Limit theorems for excursion sets of stationary random fields Evgeny Spodarev | 23.01.2013 WIAS, Berlin
page 2 LT for excursion sets of stationary random fields | Overview | 23.01.2013 Overview ◮ Motivation ◮ Excursion sets of random fields ◮ Their geometric functionals ◮ Minkowski functionals of excursion sets: state of art ◮ CLT for the volume of excursion sets of stationary random fields ◮ Second order quasi-associated fields ◮ Examples: Shot noise, Gaussian case ◮ PA - or NA -fields (possibly not second order!) ◮ Examples: infinitely divisible, max- and α -stable fields ◮ Multivariate CLT with a Gaussianity test ◮ Asymptotics of the mean Minkowski functionals of excursions of non-stationary Gaussian random fields ◮ Open problems
page 3 LT for excursion sets of stationary random fields | Motivation | 23.01.2013 Motivation Paper surface Simulated Gaussian field (Voith Paper, Heidenheim) E X ( t ) = 126 � � − � t � 2 r ( t ) = 491 exp 56 ◮ Is the paper surface Gaussian?
page 4 LT for excursion sets of stationary random fields | Excursion sets and their geometric functionals | 23.01.2013 Excursion sets Let X be a measurable real-valued random field on R d , d ≥ 1 and let W ⊂ R d be a measurable subset. Then for u ∈ R A u ( X , W ) := { t ∈ W : X ( t ) ≥ u } is called the excursion set of X in W over the level u.
page 5 LT for excursion sets of stationary random fields | Excursion sets and their geometric functionals | 23.01.2013 Centered Gaussian random field on [ 0 , 1 ] 2 , r ( t ) = exp ( − � t � 2 / 0 . 3 ) , Levels: u = − 1 . 0, 0 . 0, 1 . 0
page 6 LT for excursion sets of stationary random fields | Excursion sets and their geometric functionals | 23.01.2013 Geometric functionals of excursion sets Minkowski functionals V j , j = 0 , . . . , d : ◮ d = 1: ◮ Length of excursion intervals V 1 ( A u ( X , W )) ◮ Number of upcrossings V 0 ( A u ( X , W )) ◮ d ≥ 2: ◮ Volume | A u ( X , W ) | = V d ( A u ( X , W )) ◮ Surface area H d − 1 ( ∂ A u ( X , W )) = 2 V d − 1 ( A u ( X , W )) ◮ . . . ◮ Euler characteristic V 0 ( A u ( X , W )) , topological measure of “porosity” of A u ( X , W ) . In d = 2: V 0 ( A ) = # { connented components of A } − # { holes of A } V j , j = 0 , . . . , d − 2 are well defined for excursion sets of sufficiently smooth (at least C 2 ) random fields, see Adler and Taylor (2007).
page 7 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 ◮ Gaussian random fields ◮ Moments: ◮ Number of upcrossings, d = 1: Kac (1943), Rice (1945); Bulinskaya (1961); Cramer & Leadbetter (1967); Belyaev (1972) ◮ Minkowski functionals, d > 1: Adler (1976, 1981); Wschebor (1983); Adler & Taylor (2007); Azais & Wschebor (2009); S. & Zaporozhets (2012) ◮ CLTs: ◮ Stationary processes, d = 1: Malevich (1969); Cuzick (1976); Piterbarg (1978); Elizarov (1988); Slud (1994); Kratz (2006) ◮ Volume, d ≥ 2: Ivanov & Leonenko (1989) ◮ Surface area, d ≥ 2: Kratz & Leon (2001, 2010) ◮ Surface area, d ≥ 2, FCLT: Meschenmoser & Shashkin (2011-12), Shashkin (2012) ◮ Non-Gaussian random fields ◮ Moments: Adler, Samorodnitsky & Taylor (2010) ◮ CLTs: Bulinski, S. & Timmermann (2012); Karcher (2012); Demichev & Schmidt (2012)
page 8 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Growing sequence of observation windows A sequence of compact Borel sets ( W n ) n ∈ N is called a Van Hove sequence (VH) if W n ↑ R d with | ∂ W n ⊕ B r ( 0 ) | n →∞ | W n | = ∞ lim and lim = 0 , r > 0 . | W n | n →∞
page 9 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Theorem (CLT for the volume of A u at a fixed level u ∈ R ) Let X be a strictly stationary random field satisfying some additional conditions and u ∈ R fixed. Then, for any sequence of VH -growing sets W n ⊂ R d , one has | A u ( X , W n ) | − P ( X ( 0 ) ≥ u ) · | W n | d � � 0 , σ 2 ( u ) − → N � | W n | as n → ∞ . Here � σ 2 ( u ) = R d cov ( 1 { X ( 0 ) ≥ u } , 1 { X ( t ) ≥ u } ) dt .
page 10 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Second order quasi-associated random fields Let X = { X ( t ) , t ∈ R d } have the following properties: ◮ square-integrable ◮ has a continuous covariance function r ( t ) = Cov ( X ( o ) , X ( t )) , t ∈ R d � t � − α ◮ | r ( t ) | = O � � for some α > 3 d as � t � 2 → ∞ 2 ◮ X ( 0 ) has a bounded density ◮ quasi-associated. Then σ 2 ( u ) ∈ ( 0 , ∞ ) (Bulinski, S., Timmermann (2012)).
page 11 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Quasi-association X ( t ) , t ∈ R d � � A random field X = with finite second moments is called quasi-associated if � � | cov ( f ( X I ) , g ( X J )) | ≤ Lip i ( f ) Lip j ( g ) | cov ( X ( i ) , X ( j )) | i ∈ I j ∈ J for all finite disjoint subsets I , J ⊂ R d , and for any Lipschitz functions f : R card ( I ) → R , g : R card ( J ) → R where X I = { X ( t ) , t ∈ I } , X J = { X ( t ) , t ∈ J } . Idea of the proof of the Theorem: apply a CLT for ( BL , θ ) -dependent stationary centered square-integrable random fields on Z d (Bulinski & Shashkin, 2007).
page 12 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 ( BL , θ ) -dependence A real-valued random field X = { X ( t ) , t ∈ R d } is called ( BL , θ ) -dependent, if there exists a sequence θ = { θ r } r ∈ R + 0 , θ r ↓ 0 as r → ∞ such that for any finite disjoint sets I , J ⊂ T with dist ( I , J ) = r ∈ R + 0 , and any functions f ∈ BL ( | I | ) , g ∈ BL ( | J | ) , one has � � | cov ( f ( X I ) , g ( X J )) | ≤ Lip i ( f ) Lip j ( g ) | cov ( X ( i ) , X ( j )) | θ r , i ∈ I j ∈ J where � | cov ( X ( k ) , X ( t )) | dt . θ r = sup R d \ B r ( k ) k ∈ R d
page 13 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 CLT for ( BL , θ ) -dependent stationary random fields Theorem (Bulinski & Shashkin, 2007) Let Z = { Z ( j ) , j ∈ Z d } be a ( BL , θ ) -dependent strictly stationary centered square-integrable random field. Then, for any sequence of regularly growing sets U n ⊂ Z d , one has | U n | d � 0 , σ 2 � � S ( U n ) / → N − as n → ∞ , with σ 2 = � cov ( Z ( 0 ) , Z ( j )) . j ∈ Z d
page 14 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Special case - Shot noise random fields The above CLT holds for a stationary shot noise random field X = { X ( t ) , t ∈ R d } given by X ( t ) = � i ∈ N ξ i ϕ ( t − x i ) where ◮ { x i } is a homogeneous Poisson point process in R d with intensity λ ∈ ( 0 , ∞ ) ◮ { ξ i } is a family of i.i.d. non–negative random variables with E ξ 2 i < ∞ and the characteristic function ϕ ξ ◮ { ξ i } , { x i } are independent ◮ ϕ : R d → R + is a bounded and uniformly continuous Borel function with � t � − α � � ϕ ( t ) ≤ ϕ 0 ( � t � 2 ) = O as � t � 2 → ∞ 2 for a function ϕ 0 : R + → R + , α > 3 d , and � � �� � � � � � exp λ R d ( ϕ ξ ( s ϕ ( t )) − 1 ) dt � ds < ∞ . � � R d
page 15 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Special case - Gaussian random fields Consider a stationary Gaussian random field X = { X ( t ) , t ∈ R d } with the following properties: a , τ 2 � ◮ X ( 0 ) ∼ N � ◮ has a continuous covariance function r ( · ) � t � − α ◮ ∃ α > d : | r ( t ) | = O � � as � t � 2 → ∞ 2
page 16 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Special case - Gaussian random fields Let X be the above Gaussian random field and u ∈ R . Then, � ρ ( t ) − ( u − a ) 2 σ 2 ( u ) = 1 � 1 τ 2 ( 1 + r ) dr dt , √ 1 − r 2 e 2 π R d 0 where ρ ( t ) = corr ( X ( 0 ) , X ( t )) . In particular, for u = a σ 2 ( a ) = 1 � R d arcsin ( ρ ( t )) dt . 2 π
page 17 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Positively or negatively associated random fields Let X = { X ( t ) , t ∈ R d } have the following properties: ◮ stochastically continuous (evtl. not second order!) ◮ σ 2 ( u ) ∈ ( 0 , ∞ ) ◮ P ( X ( 0 ) = u ) = 0 for the chosen level u ∈ R ◮ positively ( PA ) or negatively ( NA ) associated. Then then above CLT holds (Karcher (2012)).
page 18 LT for excursion sets of stationary random fields | LTs for geometric functionals of excursion sets | 23.01.2013 Association � X ( t ) , t ∈ R d � A random field X = is called positively ( PA ) or negatively ( NA ) associated if cov ( f ( X I ) , g ( X J ))) ≥ 0 ( ≤ 0 , resp. ) for all finite disjoint subsets I , J ⊂ R d , and for any bounded coordinatewise non–decreasing functions f : R card ( I ) → R , g : R card ( J ) → R where X I = { X ( t ) , t ∈ I } , X J = { X ( t ) , t ∈ J } .
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