). Theorem 2. For a stationary, stochastically continuous max-stable process η the following conditions are equivalent: (a) η is mixing; (b) η is mixing of all orders; (c) lim E[Y(x) Y(0)]=0; x→∞ ∧ (d) lim Y(x)=0 in probability. x→∞ Theequivalenceof(a),(b),(c)inTheorem3wasknownbefore(seeTheorem3.4 in[22]forthe equivalenceof(a)and(c), andTheorem1.1in[8]forthe equivalence of(a)and(b)). We shallproveinSection4that(c) isequivalentto (d). Moreover, we shall introduce a decomposition of the process η into a mixing part and a part containing no mixing components. Finally, we can characterize the mixed moving maximum property. The defini- tion of this property will be recalled in Section 3. Theorem 3. For a stationary, stochastically continuous max-stable process η with locally bounded sample paths, the following conditions are equivalent: (a) η has a mixed moving maximum representation; (b) η has noconservative component in its spectralrepresentation, that is η = C 0; (c) Y(x)λ(dx) <+ almost surely; X ∞ (d) lim Y(x)=0 almost surely. x→∞ R The equivalence of (a), (b), (c) in Theorem 3 was known before and holds even without the assumption of local boundedness (see Sections 3.1, 3.2 and the ref- erences therein). Our main contribution is an alternative characterization of the conservative/dissipative decomposition stated in Proposition 10 that implies the equivalence of (c) and (d). This equivalence may look strange at a first glance because neither (c) implies (d) nor it is implied by (d) for a general stochastic process Y. However,the process Y appearingin Theorems 1, 2, 3 is subject to the restriction that it leads to a stationary process η. Processes Y with this property werecalledBrown–Resnickstationaryin[9]. AnotherrestrictionappearinginThe- orem3 is the localboundedness ofη. This conditioncannotbe removed,as willbe 4 CLÉMENTDOMBRYANDZAKHARKABLUCHKO shown in Example 11. A special case of the implication (d) (c) when logY is a ⇒ Gaussianprocess with stationaryincrements and certain drift was obtained in [26, Theorem 7.1]. The rest of the paper is structured as follows. Section 2 is devoted to pre- liminaries on non-singular ergodic theory and cone decompositions for max-stable processes. Section 3 reviews known results on the conservative/dissipative decom- positions and provides an alternative definition via a simple cone decomposition with an emphasis on the case of locally bounded max-stable processes. Section 4 introducesthepositive/nulldecompositionandproposesanalternativeconstruction via another simple cone decomposition. In Section 5 we study mixing. 2. Preliminaries 2.1. Non-singular flow representations of max-stable processes. We recall some information on non-singular flow representations of stationary max-stable processes. For more details on non-singular ergodic theory, the reader should refer to Krengel [10], Aaronson [1] or Danilenko and Silva [2]. Definition 4. A measurable non-singular flow on a measure space (S, ,µ) is a B family of functions φ :S S, x , satisfying x → ∈X (i) (flow property) for all s S and x ,x , 1 2 ∈ ∈X φ (s)=s and φ (s)=φ (φ (s)); 0 x1+x2 x2 x1 (ii) (measurability) the mapping (x,s) φ (s) is measurable from S to S; x 7→ X × (iii) (non-singularity) for all x , the measures µ φ−1 and µ are equivalent, ∈X ◦ x i.e. for all A , µ(φ−1(A))=0 if and only if µ(A)=0. ∈B x The non-singularity property ensures that one can define the Radon–Nikodym derivative d(µ φ ) x (2) ω (s)= ◦ (s). x dµ By the measurability property, one may assume that the mapping (x,s) ω (s) x 7→ is jointly measurable on S. X × According to de Haan and Pickands [5], see also [7] and [26], any stochastically continuousstationary max-stableprocessηadmitsa(distributional)representation of the form (3) η(x)= U f (s ), x , i x i ∈X i≥1 _ where f (s)=ω (s)f (φ (s)) and x x 0 x - (φ ) is a measurable non-singular flow on some σ-finite measure space x x∈X (S, ,µ), with ω (s) defined by (2), x B - f L1(S, ,µ) is non-negative such that the set f = 0 contains no 0 0 ∈ B { } (φ ) –invariant set B of positive measure, x x∈X ∈B - (s ,U ) issomeenumerationofthepointsofthePoissonpointprocess i i i≥1 { } on S (0,+ ) with intensity µ(ds) u−2du. × ∞ × If(S, ,µ)isaprobability space,thepointprocess (s ,U ) canbegenerated i i i≥1 B { } by taking (s ) to be i.i.d. randomelements in S with probability distribution µ, i i≥1 that are independent from (U ) . Thus, one easily recovers the de Haan repre- i i≥1 sentation (1) by considering the i.i.d. stochastic processes Y (x)=f (s ), i 1. i x i ≥ ERGODIC DECOMPOSITIONS OF STATIONARY MAX-STABLE PROCESSES 5 The flow representation (3) is comonly written as an extremal integral e (4) η(x)= f (s)M(ds), x , x ∈X ZS whereM(ds)denotes a 1-Fréchetrandomsup-measureon (S, )withcontrolmea- B sureµ. ThereadershouldrefertoStoevandTaqqu[23]formoredetailsonextremal integrals. In the present paper, one can simply view the extremal integral (4) as a shorthand for the pointwise maximum over a Poisson point process (3). 2.2. Cone-baseddecompositions. InthespiritofWangandStoev[26,Theorem 4.2]andDombryandKabluchko[6,Lemma16],weshallusedecompositionsofmax- stable processes based on cones. We denote by = ( ,[0,+ )) 0 the set 0 F F X ∞ \{ } of non-negative measurable functions on excluding the zero function. A subset X is called a cone if for all f and u>0, uf . The cone is said to be 0 C ⊂F ∈C ∈C C shift-invariant if for all f and x we have f( +x) . ∈C ∈X · ∈C Lemma 5 (Lemma 16in[6]). Let and be two shift-invariant cones such that 1 2 = and = ∅. LeCt η be aCstationary max-stable process given by 0 1 2 1 2 F C ∪C C ∩C representation (1) such that the events Y and Y are measurable. i 1 i 2 { ∈ C } { ∈ C } Consider the decomposition η =η η with 1 2 ∨ η (x)= U Y (x) and η (x)= U Y (x) . 1 i i 1{Yi∈C1} 2 i i 1{Yi∈C2} i≥1 i≥1 _ _ Then, η and η are stationary and independent max-stable processes whose distri- 1 2 bution depends only on the distribution of η and not on the specific representation (1). 3. Conservative/dissipative decomposition 3.1. Definition of the conservative/dissipative decomposition. We recall theHopf(orconservative/dissipative)decompositionfromnon-singularergodicthe- ory; see Aaronson [1]. We start with the discrete case =Zd. X DAemfienaistuioranbl6e.sCetonWsiderSaimsesaasiudretospbaecew(aSn,dBer,iµn)gainfdthaensoent-ssiφn−gu1(laWr)fl,oxw(φZx)dx,∈aZrde. ⊂ x ∈ disjoint. The Hopf decomposition theorem states that there exists a partition of S into two disjoint measurable sets S =C D, C D =∅, such that ∪ ∩ (i) C and D are (φx)x∈Zd–invariant, (ii) there exists no wandering set W C with positive measure, ⊂ (iii) there exists a wandering set W0 ⊂D such that D =∪x∈Zdφx(W0). This decomposition is unique mod µ and is called the Hopf decomposition of S associated with the flow (φx)x∈Zd; the sets C and D are called the conservative and dissipative parts respectively. In the case when =Rd, we follow Roy [17] by X defining the Hopf decomposition of S associated with a measurable flow (φx)x∈Rd as the Hopf decomposition associated with the discrete skeleton flow (φx)x∈Zd. One can then introduce the conservative/dissipative decomposition of the max- stable process η given by (3), (4): we have η =η η with C D ∨ e e (5) η (x)= f (s)M(ds) and η (x)= f (s)M(ds), x . C x D x ∈X ZC ZD 6 CLÉMENTDOMBRYANDZAKHARKABLUCHKO The processes η and η are independent and their distribution depends only on C D the distribution of η and not on the particular choice of the representation (3). The importance of the conservative/dissipative decomposition comes from the notion of mixed moving maximum representation. Definition 7. A stationary max-stable process (η(x)) is said to have a mixed x∈X moving maximum representation (shortly M3-representation) if f.d.d. η(x) = V Z (x X ), x , i i i − ∈X i≥1 _ where - (X ,V ),i 1 is a Poisson point process on (0,+ ) with intensity i i { ≥ } X × ∞ λ(dx) u−2du, × - (Z ) are i.i.d. copies of a non-negative measurable stochastic process Z i i≥1 on satisfying E[ Z(x)λ(dx)] <+ , X X ∞ - (X ,V ),i 1 and (Z ) are independent. i i i i≥1 { ≥ } R The following important theorem relates the dissipative/conservative decompo- sitionandtheexistenceofanM3-representation;seeWangandStoev[26,Theorem 6.4] in the max-stable case with d=1 or Roy [17, Theorem 3.4] in the sum-stable case with d 1. ≥ Theorem8. Letηbeastationarymax-stableprocessgivenbythenon-singularflow representation (3). Then, η has an M3-representation if and only if η is generated by a dissipative flow. 3.2. Characterization using spectral functions. Thefollowingsimpleintegral test on the spectral functions allows us to retrieve the conservative/dissipative de- composition; see Roy and Samorodnitsky [18, Proposition], Roy [17, Proposition 3.2] and Wang and Stoev [26, Theorem 6.2]. Theorem 9. We have (i) f (s)λ(dx)= µ(ds)–a.e. on C; X x ∞ (ii) f (s)λ(dx)< µ(ds)–a.e. on D. RX x ∞ ConsiRderastationarymax-stableprocessηgivenbydeHaan’srepresentation(1). In view of Theorem 9, we introduce the cones of functions (6) = f ; f(x)λ(dx) = , C 0 F ∈F ∞ (cid:26) ZX (cid:27) (7) = f ; f(x)λ(dx) < . D 0 F ∈F ∞ (cid:26) ZX (cid:27) These cones are clearly shift-invariant and, assuming that Y is jointly measur- able and separable, the events Y and Y are measurable. Using C D { ∈ F } { ∈ F } Lemma 5, we define (8) η (x)= U Y (x) and η (x)= U Y (x) . C i i 1{Yi∈FC} D i i 1{Yi∈FD} i≥1 i≥1 _ _ Using Theorem 9 and Lemma 5 one can easily prove that we retrieve (in distribu- tion) the conservative/dissipative decomposition (5) based on the flow representa- tion (3). ERGODIC DECOMPOSITIONS OF STATIONARY MAX-STABLE PROCESSES 7 The main contribution of this section concerns the case when the max-stable process η has locally bounded sample paths, which is usually the case in applica- tions. Interestingly, one can then introduce another, more simple and convenient, cone decomposition equivalent to (8). Consider ˜ = f ; limsupf(x)>0 , C 0 F ∈F (cid:26) x→∞ (cid:27) ˜ = f ; lim f(x)=0 . D 0 F ∈F x→∞ n o Note that since the process Y is assumed to be separable, the events Y ˜ C { ∈ F } and Y ˜ are measurable. C { ∈F } Proposition 10. Let η be a stationary max-stable process given by de Haan’s rep- resentation (1) and assume that η has locally bounded sample paths. Then, modulo null sets, Y = Y ˜ and Y = Y ˜ . C C D D { ∈F } { ∈F } { ∈F } { ∈F } We deduce that the decomposition η˜ (x)= U Y (x) and η˜ (x)= U Y (x) . C i i 1{Yi∈F˜C} D i i 1{Yi∈F˜D} i≥1 i≥1 _ _ is almost surely equal to the decomposition (8). Proof. We consider first the discrete setting = Zd. The convergence of the X series f(x) implies the convergence lim f(x) = 0 so that the inclusion x∈Zd x→∞ Y Y ˜ is trivial. We need only to prove the converse inclusion { ∈ FPD} ⊂ { ∈ FD} Y ˜ Y . Then, the equality Y = Y ˜ (modulo null D D D D { ∈F }⊂ { ∈F } { ∈F } { ∈F } sets) implies the equality of the complementary sets, i.e. Y = Y ˜ . C C { ∈F } { ∈F } Proof of the inclusion Y ˜ Y . Let Y˜ = Y and η˜ = { ∈ FD} ⊂ { ∈ FD} D 1{Y∈F˜D} D U Y . We shall show that η˜ admits an M3-representation. By The- ∨i≥1 i i1{Yi∈F˜D} D orem8, this implies that Y˜ belongs a.s. to and hence Y ˜ Y D D D D F { ∈F }⊂{ ∈F } modulo null sets. For the sake of notational convenience, we assume that Y ˜ D a.s. so that Y˜ = Y and η˜ = η. We prove that η has an M3-representation∈wFith D D a strategy similar to the proof of Theorem 14 in Kabluchko et al. [9]. We sketch only the main lines. We introduce the random variables Y (X + ) i i (9) X =argmaxY (x), Z ()= · , V =U maxY (x). i i i i i i x∈X · maxx∈X Yi(x) x∈X If the argmax is not unique, we use the lexicographically smallest value. Clearly, we have U Y (x)=V Z (x X ) for all x so that i i i i i − ∈X η(x)= V Z (x X ). i i i − i≥1 _ It remains to check that (X ,V ,Z ) has the properties required in Definition 7, i i i i≥1 i.e. is a Poisson point process on (0, ) with intensity measure λ(dx) 0 X × ∞ ×F × u−2du Q(df), where Q is a probability measure on . Clearly, (X ,V ,Z ) 0 i i i i≥1 × F is a Poisson point process as the image of the original point process (U ,Y ) . i i i≥1 Its intensity is the image of the intensity of the original point process. With a 8 CLÉMENTDOMBRYANDZAKHARKABLUCHKO straightforward transposition of the arguments of [9, Theorem 14], one can check that it has the required form. We now turn to the case =Rd. The convergenceof the integral f(x)λ(dx) X X does not imply the convergence lim f(x) = 0. But it is easy to prove that for x→∞ R K =[ 1/2,1/2]d, the convergenceofthe integral sup f(x+u)λ(dx) implies − X u∈K the convergence lim f(x)=0. We introduce the cone x→∞ R ′ = f ; supf(x+u)λ(dx)< . FD ∈F0 ∞ (cid:26) ZX u∈K (cid:27) The inclusions of cones ′ and ′ ˜ imply the trivial inclusions of FD ⊂ FD FD ⊂ FD events Y ′ Y and Y ′ Y ˜ . { ∈FD}⊂{ ∈FD} { ∈FD}⊂{ ∈FD} We shall prove below that, modulo null sets, Y Y ′ and Y ˜ Y { ∈FD}⊂{ ∈FD} { ∈FD}⊂{ ∈FD} whence we deduce the equalities, modulo null sets, Y = Y ′ = Y ˜ , { ∈FD} { ∈FD} { ∈FD} proving the proposition. Proof of the inclusion Y Y ′ . Let Y = Y and η = { ∈ FD} ⊂ { ∈ FD} D 1{Y∈FD} D U Y be the dissipative part of η. Theorem 8 implies that η has an ∨i≥1 i i1{Yi∈FD} D M3-representationof the form f.d.d. η (x) = V Z (x X ), x . D i D,i i − ∈X i≥1 _ The fact that η is locally bounded implies that η is a.s. finite on K and D θ (K) (10) P supη (x) z =exp D D ≤ − z (cid:20)x∈K (cid:21) (cid:18) (cid:19) with θ (K)=E supZ (x y)λ(dy) < . D D − ∞ (cid:20)ZX x∈K (cid:21) We deduce that sup Z (x y)λ(dy) is a.s. finite and hence, Z belongs X x∈K D − D a.s. to the cone ′ . This implies that Y ′ almost surely, whence FRD 1{Y∈FD} ∈ FD Y Y ′ modulo null sets. { ∈FD}⊂{ ∈FD} Proof of the inclusion Y ˜ Y . With the same notation as in the D D { ∈ F } ⊂ { ∈ F } dicrete case, we show that η˜ is generated by a dissipative flow and hence has an D M3-representation. By Theorem 8, this implies that Y˜ belongs a.s. to and D D proves the inclusion Y ˜ Y . Note that the discrete sFkeleton D D { ∈ F } ⊂ { ∈ F } Y˜Dskel =(Y˜D(x))x∈Zd satisfieslimx→∞Y˜Dskel =0. We deduce Y˜Dskel ∈F˜D a.s.which is equivalent to Y˜skel a.s. (see the proof above in the discrete case). Hence D ∈ FD (η˜D(x))x∈Zd is generatedby a dissipative flow and this implies that (η˜D(x))x∈Rd is generated by a dissipative flow (see [17, Section 2]). (cid:3) Proof of Theorem 3. Theequivalenceof(a),(b),(c)inTheorem3wasknownbefore and holds even without the assumption of local boundedness (see Section 3.1 and the reference therein). The equivalence of (c) and (d) holds under the assumption of local boundedness and is a straightforwardconsequence of Proposition 10. (cid:3) ERGODIC DECOMPOSITIONS OF STATIONARY MAX-STABLE PROCESSES 9 Example11. Theassumptionthatthesamplepathsofηshouldbelocallybounded cannotbe removedfromProposition 10. To see this, consider the following (deter- ministic) process Z: ∞ Z(x)= f(n2(x n)), x R, − ∈ n=1 X where f(t)=(1 t2) . The process Z is non-zero only on the intervals of the |t|≤1 − 1 form (n 1 ,n+ 1 ), n N. Its sample paths are continuous and bounded on R. − n2 n2 ∈ The M3-process η corresponding to Z is well-defined because Z(x)dx< . On R ∞ the other hand, P[Z ˜ ]=0 and hence, P[Y ˜ ]=0, where Y is the spectral ∈FD ∈FD R function of η from the de Haan representation (1). It is easy to check that θ P sup η(x) z =exp [0,1] , z >0, "x∈[0,1] ≤ # (cid:18)− z (cid:19) with θ = sup Z(x y) dy =+ , [0,1] ZR x∈[0,1] − ! ∞ whencesup η(x)=+ a.s.andthesamplepathsofηarenotlocallybounded. x∈[0,1] ∞ 4. Positive/null decomposition 4.1. Definition of the positive/null decomposition. We startby defining the Neveu decompositionof the non-singularflow (φ ) ; see, e.g., Krengel[10, The- x x∈X orem 3.9], Samorodnitsky [20] or Wang et al. [24, Theorem 2.4]. Definition 12. Consider a measure space (S, ,µ) and a measurable non-singular B flow (φ ) on S. A measurable set W S is said to be weakly wandering with x x∈X ⊂ respect to (φx)x∈X if there exists a sequence {xn}n∈N ⊂ X such that φ−xn1(W)∩ φ−1(W)=∅ for all n=m. xm 6 The Neveu decomposition theorem states that there exists a partition of S into two disjoint measurable sets S =P N, P N =∅, such that ∪ ∩ (i) P and N are (φ ) –invariant for all x , x x∈X ∈X (ii) P has no weakly wandering set of positive measure, (iii) N is a union of countably many weakly wandering sets. This decomposition is unique mod µ and is called the Neveu decomposition of S associatedwith(φ ) ;P andN arecalledthepositive andnull componentswith x x∈X respect to (φ ) , respectively. It can be shown that P is the largest subset of S x x∈X supporting a finite measure which is equivalent to µ and invariant under the flow (φ ) ([24,Lemma2.2]). Hence,thereexistsafinitemeasurewhichisequivalent x x∈X to µ and invariant under the flow if and only if N =∅ mod µ. Thecorrespondingpositive/nulldecompositionofthestationarymax-stablepro- cess η represented as in (3), (4) is given by η =η η with P N ∨ e e (11) η (x)= f (s)M(ds) and η (x)= f (s)M(ds), x . P x N x ∈X ZP ZN The positive and null components η and η are independent, stationary max- P N stable processes,and their distribution does notdepend on the particularchoice of the representation(3). 10 CLÉMENTDOMBRYANDZAKHARKABLUCHKO 4.2. Characterization using spectral functions. An integraltest on the spec- tral functions which allows to retrieve the positive/null decomposition is known in theone-dimensionalcase(seeSamorodnitsky[20]orWangandStoev[26,Theorem 5.3]). Theorem13. Consider thecased=1andintroducetheclass ofpositiveweight W functions w: (0,+ ) such that w(x)λ(dx) < and w(x) and w( x) are X → ∞ X ∞ − non-decreasing on [0,+ ). Then we have X ∩ ∞ R (i) For all w , f (s)w(x)λ(dx) = µ(ds)–a.e. on P; ∈W X x ∞ (ii) For some w , f (s)w(x)λ(dx) < µ(ds)–a.e. on N. ∈WR X x ∞ The next theorem is a nRew integral test characterizing the positive/null decom- position. This test is simpler than Theorem 13 and is valid for all d 1. Recall ≥ thatwe write B =[ r,r]d forr >0. In the nexttheoremandits corollarywe r − ∩X do not require the sample paths of η to be locally bounded. Theorem 14. Let η be a stationary, stochastically continuous max-stable process given by the non-singular flow representation (3). We have (i) lim 1 f (s)λ(dx) exists and is positive µ(ds)–a.e. on P; r→∞ λ(Br) Br x (ii) liminf 1 f (s)λ(dx)=0 µ(ds)–a.e. on N. r→∞ λ(BRr) Br x Proof. We consider the pRositive case and the null case separately. Case 1. Assume first that η is generated by a positive flow. Then, there is a probability measure µ∗ on (S, ) which is equivalent to µ and which is invariant B under the flow. Note that any property holds µ–a.e. if and only if it holds µ∗–a.e. We denote by D(s)= dµ (s) (0, ) the Radon–Nikodym derivative and observe dµ∗ ∈ ∞ that for every x , the function f∗(s):=f (s)D(s) satisfies ∈X x x (12) f∗(s)=f∗(φ (s)) for λ µ–a.e. (x,s) S. x 0 x × ∈X × Indeed, by definition of f∗ and ω , we have x x D(s)ω (s) f∗(s)=D(s)f (s)=D(s)ω (s)f (φ (s))= x f∗(φ (s)). x x x 0 x D(φ (s)) 0 x x However,recalling the definition (2) of ω (s) and that D(s)= dµ (s) (0, ), we x dµ∗ ∈ ∞ obtain D(s)ω (s) dµ d(µ φ ) d(µ∗ φ ) d(µ∗ φ ) x x x x = (s) ◦ (s) ◦ (s)= ◦ (s)=1 D(φ (s)) dµ∗ dµ d(µ φ ) dµ∗ x x ◦ µ–a.e. for every x because the measure µ∗ is invariant. This yields (12). By ∈ X the multiparameter Birkhoff Theorem (see [24, Theorem 2.8]), we have 1 (13) lim f∗(s)λ(dx)=E[f∗ ] µ∗–a.e., r→∞λ(Br)ZBr x 0|I where is the σ-algebra of (φ ) –invariant measurable sets and E denotes the x x∈X I expectationw.r.t.µ∗. We provethatthe conditionalexpectationonthe right-hand sideisa.e.strictlypositive. ThesetB = E[f∗ ]=0 ismeasurableand(φ ) – { 0|I } x x∈X invariant. Moreover,f∗ (andhence,f )vanishesa.e.onB sincef∗ isnon-negative. 0 0 0 This implies that µ(B) = 0 by the second condition in the definition of the flow