Invariance Principle for weakly dependent random fields Jana - PowerPoint PPT Presentation

Invariance Principle for weakly dependent random fields Jana Klicnarová University of South Bohemia České Budějovice Czech Republic CINCINNATI SYMPOSIUM 2014 Joint work with Dalibor Volný and Yizao Wang Supported by Czech Science Foundation (project n. P201/11/P164). Jana Klicnarová Cincinnati, September 21, 2014 1 / 26

Problem ( X i ) { i ∈ Z d } – a stationary random field with a zero mean and a finite second moment. CLT problem Put � S Γ n = X i , i ∈ Γ n where Γ n ⊂ Z d , | Γ n | → ∞ as n → ∞ . When does � | Γ n | → N (0 , σ 2 )? S Γ n / Jana Klicnarová Cincinnati, September 21, 2014 2 / 26

IP problem Let � S n ( A ) = λ ( nA ∩ R i ) X i , i ∈ [ 0 , n ] where R i = ( i − 1 , i ] and nA = { n x : x ∈ A } . When does { n − d / 2 S n ( A ); A ∈ A} → σ W C ( A ) , in where W is a standard Brownian motion indexed by A ? Jana Klicnarová Cincinnati, September 21, 2014 3 / 26

Basic notation ( R Z d , B Z d , P Z d ) (Ω , A , P ) = ( ǫ k ) k ∈ Z d – iid random variables, ǫ k ( ω ) = ω k – a measurable function on Ω such that f ∈ L 2 ( µ ), f regular and with a zero mean. { T k } k ∈ Z d shift operators on R d : ( T k ω ) l = ω k + l , – it is a (strictly) stationary process ( X i = f ◦ T i ). ( X i ) i – a unitary operator on L 2 , such that U i f = f ◦ T i , – U ( F k ) k – a filtration, F k = σ ( ǫ l : l ≤ k ) σ 2 E( S Γ n ( f )) 2 . = Γ n Jana Klicnarová Cincinnati, September 21, 2014 4 / 26

Projection and Hannan’s condition In a 1-dimensional case: P i ( X ) = E( X |F i ) − E( X |F i − 1 ) . Hannan’s condition (1973) ∞ � || P 0 ( X i ) || 2 < ∞ . i =1 Hannan (1973) proved a CLT and an IP for stationary processes under this condition and with some more conditions. Later, Dedecker, Merlevede and Volný (2007) proved an IP for stationary processes under only this condition. Jana Klicnarová Cincinnati, September 21, 2014 5 / 26

Hannan’s condition in a high dimension The definition of P i in a d-dimensional case is more complicated, for more details see Volný and Wang (2014). The idea of the definition: we suppose a commuting filtration (see Khoshnevisan (2002)): for every bounded F l -measurable r.v. Y: E( Y |F k ) = E( Y |F k ∧ l ) a . s . the projection operator P i is defined as: q =1 P q P i = Π d j q , where P q j q are "marginal" projections. Jana Klicnarová Cincinnati, September 21, 2014 6 / 26

Hannan’s condition in a multidimensional case and limit theorems Volný and Wang (2014) established an Invariance Principle under Hannan’s condition (they suppose a finite second moment): � || P 0 X i || 2 < ∞ . i ∈ Z d Their result is for summation over rectangles. We are interested in limit theorems where a summation is over more general sets. Jana Klicnarová Cincinnati, September 21, 2014 7 / 26

El Machkouri, Volný, Wu (2013) p-stability – notation Let us have X i = g ( ε i − j ; j ∈ Z d ) , i ∈ Z d , (1) ′ where ( ε i ) i are i.i.d., and ( ε i ) i are i.i.d. copies of ( ε i ) i . Then X ∗ i is a version of X i such that X ∗ i = g ( ε ∗ i − j ; j ∈ Z d ) , i ∈ Z d , where ε ∗ = ε i for all i � = 0 , i ′ = ε 0 for i = 0 . Then we can put δ i , p = || X i − X ∗ i || p and ∆ p = � i ∈ Z d δ i , p . Definition We say, that the process is p-stable if ∆ p < ∞ . Jana Klicnarová Cincinnati, September 21, 2014 8 / 26

El Machkouri, Volný, Wu (2013) – notation Ψ -stability – Young function A function Ψ is a Young function if it is a real convex nondecreasing function defined on R + which satisfies t →∞ Ψ( t ) lim = ∞ Ψ(0) = 0 . The Orlicz space L Ψ is defined as a space of real random variables Z defined on a probability space (Ω , A , P ) such that E[Ψ( | Z | / c )] < ∞ for some c > 0. For more details see for example Ledoux and Talagrand (1991). Jana Klicnarová Cincinnati, September 21, 2014 9 / 26

El Machkouri, Volný, Wu (2013) – notation Luxembourg norm The Orlicz space L Ψ is equipped with Luxemburg norm || · || Ψ defined for real random variable by || Z || Ψ = inf { c > 0; E[Ψ( | Z | / c )] ≤ 1 } . So, it is possible to generalize the definition of p -stable processes to Ψ-stable processes. Then we can put δ i , Ψ = || X i − X ∗ i || Ψ and ∆ Ψ = � i ∈ Z d δ i , Ψ . Definition We say, that the process is Ψ-stable if ∆ Ψ < ∞ . Jana Klicnarová Cincinnati, September 21, 2014 10 / 26

CLT – El Machkouri, Volný, Wu (2013) Theorem Let ( X i ) i ∈ Z d be a stationary centred random field defined by (1) satisfying ∆ 2 < ∞ . Assume that (Γ n ) n is a sequence of finite subsets of Z d such that | Γ n | → ∞ and σ Γ n = E( S 2 Γ n ) → ∞ , then Levy distance � � � S Γ n / | Γ n | , N (0 , σ Γ 2 n / | Γ n | ) → 0 as n → ∞ . L Corollary If | ∂ Γ n | / | Γ n | → 0 and σ 2 = � k ∈ Z d E( X 0 X k ) > 0 then S Γ n | Γ n | → N (0 , σ 2 ) . � Jana Klicnarová Cincinnati, September 21, 2014 11 / 26

El Machkouri, Volný, Wu (2013) – notation To introduce an Invariance Principle given by El Machkouri, Volný and Wu we need also to recall some definitions about entropy and VC-classes. For more details see for example van der Vaart and Wellner (1996). Covering number and entropy Let us have a collection A of Borel subsets on [0 , 1] d . We can equip a collection with a pseudometric ρ : � ρ ( A , B ) = λ ( A ∆ B ) . To measure a size of A it is possible to use a metric entropy. Let us recall, that the entropy H ( A , ρ, ε ) is the logarithm of N ( A , ρ, ε ), where N ( A , ρ, ε ) is so called a covering number – it is the smallest number of open balls of radius ε with respect to ρ which cover A . Jana Klicnarová Cincinnati, September 21, 2014 12 / 26

El Machkouri, Volný, Wu (2013) – notation Vapnik-Chervonenkis classes Let C be a collection of subsets of a set X . And let F ⊂ X . We say that C picks out a certain subset of F if this can be formed as F ∩ C for some C ∈ C . The collection C is said to shatter F if it picks out each of its 2 | F | subsets. The VC-index V ( C ) of the class C is the smallest n for which no set of size n is shattered by C . Formally, � � x 1 ,..., x n ∆ n ( C , x 1 , . . . , x n ) < 2 n V ( C ) = inf n ; max , where ∆ n ( C , x 1 , . . . , x n ) = |{ C ∩ { x 1 , . . . , x n } ; C ∈ C}| . Jana Klicnarová Cincinnati, September 21, 2014 13 / 26

El Machkouri, Volný, Wu (2013) – notation Young function Let β > 0 and 1 β I { 0 <β< 1 } . h β = ((1 − β ) β ) Then we denote by ψ β the Young function: ψ β ( x ) = exp { ( x + h β ) β } − exp { h β x ∈ R + . β } , Jana Klicnarová Cincinnati, September 21, 2014 14 / 26

IP – El Machkouri, Volný, Wu (2013) IP Let ( U i f ) i ∈ Z d be a stationary centered random field and let A be a collection of regular Borel subsets of [0 , 1] d . Assume that one of the following conditions holds: (i) The collection A is a Vapnik-Chervonenkis class with an index V and there exists p > 2( V − 1) such that f ∈ L p and ∆ p < ∞ . (ii) There exists a positive θ and 0 < q < 2: E[exp ( θ | f | β ( q ) )] < ∞ , where β ( q ) = 2 q / (2 − q ) and ∆ Ψ( β ( q )) < ∞ and such that the class A satisfies condition � 1 ( H ( A , ρ, ε )) 1 / q d ε < ∞ . 0 Jana Klicnarová Cincinnati, September 21, 2014 15 / 26

IP – El Machkouri, Volný, Wu (2013) � 1 0 ( H ( A , ρ, ε )) 1 / 2 d ε < ∞ and (iii) f ∈ L ∞ , ∆ ∞ < ∞ . Then the sequence of processes { n − d / 2 S n ( A ); A ∈ A} , where � S n ( A ) = λ ( nA ∩ R i ) U i f i ∈ [ 0 , n ] with R i = ( i − 1 , i ], converges in distribution in C ( A ) to σ W , where W is a standard Brownian motion indexed by A and σ 2 = � i ∈ Z d E( fU i f ). Jana Klicnarová Cincinnati, September 21, 2014 16 / 26

CLT with general summation under Hannan’s condition Problem Can we formulate a limit theorem for general (Γ n ) also under Hannan’s condition? CLT for martingale differences If ( f ◦ T i ) i ∈ Z d is a martingale difference field and f ∈ L 2 then the Central Limit Theorem holds: for Γ n ⊂ Z d : | Γ n | → ∞ we have S Γ n ( f ) | Γ n | → N (0 , || f || 2 2 ) . � Jana Klicnarová Cincinnati, September 21, 2014 17 / 26

Important condition on (Γ n ) To obtain a Central Limit Theorem for general sets (Γ n ), we need a collection of (Γ n ) to satisfy some condition. | ∂ Γ n | If we suppose (Γ n ) such that lim n →∞ | Γ n | = 0 then under ( L 2 ) Hannnan’s condition: E( S 2 Γ n ( f )) � lim = E( fU k f ) . | Γ n | n →∞ k ∈ Z d At least we need E( S 2 Γ n ( f )) lim inf > 0 . | Γ n | n →∞ Jana Klicnarová Cincinnati, September 21, 2014 18 / 26

CLT under Hannan’s condition with | ∂ Γ n | / | Γ n | → 0 Theorem Let a zero-mean f ∈ L 2 be regular and satisfy Hannnan’s condition. Let (Γ n ) n ∈ N be such that | ∂ Γ n | lim | Γ n | = 0 and | Γ n | → ∞ . n →∞ | Γ n | → N (0 , σ 2 ) , where σ 2 = � � then S Γ n ( f ) / k Z d E( fU k f ). Jana Klicnarová Cincinnati, September 21, 2014 19 / 26