Extremal behaviour of chaotic dynamics

Transcription

1 Extremal behaviour of chaotic dynamics Ana Cristina Moreira Freitas CMUP & FEP, Universidade do Porto joint work with Jorge Freitas and Mike Todd (CMUP & FEP) 1 / 24

2 Extreme Value Theory Consider a stationary stochastic process X 0, X 1, X 2,... with marginal d.f. F. Let F = 1 F and u F = sup{x : F (x) < 1}. We are interested in the study of the distributional properties of the maximum M n = max{x 0,..., X n 1 } (1) as n. (CMUP & FEP) 2 / 24

3 Extreme Value Laws Definition We say that we have an Extreme value law (EVL) for M n if there is a non-degenerate d.f. H : R [0, 1] (with H(0) = 0) and for all τ > 0, there exists a sequence of levels u n = u n (τ) such that and for which the following holds:. np(x 0 > u n ) τ as n, (2) P(M n u n ) H(τ) as n. (3) (CMUP & FEP) 3 / 24

4 The independent case In the case X 0, X 1, X 2,... are i.i.d. r.v. then since P(M n u n ) = (F(u n )) n we have that if (2) holds, then (3) holds with H(τ) = e τ : P(M n u n ) = (1 P(X 0 > u n )) n and vice-versa. ( 1 τ n ) n e τ as n, When X 0, X 1, X 2,... are not i.i.d. but satisfy some mixing condition D(u n ) introduced by Leadbetter then something can still be said about H. (CMUP & FEP) 4 / 24

5 Condition D(u n ) from Leadbetter Let F i1,...,i n denote the joint d.f. of X i1,..., X in, and set F i1,...,i n (u) = F i1,...,i n (u,..., u). Condition (D(u n )) We say that D(u n ) holds for the sequence X 0, X 1,... if for any integers i 1 <... < i p and j 1 <... < j k for which j 1 i p > t, and any large n N, F i1,...,i p,j 1,...,j k (u n ) F i1,...,i p (u n )F j1,...,j k (u n ) γ(n, t), where γ(n, t n ) n 0, for some sequence t n = o(n). (CMUP & FEP) 5 / 24

6 Extremal Index Theorem ([C81], see also [LLR83]) If D(u n ) holds for X 0, X 1,... and the limit (3) exists for some τ > 0 then there exists 0 θ 1 such that H(τ) = e θτ for all τ > 0. Definition We say that X 0, X 1,... has an Extremal Index (EI) 0 θ 1 if we have an EVL for M n with H(τ) = e θτ for all τ > 0. (CMUP & FEP) 6 / 24

7 Hitting Times and Kac s Lemma Consider the system (X, B, µ, f ), where X is a topological space, B is the Borel σ-algebra, f : X X is a measurable map and µ is an f -invariant probability measure, i.e., µ(f 1 (B)) = µ(b), for all B B. For a set A X let r A (x) the first hitting time to A of the point x, i.e. r A (x) = min{j N : f j (x) A}. Let µ A denote the conditional measure on A, i.e. µ A := µ A µ(a). By Kac s Lemma, the expected value of r A with respect to µ A is r A dµ A = 1/µ(A). A (CMUP & FEP) 7 / 24

8 Hitting Time Statistics and Return Time Statistics Definition Given a sequence of sets (U n ) n N so that µ(u n ) 0, the system has RTS G for (U n ) n N if for all t 0 ( µ Un r Un t ) µ(u n ) G(t) as n. (4) and the system has HTS G for (U n ) n N if for all t 0 ( µ r Un t ) G(t) as n, (5) µ(u n ) (CMUP & FEP) 8 / 24

9 Stationary stochastic processes arising from chaotic dynamics Consider a discrete dynamical system (X, B, µ, f ), where X is a d-dimensional Riemannian manifold, B is the Borel σ-algebra, f : X X is a map, µ is an f -invariant probability measure, absolutely continuous with respect to Lebesgue (acip). (CMUP & FEP) 9 / 24

10 In this context, we consider the stochastic process X 0, X 1,... given by X n = ϕ f n, for each n N, (6) where ϕ : X R {± } is an observable (achieving a global maximum at ξ X ) of the form ϕ(x) = g(dist(x, ξ)), where ξ X, dist denotes a Riemannian metric in X and the function g : [0, + ) R {+ } has a global maximum at 0 and is a strictly decreasing bijection for a neighborhood V of 0.. (CMUP & FEP) 10 / 24

11 Observe that if at time j N we have an exceedance of the level u sufficiently large, i.e. X j (x) > u, then we have an entrance of the orbit of x in the ball of radius g 1 (u) around ξ, at time j. {X 0 > u} = {g(dist(x, ξ)) > u} = {dist(x, ξ) < g 1 (u)} = B g 1 (u) (ξ). and ) 1 F(u) = µ (B g 1 (u) (ξ). (CMUP & FEP) 11 / 24

12 Connection between EVL and HTS (X, B, µ, f ) is a dynamical system where µ is an acip. Motivated by Collet s work, [C01], we obtained: Theorem ([FFT10]) If we have HTS G for balls centred on ξ X, then we have an EVL for M n with H = G. Theorem ([FFT10]) If we have an EVL H for M n, then we have HTS G = H for balls centred on ξ. (CMUP & FEP) 12 / 24

13 Idea of the proof: n 1 {x : M n (x) u n } = {x : X j (x) u n } j=0 n 1 = {x : g(dist(f j (x), ξ)) u n } j=0 n 1 = {x : dist(f j (x), ξ) g 1 (u n )} = {x : r Bg 1 (u n) (ξ) (x) n} j=0 Thus, µ{x : M n (x) u n } = µ{x : r Bg 1 (u n) (ξ) (x) n} (CMUP & FEP) 13 / 24

14 Note that τ ) n 1 F(u n) = µ (B g 1 (u (ξ) n) n τ ) µ (B g 1 (u (ξ) n) and so µ {x : M n (x) u n } µ x : r B g 1 (u n) (ξ) (x) τ ) µ (B g 1 (u (ξ) 1 G(τ) n) (CMUP & FEP) 14 / 24

15 Assuming D(u n ) holds, let (k n ) n N be a sequence of integers such that k n and k n t n = o(n). (7) Condition (D (u n )) We say that D (u n ) holds for the sequence X 0, X 1,... if lim sup n Theorem (Leadbetter) [n/k] n j=1 P{X 0 > u n and X j > u n } = 0. (8) Let {u n } be such that n(1 F(u n )) τ, as n, for some τ 0. Assume that conditions D(u n ) and D (u n ) hold. Then P(M n u n ) e τ as n. (CMUP & FEP) 15 / 24

16 Assuming D(u n ) holds, let (k n ) n N be a sequence of integers such that k n and k n t n = o(n). (7) Condition (D (u n )) We say that D (u n ) holds for the sequence X 0, X 1,... if lim sup n Theorem (Leadbetter) [n/k] n j=1 P{X 0 > u n and X j > u n } = 0. (8) Let {u n } be such that n(1 F(u n )) τ, as n, for some τ 0. Assume that conditions D(u n ) and D (u n ) hold. Then P(M n u n ) e τ as n. (CMUP & FEP) 15 / 24

17 Condition (D 2 (u n )) We say that D 2 (u n ) holds for the sequence X 0, X 1,... if for any integers l, t and n P {X 0 > u n max{x t,..., X t+l 1 u n }} P{X 0 > u n }P{M l u n } γ(n, t), where γ(n, t) is nonincreasing in t for each n and nγ(n, t n ) 0 as n for some sequence t n = o(n). Theorem ([FF08a]) Let {u n } be such that n(1 F(u n )) τ, as n, for some τ 0. Assume that conditions D 2 (u n ) and D (u n ) hold. Then P(M n u n ) e τ as n. (CMUP & FEP) 16 / 24

18 Condition (D 2 (u n )) We say that D 2 (u n ) holds for the sequence X 0, X 1,... if for any integers l, t and n P {X 0 > u n max{x t,..., X t+l 1 u n }} P{X 0 > u n }P{M l u n } γ(n, t), where γ(n, t) is nonincreasing in t for each n and nγ(n, t n ) 0 as n for some sequence t n = o(n). Theorem ([FF08a]) Let {u n } be such that n(1 F(u n )) τ, as n, for some τ 0. Assume that conditions D 2 (u n ) and D (u n ) hold. Then P(M n u n ) e τ as n. (CMUP & FEP) 16 / 24

19 Periodic points From here on ξ is a repelling p-periodic point, which implies that f p (ξ) = ξ, f p is differentiable at ξ and 0 < det D(f p )(ξ) < 1. Note that {X 0 > u} {X p > u} and for all u sufficiently large P({X 0 > u} {X p > u}) det D(f p )(ξ) P(X0 > u). Consequently, D (u n ) does not hold since [n/k n] n j=1 P(X 0 > u n, X j > u n ) np(x 0 > u n, X p > u n ) det D(f p )(ξ) τ (CMUP & FEP) 17 / 24

20 For repelling periodic points as above, let θ = 1 det D(f p )(ξ). Then the following condition holds: Condition (MP p,θ (u n )) We say that X 0, X 1, X 2,... satisfies the condition MP p,θ (u n ) for p N and θ [0, 1] if lim sup P(X j > u n X 0 > u n ) = 0, lim P(X p > u n X 0 > u n ) = (1 θ) n 1 j<p n and P(X p > u n, X 2p > u n,..., X ip > u n X 0 > u n ) lim sup n i (1 θ) i = 1. (CMUP & FEP) 18 / 24

21 Define the event Q p,0 (u) := {X 0 > u, X p u}. Observe that for u sufficiently large, Q p,0 (u) corresponds to an annulus centred at ξ. Define the events: Q p,i (u) := {X i > u, X i+p u}, Qp,i (u) := {X i > u} \ Q p,i (u) and Q p,s,l (u) = s+l 1 i=s Qp,i c (u). (CMUP & FEP) 19 / 24

22 Theorem ([FFT12]) Let (u n ) n N be such that np(x 0 > u n ) τ, for some τ 0. Suppose X 0, X 1,... is as in (6) and satisfies MP p,θ (u n ) for p N, and θ (0, 1). Then lim P(M n u n ) = lim P(Q p,0,n (u n )) (9) n n First observe that {M n u n } Q p,0,n (u n ). Moreover, Q p,0,n (u n ) \ {M n u n } n 1 i=0 {X i > u n, X i+p > u n,..., X i+si p > u n }, where s i = [ n 1 i p ]. It follows by MP p,θ (u n ) and stationarity that. n 1 P(Q p,0,n (u n )\{M n u n }) P ( ) X i > u n, X i+p > u n,..., X i+si p > u n i=0 [n/p] p P ( ) X 0 > u n, X p > u n, X 2p > u n,..., X ip > u n 0. n i=0 (CMUP & FEP) 20 / 24

23 Condition (D p (u n )) We say that D p (u n ) holds forx 0, X 1,... if for any l, t and n P ( Qp,0 (u n ) Q p,t,l (u n ) ) P(Q p,0 (u n ))P(Q p,0,l (u n )) γ(n, t), where γ(n, t) is nonincreasing in t for each n and nγ(n, t n ) 0 as n for some sequence t n = o(n). Let (k n ) n N be a sequence of integers such that k n and k n t n = o(n). Condition (D p(u n )) We say that D p(u n ) holds for the sequence X 0, X 1, X 2,... if there exists a sequence {k n } n N satisfying (7) and such that [n/kn] lim n P(Q p,0 (u n ) Q p,j (u n )) = 0. (10) n j=1 (CMUP & FEP) 21 / 24

24 Theorem ([FFT12]) Let (u n ) n N be such that np(x > u n ) τ, as n for some τ 0. Consider a stationary stochastic process X 0, X 1, X 2,... satisfying MP p,θ (u n ) for some p N, and θ (0, 1). Assume further that conditions D p (u n ) and D p(u n ) hold. Then Note that lim P(M n u n ) = lim P(Q p,0,n (u n )) = e θτ. (11) n n P(Q p,0 (u)) = P(X 0 > u, X p u) = = P(X 0 > u) P(X 0 > u, X p > u) = P(X 0 > u) (1 θ)p(x 0 > u) = θp(x 0 > u), and so θ P(Q p,0(u)) P(X 0 > u). (CMUP & FEP) 22 / 24

25 M. R. Chernick, A limit theorem for the maximum of autoregressive processes with uniform marginal distributions, Ann. Probab. 9 (1981), no. 1, P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergodic Theory Dynam. Systems 21 (2001), no. 2, A. C. M. Freitas and J. M. Freitas, On the link between dependence and independence in extreme value theory for dynamical systems, Statist. Probab. Lett. 78 (2008), no. 9, A. C. M. Freitas, J. M. Freitas, and M. Todd, Hitting time statistics and extreme value theory, Probab. Theory Related Fields 147 (2010), no. 3, A. C. M. Freitas, J. M. Freitas, and M. Todd, The extremal index, hitting time statistics and periodicity, Adv. Math. 231 (2012), no. 5, (CMUP & FEP) 22 / 24

26 M. R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics, New York: Springer-Verlag (1983). (CMUP & FEP) 23 / 24

27 Decay of correlations implies D 2 (u n ) Suppose that there exists a nonincreasing function γ : N R such that for all φ : X R with bounded variation and ψ : X R L, there is C > 0 independent of φ, ψ and n such that φ (ψ f t )dµ φdµ ψdµ CVar(φ) ψ γ(t), n 0, where Var(φ) denotes the total variation of φ and nγ(t n ) 0, as n for some sequence t n = o(n). (12) Taking φ = 1 {X>un} and ψ = 1 {Ml u n}, then (12) D 2 (u n ), (with γ(n, t) = CVar(1 {X>un}) 1 {Ml u n} γ(t) C γ(t) and for the sequence {t n } such that t n /n 0 and nγ(t n ) 0 as n ). (CMUP & FEP) 23 / 24

28 Decay of correlations against L 1 implies D p(u n ) Suppose that there exists a nonincreasing function γ : N R such that for all φ : X R with bounded variation and ψ : X R L 1, there is C > 0 independent of φ, ψ and n such that φ (ψ f t )dµ φdµ ψdµ CVar(φ) ψ 1γ(t), n 0, where Var(φ) denotes the total variation of φ and nγ(t n ) 0, as n for some sequence t n = o(n). Taking φ = 1 Qp(u n) and ψ = 1 Qp(u n), then (13) D p(u n ), P(Q p,0 (u n ) Q p,j (u n )) P(Q p,0 (u n )) 2 + C P(Q p,0 (u n ))γ(j) (τ/n) 2 + C (τ/n)γ(j). (13) (CMUP & FEP) 24 / 24