On the structure of maxstable processes Yizao Wang and Stilian Stoev - PowerPoint PPT Presentation

Outline Representations of Max–Stable Processes Structure and Classification Examples On the structure of max–stable processes Yizao Wang and Stilian Stoev University of Michigan, Ann Arbor Graybill VIII: Extreme Value Analysis Fort Collins, Colorado, June 24, 2009

Outline Representations of Max–Stable Processes Structure and Classification Examples Representations of Max–Stable Processes 1 Structure and Classification 2 Examples 3

Outline Representations of Max–Stable Processes Structure and Classification Examples Preliminaries Motivation: Consider i.i.d. processes { Y ( i ) t } t ∈ T . If for some a n ∼ n 1 /α ℓ ( n ) ( α > 0), we have � 1 � � Y ( i ) f . d . d . − → { X t } t ∈ T , ( n → ∞ ) , t a n t ∈ T 1 ≤ i ≤ n then X = { X t } t ∈ T is max–stable ( α − Fr´ echet). Def The process X = { X t } t ∈ T is max–stable ( α − Fr´ echet) if: d { X (1) ∨ · · · ∨ X ( n ) = n 1 /α { X t } t ∈ T , } t ∈ T t t for all n ∈ N , where X ( i ) = { X ( i ) t } t ∈ T are independent copies of X and where α > 0 . ◦ The margins of X are then α − Fr´ echet ( α > 0), namely: P { X t ≤ x } = exp {− σ α t x − α } , x > 0 , with σ α t > 0. Fine print: For simplicity, we focus on max–stable processes with α − Fr´ echet marginals. By transforming the margins, our theory applies to the most general definition of max–stable processes where the margins can be extreme value distributions of different types.

Outline Representations of Max–Stable Processes Structure and Classification Examples Spectral Representations For an α − Fr´ echet process ( α > 0) X = { X t } t ∈ T , we have: (a) de Haan’s spectral representation: ∞ � f t ( U i ) / Γ 1 /α X t = , ( t ∈ T ) i i =1 for a Poisson point process (Γ i , U i ) on (0 , ∞ ) × U with intensity dx × µ ( du ). (b) Extremal integrals: � e X t = f t ( u ) M α ( du ) , ( t ∈ T ) U for an α − Fr´ echet random sup–measure M α ( du ) on ( U , µ ). • The deterministic functions f t ( u ) ≥ 0 are called spectral functions of X and satisfy: � f t ( u ) α µ ( du ) < ∞ , ( t ∈ T ) . U Fine print: The measure space ( U , µ ) can be chosen ([0 , 1] , dx ) if the process { X t } t ∈ T is separable in probability , in particular, continuous in probability when T is a separable metric space.

Outline Representations of Max–Stable Processes Structure and Classification Examples Extremal integrals Let M α be a random α − Fr´ echet sup–measure on ( U , µ ). ◦ For simple functions f ( u ) = � n i =1 a i 1 A i ( u ), f ( u ) ≥ 0: � � e f ( u ) M α ( du ) = a i M α ( A i ) . U 1 ≤ i ≤ n � e U fdM α extends to all f ∈ L α ◦ The def of + ( µ ) and � � e f α d µ x − α } , P { fdM α ≤ x } = exp {− x > 0 . U U ◦ For f , g ∈ L α + ( µ ): � � � e e e ( af ∨ bg ) dM α = a fdM α ∨ b gdM α (max–linearity) U U U � � e e ◦ U fdM α and U gdM α are independent if and only if fg = 0, (mod µ ).

Outline Representations of Max–Stable Processes Structure and Classification Examples Benefits: For any f t ∈ L α + ( µ ) , t ∈ T , we get a max–stable process: � e X t := f t dM α U ◦ For the finite–dimensional distributions, we have: � e ( ∨ 1 ≤ i ≤ d x − 1 P { X t i ≤ x i , 1 ≤ i ≤ d } P { f t i ) dM α ≤ 1 } = i U � ( ∨ f α t i / x α = exp {− i ) d µ } . U Examples: � e • (moving maxima) X t := R f ( t − x ) M α ( dx ), with ( U , µ ) ≡ ( R , dx ) and f ∈ L α + ( dx ). • (mixed moving maxima) With ( U , µ ) = ( R × V , dx × d ν ): � e f ( x , v ) ∈ L α X t := f ( t − x , v ) M α ( dx , dv ) , + ( dx , d ν ) . R × V • (Brown–Resnick) With ( U , µ ) a probability space and { w t ( u ) } t ∈ R a standard Brownian motion on ( U , µ ): � e e w t ( u ) −| t | / 2 α M α ( du ) , X t := t ∈ R . U

Outline Representations of Max–Stable Processes Structure and Classification Examples Max-linear isometries Consider a max–stable process: � e X t = f t dM α , ( t ∈ T ) . U Some Natural Questions: • How does the structure of the f t ( u )’s determine the structure of X = { X t } t ∈ T and vice versa? • Given another representation { g t } ⊂ L α + ( V , ν ) � � � e d g t d � { X t } t ∈ T = M α , V what is the relationship between { f t } t ∈ T and { g t } t ∈ T ? Answer: There exists a max–linear isometry I : L α + ( U , µ ) → L α + ( V , ν ) , such that for all t ∈ T . I ( f t ) = g t ,

Outline Representations of Max–Stable Processes Structure and Classification Examples That is: for all f i ∈ L α I ( a 1 f 1 ∨ a 2 f 2 ) = a 1 I ( f 1 ) ∨ a 2 I ( f 2 ) , + ( U , µ ) , a i ≥ 0 , (max–linear) and � � I ( f ) α d ν = f α d µ (isometry) . V U ◦ Conversely, any max–linear isometry I : L α + ( U , µ ) → L α + ( V , ν ) yields an equivalent spectral represenation g t := I ( f t ) of the process X over ( V , ν ). ◦ Understanding the structure of the max–linear isometries is key! • In Wang & S., 2009, we extend results of Hardin, 1981/82 on linear isometries to max–linear isometries. • The role of these results in the theory of max–stable processes is best understood via the notion of a minimal representation.

Outline Representations of Max–Stable Processes Structure and Classification Examples Minimal spectral representations Def The spectral representation { f t } t ∈ T ⊂ L α + ( U , µ ) of X is minimal if: (i) (full support) supp { f t , t ∈ T } = U (mod µ ) (ii) (non–redundance) For any measurable A ⊂ U, there exists B ∈ ρ { f t , t ∈ T } ≡ σ { f t / f s , t , s ∈ T } , such that µ ( A ∆ B ) = 0 . ◦ This def is identical to the one of Rosi´ nski (1995) in the sum–stable case and similar to Hardin (1982) and to the proper rep in de Haan and Pickands (1986). • Why are minimal representations minimal? ◦ The full–support condition is natural. ◦ Consider the process � e t 2 sin 2 ( u ) M α ( du ) = t 2 Z , X t := [0 , 1] � e [0 , 1] sin 2 ( u ) M α ( du ). where Z = • This representation is clearly redundant. Note that ρ ( f t , t ∈ T ) = {∅ , [0 , 1] } �≃ B [0 , 1] .

Outline Representations of Max–Stable Processes Structure and Classification Examples ◦ We have a natural, simpler representation: � e d f t ( u ) � with f t ( u ) = t 2 , X t = M α ( du ) , U � [0 , 1] sin 2 α ( x ) dx . and trivial U = { 0 } , and µ ( du ) = c δ 0 ( du ), c := • The ratio σ − algebra ρ ( f t , t ∈ T ) captures best the ’minimal information’ needed to represent the process. • What are the benefits of minimal representations? Thm 1. (Wang & S.(2009)) Let { f t } t ∈ T ⊂ L α + ( µ ) be a minimal measurable rep of X. If { g t } t ∈ T ⊂ L α + ( V , ν ) is another measurable rep of X, then: ν − a . e . g t ( v ) = h ( v ) f t ( φ ( v )) , for some measurable h ≥ 0 and φ : V → U . The map φ is unique (mod ν ). If { g t } is also minimal, then φ is bimeasurable, ν ∼ µ ◦ φ and d µ ◦ φ ( v ) = h α ( v ) > 0 . d ν

Outline Representations of Max–Stable Processes Structure and Classification Examples Minimal representations with standardized support Consider the sets S I , N where I = 0 , 1 and 0 ≤ N ≤ ∞ : ◦ If I = 1, set S 1 , N = (0 , 1) ∪ { 1 , · · · , N } , (0 ≤ N ≤ ∞ ) ◦ If I = 0, set S 0 , N = { 1 , · · · , N } , (0 ≤ N ≤ ∞ ) ◦ By convention: S 1 , 0 = (0 , 1) , S 0 , ∞ = N and S 0 , 0 = ∅ . • Equip S I , N with the measure N � λ I , N ( x ) = dx + δ { i } ( dx ) . i =1 Fine print: Every standard Lebesgue space is isomorphic to some ( S I , N , λ I , N ). Def A minimal representation { f t } t ∈ T ⊂ L α + ( U , µ ) is said to have standardized support if, for some I , N: ( U , µ ) ≡ ( S I , N , λ I , N ) . Thm 2. (Wang & S., 2009) Every separable in probability α − Fr´ echet process X has a minimal representation with standardized support: � � � e d { X t } t ∈ T = f t dM α t ∈ T . S I , N

Outline Representations of Max–Stable Processes Structure and Classification Examples Continuous–Discrete Decomposition Consider an α − Fr´ echet process X with the minimal rep of standardized support: � � � e d { X t } t ∈ T = f t dM α t ∈ T . S I , N By setting � � e e X I X N t := f t dM α and t := f t dM α , S I , N ∩ (0 , 1) S I , N ∩ N we obtain the continuous–discrete decomposition: d = { X I t ∨ X N { X t } t ∈ T t } t ∈ T . The components X I t and X N are independent. t Intuition: Suppose I = 1 and N > 0. Then, � N � e X I X N t = f t dM α and t = f t ( i ) Z i , (0 , 1) i =1 echet, independent of X I where Z i = M α { i } are i.i.d. standard α − Fr´ t .