Persistence of Gaussian stationary processes Naomi D. Feldheim - PowerPoint PPT Presentation

Introduction Persistence Probability Ideas from the Proofs Persistence of Gaussian stationary processes Naomi D. Feldheim Joint work with Ohad N. Feldheim Department of Mathematics Tel-Aviv University Darmstadt July, 2014

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Real Gaussian Stationary Processes (GSP) Let T ∈ { Z , R } . A GSP is a random function f : T → R s.t. It has Gaussian marginals: ∀ n ∈ N , x 1 , . . . , x n ∈ T : ( f ( x 1 ) , . . . , f ( x n )) ∼ N R n (0 , Σ) It is Stationary: ∀ n ∈ N , x 1 , . . . , x n ∈ T and ∀ t ∈ T : � � d � � f ( x 1 + t ) , . . . , f ( x n + t ) ∼ f ( x 1 ) , . . . , f ( x n ) If T = Z we call it a GSS (Gaussian Stationary Sequence) . If T = R we call it a GSF (Gaussian Stationary Function) . We assume GSFs are a.s. continuous.

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Covariance kernel For a GSP f : T → R the covariance kernel r : T → R is defined by: r ( x ) = E [ f (0) f ( x )] = E [ f ( t ) f ( x + t )] .

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Covariance kernel For a GSP f : T → R the covariance kernel r : T → R is defined by: r ( x ) = E [ f (0) f ( x )] = E [ f ( t ) f ( x + t )] . determines the process f .

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Covariance kernel For a GSP f : T → R the covariance kernel r : T → R is defined by: r ( x ) = E [ f (0) f ( x )] = E [ f ( t ) f ( x + t )] . determines the process f . positive-definite: � 1 ≤ i , j ≤ n c i c j r ( x i − x j ) ≥ 0.

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Covariance kernel For a GSP f : T → R the covariance kernel r : T → R is defined by: r ( x ) = E [ f (0) f ( x )] = E [ f ( t ) f ( x + t )] . determines the process f . positive-definite: � 1 ≤ i , j ≤ n c i c j r ( x i − x j ) ≥ 0. symmetric: r ( − x ) = r ( x ).

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Covariance kernel For a GSP f : T → R the covariance kernel r : T → R is defined by: r ( x ) = E [ f (0) f ( x )] = E [ f ( t ) f ( x + t )] . determines the process f . positive-definite: � 1 ≤ i , j ≤ n c i c j r ( x i − x j ) ≥ 0. symmetric: r ( − x ) = r ( x ). continuous.

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Spectral measure Bochner’s Theorem Write Z ∗ = [ − π, π ] , R ∗ = R . Then � T ∗ e − ix λ d ρ ( λ ) , r ( x ) = � ρ ( x ) = where ρ is a finite, symmetric, non-negative measure on T ∗ . We call ρ the spectral measure of f .

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Spectral measure Bochner’s Theorem Write Z ∗ = [ − π, π ] , R ∗ = R . Then � T ∗ e − ix λ d ρ ( λ ) , r ( x ) = � ρ ( x ) = where ρ is a finite, symmetric, non-negative measure on T ∗ . We call ρ the spectral measure of f . We assume: � | λ | δ d ρ ( δ ) < ∞ . ∃ δ > 0 :

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Toy-Example Ia - Gaussian wave ζ j i.i.d. N (0 , 1) f ( x ) = ζ 0 sin( x ) + ζ 1 cos( x ) Covariance Kernel 1 r ( x ) = cos( x ) 0.8 0.6 ρ = 1 2 ( δ 1 + δ − 1 ) 0.4 0.2 0 −0.2 Three Sample Paths −0.4 1 −0.6 −0.8 0.8 −1 −10 −5 0 5 10 0.6 0.4 Spectral Measure 0.2 0.5 0.45 0 0.4 −0.2 0.35 0.3 −0.4 0.25 −0.6 0.2 0.15 −0.8 0.1 0.05 −1 0 1 2 3 4 5 6 7 8 9 10 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Toy-Example Ib - Almost periodic wave f ( x ) = ζ 0 sin( x ) + ζ 1 cos( x ) Covariance Kernel √ √ 1 + ζ 2 sin( 2 x ) + ζ 3 cos( 2 x ) 0.8 0.6 √ 0.4 r ( x ) = cos( x ) + cos( 2 x ) 0.2 � � 0 ρ = 1 δ 1 + δ − 1 + δ √ √ 2 + δ − −0.2 2 2 −0.4 −0.6 −0.8 Three Sample Paths −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 2.5 2 Spectral Measure 1.5 0.5 1 0.45 0.5 0.4 0.35 0 0.3 −0.5 0.25 −1 0.2 0.15 −1.5 0.1 −2 0.05 −2.5 0 0 1 2 3 4 5 6 7 8 9 10 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Example II - i.i.d. sequence f ( n ) = ζ n Covariance Kernel 1 r ( n ) = δ n , 0 0.8 1 0.6 d ρ ( λ ) = 2 π 1 I [ − π,π ] ( λ ) d λ 0.4 0.2 0 Three Sample Paths −0.2 2 −0.4 −0.6 1.5 −0.8 −1 −5 −4 −3 −2 −1 0 1 2 3 4 5 1 Spectral Measure 0.5 0.2 0.18 0 0.16 0.14 −0.5 0.12 0.1 −1 0.08 0.06 −1.5 0 1 2 3 4 5 6 7 8 9 10 0.04 0.02 0 −5 −4 −3 −2 −1 0 1 2 3 4 5

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Example IIb - Sinc Kernel f ( x ) = � n ∈ N ζ n sinc( x − n ) Covariance Kernel r ( x ) = sin( π x ) 1 = sinc( x ) π x 0.8 1 d ρ ( λ ) = 2 π 1 I [ − π,π ] ( λ ) d λ 0.6 0.4 0.2 Three Sample Paths 0 2 −0.2 1.5 −0.4 −5 −4 −3 −2 −1 0 1 2 3 4 5 1 Spectral Measure 0.5 0.2 0.18 0 0.16 0.14 −0.5 0.12 0.1 −1 0.08 0.06 −1.5 0 1 2 3 4 5 6 7 8 9 10 0.04 0.02 0 −5 −4 −3 −2 −1 0 1 2 3 4 5

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Example III - Gaussian Covariance � x n e − x 2 √ f ( x ) = ζ n 2 n ! Covariance Kernel n ∈ N 1 r ( x ) = e − x 2 0.9 2 0.8 d ρ ( λ ) = √ π e − λ 2 0.7 2 d λ 0.6 0.5 0.4 0.3 Three Sample Paths 0.2 3 0.1 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2 Spectral Measure 1 1.8 1.6 0 1.4 1.2 −1 1 0.8 0.6 −2 0.4 0.2 −3 0 1 2 3 4 5 6 7 8 9 10 0 −5 −4 −3 −2 −1 0 1 2 3 4 5

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction Example IV - Exponential Covariance Covariance Kernel r ( x ) = e −| x | 1 0.9 2 d ρ ( λ ) = λ 2 +1 d λ 0.8 0.7 0.6 Three Sample Paths 0.5 0.4 3 0.3 0.2 2 0.1 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 1 Spectral Measure 0 2 1.8 −1 1.6 1.4 1.2 −2 1 0.8 −3 0.6 0 1 2 3 4 5 6 7 8 9 10 0.4 0.2 0 −5 −4 −3 −2 −1 0 1 2 3 4 5

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction General Construction ρ - a finite, symmetric, non-negative measure on T ∗ { ψ n } n - ONB of L 2 ρ ( T ∗ )

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction General Construction ρ - a finite, symmetric, non-negative measure on T ∗ { ψ n } n - ONB of L 2 ρ ( T ∗ ) ⇓ � T ∗ e − ix λ ψ n ( λ ) d ρ ( λ ) ϕ n ( x ) :=

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction General Construction ρ - a finite, symmetric, non-negative measure on T ∗ { ψ n } n - ONB of L 2 ρ ( T ∗ ) ⇓ � T ∗ e − ix λ ψ n ( λ ) d ρ ( λ ) ϕ n ( x ) := ⇓ � f ( t ) d = ζ n ϕ n ( t ) , where ζ n are i.i.d. N (0 , 1) . n

Introduction Definitions Persistence Probability Examples Ideas from the Proofs General Construction General Construction ρ - a finite, symmetric, non-negative measure on T ∗ { ψ n } n - ONB of L 2 ρ ( T ∗ ) ⇓ � T ∗ e − ix λ ψ n ( λ ) d ρ ( λ ) ϕ n ( x ) := ⇓ � f ( t ) d = ζ n ϕ n ( t ) , where ζ n are i.i.d. N (0 , 1) . n make sure that ϕ n are R -valued.

Definition Introduction Prehistory Persistence Probability History Ideas from the Proofs Main Result Persistence Probability Definition Let f be a GSP on T . The persistence probability of f up to time t ∈ T is � � P f ( t ) := P f ( x ) > 0 , ∀ x ∈ (0 , t ] . a.k.a. gap or hole probability (referring to gap between zeroes or sign-changes).

Definition Introduction Prehistory Persistence Probability History Ideas from the Proofs Main Result Persistence Probability Definition Let f be a GSP on T . The persistence probability of f up to time t ∈ T is � � P f ( t ) := P f ( x ) > 0 , ∀ x ∈ (0 , t ] . a.k.a. gap or hole probability (referring to gap between zeroes or sign-changes). Question: What is the behavior of P ( t ) as t → ∞ ? Guess: “typically” P ( t ) ≍ e − θ t .

Definition Introduction Prehistory Persistence Probability History Ideas from the Proofs Main Result Persistence Probability Definition Let f be a GSP on T . The persistence probability of f up to time t ∈ T is � � P f ( t ) := P f ( x ) > 0 , ∀ x ∈ (0 , t ] . a.k.a. gap or hole probability (referring to gap between zeroes or sign-changes). Question: What is the behavior of P ( t ) as t → ∞ ? Guess: “typically” P ( t ) ≍ e − θ t . ( X n ) n ∈ Z i.i.d. ⇒ P X ( N ) = 2 − N ( N +1)! ≍ e − N log N 1 Y n = X n +1 − X n ⇒ P Y ( N ) = Z n ≡ Z 0 ⇒ P Z ( N ) = P ( Z 0 > 0) = 1 2